B U L L E T I N of the International Membrane Computing Society I M C S

Transcription

1 B U L L E T I N of the International Membrane Computing Society I M C S Number 4 December 2017 Bulletin Webpage: Webmaster: Andrei Florea, andrei91ro@gmail.com

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3 Foreword We just complete now the second year of apparition of our Bulletin... Again, the December issue is longer than the June one maybe this will become a tradition : the winter volumes to be longer than the summer ones... The usual sections are present, with a special emphasis on the surveys section and the open problems section, again rich and provocative, which is rather good for the IMCS plans to start editing a journal (these surveys can be reproduced in the journal, after a possible rewriting), and to also edit in the near future some collective volumes. A comprehensive chapter is also the Miscellanea one, with a new kind of item included which could be useful to the membrane computing community, i.e., a call for position; symmetrically, position offers are welcome. A sad item of this section is the Obituary for Tom Head, a great name of DNA computing and a honorary member of IMCS, who passed away in November. Requiescat in pace! As usual, I would like to stress the fact that the IMCS Bulletin is conceived as a working material for the MC community, as a medium for communicating in a fast and efficient way any idea, news, problem, result. As it is already known, each issue gradually grows and remains available at net/imcsbulletin), also being printed. (If somebody wants to have a printed copy, s/he has to contact the IMCS secretary see information about the structure of IMCS, including addresses, in the subsequent pages.) It is important to stress the fact that the copyright of all materials remains with their authors. There is a change now in what concerns the Bulletin chair: Marian Gheorghe, Bradford, UK, m.gheorghe@bradford.ac.uk will take this task for the next period. The protocol will remain the same (the same web page, grown step by step, the same style), with the contributions to be submitted to Marian (Latex).

4 4 Foreword I am hereby inviting all people interested in membrane computing to consult the Bulletin and, also, to contribute views from outside are always of interest and useful. * The realization of the Bulletin of IMCS owns very much (i) to all contributors, (ii) to the webmaster, Andrei Florea, andrei91ro@gmail.com, and to the MC research group in Politechnica University in Bucharest, led by prof. Cătălin Buiu, where the bulletin is hosted, and (iii) to prof. Gexiang Zhang, the President of IMCS, to his group, and to the Xihua University in Chengdu, China, where the bulletin is printed. Many thanks to all. Gheorghe Păun December 2017

5 Contents Letter from The President IMCS Matters Structure of IMCS Constitution of IMCS IMCS Journal: Journal of Membrane Computing (JMEC) News from MC Research Groups Vincenzo Manca: Bioinformatics and Natural Computing at the University of Verona Francis George C. Cabarle, Jym Paul Carandang, Ren Tristan dela Cruz, Kelvin C. Buño, John Matthew Villaflores, Miguel Ángel Martínez-del-Amor, Nestine Hope Hernandez, Richelle Ann B. Juayong, Henry N. Adorna: Spiking Neural P Systems Research at Algorithms and Complexity Laboratory of the University of the Philippines Diliman Book Presentations Thomas Hinze: Computer der Natur (Computers Found in Nature).. 35 Tutorials, Surveys Luis Valencia-Cabrera, Miguel A. Martínez-del-Amor, D. Orellana-Martín, Ignacio Pérez-Hurtado, Mario J. Pérez-Jiménez: Cooperative P Systems and the P Versus NP Problem Luis Valencia-Cabrera, David Orellana-Martín, Miguel Á. Martínez-del-Amor, Mario J. Pérez Jiménez: From Super-cells to Robotic Swarms: Two Decades of Evolution in the Simulation of P Systems

6 6 Contents Petr Sosík: P Systems Attacking Hard Problems Beyond NP: A Survey Open Problems Gheorghe Păun: A Dozen of (Meta/Mega?) Research Topics Luca Manzoni, Antonio E. Porreca: Dualistic Open Problems in Membrane Computing Mike Stannett: With Theorem Provers and P systems: How to Build a New Computational Universe Artiom Alhazov, Rudolf Freund, Sergiu Ivanov, Linqiang Pan, Bosheng Song: Clock-freeness and Related Concepts in P Systems: from Definitions to Open Problems Abstracts of PhD Theses Richelle Ann B. Juayong: Communication Complexity in P Systems with Energy Calls for Participation to MC and Related Conferences/Meetings th Brainstorming Week on Membrane Computing (BWMC18), Sevilla, Spain The 19th International Conference on Membrane Computing (CMC2019), 4-7 September, 2018, Dresden, Germany Reports on MC Conferences/Meetings Marian Gheorghe: The 18 th International Conference on Membrane Computing Bradford, UK, July 25 28, Gexiang Zhang, Qiang Yang, Linqiang Pan: Summary of the 6th Asian Conference on Membrane Computing (ACMC 2017) Miscellanea Prithwineel Paul: Call for a Position IMCS Prize Arriving in Sevilla Thomas Hinze: Interview with Gheorghe Păun Natasha Jonoska, Gheorghe Păun, Grzegorz Rozenberg: Transdisciplinarity, Creativity, Elegance (Obituary for Tom Head) Contents of Previous Volumes

7 Letter from The President Dear IMCS members, Merry Christmas and Happy New Year! On the eve of Christmas and the New Year, I would like to share a piece of news with you and let you feel already the pressure this news brings to us. The new journal and also the first membrane computing journal, Journal of Membrane Computing (JMC), has started to call for high-quality papers. Both the journal web site, and the submission web site, will be available soon. There are four issues each year. The first issues will appear online in the coming year and will be officially published in 2019 by Springer. All the papers published in JMC have free access in 2019 and Our goal is to create and run a high-profile journal. Our ambitious short-term goal is to let it be indexed by Institute of Scientific Information (ISI) in the first round of evaluation after three years publication. It is no doubt that the appearance of JMC in 2018 is absolutely a big event in the history of membrane computing. JMC might mark and symbolize the 20th anniversary of the proposal of membrane computing. There is much pressure to run this journal well, especially before JMC is indexed by ISI. On November 28, Linqiang and I attended 2017 Springer copublishing journal seminar organized by Springer for the Editor-in-Chiefs of all the co-publishing journals by Springer and China. The topics were about how to start to run a new Springer journal, how to promote the new journal, etc. JMC is a quarterly journal with at least six papers for each issue. Thus, there are 24 high quality papers a year. To push JMC forward to be evaluated by ISI using a set of strict criteria after three years, we are planning to publish two survey papers on various aspects of membrane computing and four regular papers in each issue to enhance the impact of the journal. The criteria concern not only for the quality of papers, but also each author and each member of editorial board. We are worrying about getting enough and timely high quality submissions. From 2018, it is

8 8 Letter from The President planned to select several best papers to be published in JMC from the three conferences: BWMC, CMC and ACMC. I really hope you all would like to contribute to this new journal. As discussed with the IMCS Board through , Dr. Qiang Yang, Robotics Research Center at Xihua University, China, will take over the responsibility of being the Treasurer of IMCS. Prof. Marian Gheorghe will be in charge of the Bulletin in the next years. Finally, I would like to thank our Bulletin Chair, Gheorghe Păun, and his committee members, for their efficient work in the past two years. Special thanks is given to all the contributors. All of them have set a very good example to start and run the Bulletin. Gexiang Zhang Chengu, China December 22, 2017

9 IMCS Matters Structure of IMCS

10 10 Structure of IMCS The Board of IMCS The Executive Board: President: Gexiang Zhang, China, Vice President: Alberto Leporati, Italy, Treasurer: Qiang Yang, China, Secretary: Tao Song, China, Bulletin Committee Chair: Gheorghe Păun, Romania, Website Committee Chair: Xiangxiang Zeng, China, PR Committee Chair: Marian Gheorghe, U.K. Publication Committee Chair: Linqiang Pan, China, Conferences Committee Chair: Claudio Zandron, Italy, Awards Committee Chair: Mario Pérez-Jiménez, Spain, The Advisory Board: Erszébet Csuhaj-Varjú, Hungary Chair, Yu (Kevin) Cao, U.S.A. Svetlana Cojocaru, Rep. Moldova Marian Gheorghe, U.K. Xiyu Liu, China Vincenzo Manca, Italy Giancarlo Mauri, Italy Radu Nicolescu, New Zealand Taishin T. Nishida, Japan Mario Pérez-Jiménez, Spain K.G. Subramanian, India Jun Wang, China Xingyi Zhang, China Honorary President: Gheorghe Păun, Romania Honorary Members: Arto Salomaa, Finland Grzegorz Rozenberg, The Netherlands Kamala Krithivasan, India Oscar H. Ibarra, U.S.A. Takashi Yokomori, Japan Tom Head, U.S.A. Jürgen Dassow, Germany Lila Kari, Canada Cristian S. Calude, New Zealand

11 Bulletin Committee Structure of IMCS 11 Gheorghe Păun, Romania Chair, Henry N. Adorna, Philippines, Catalin Buiu, Romania, Matteo Cavaliere, U., Gabriel Ciobanu, Romania, Michael J. Dinneen, New Zealand, Svetlana Cojocaru, Rep. Moldova, Rudi Freund, Austria, Marian Gheorghe, U.K., Ping Guo, China, guoping Thomas Hinze, Germany, Florentin Ipate, Romania, Tseren-Onolt Ishdorj, Mongolia, Alberto Leporati, Italy, Vincenzo Manca, Italy, Taishin T. Nishida, Japan, Agustín Riscos-Núñez, Spain, José Maria Sempere, Spain, Petr Sosík, Czech Rep., K.G. Subramanian, India, György Vaszil, Hungary, Sergey Verlan, France, Claudio Zandron, Italy, Xingyi Zhang, China, Zhiqiang Zhang, China, Website Committee Xiangxiang Zeng, Xiamen, China Chair, Cătălin Buiu, Bucharest, Romania, Luis Valencia Cabrera, Seville, Spain, Hong Peng, Xihua, China, Xingyi Zhang, Anhui, China, Gaoshan Deng, Xiamen, China Andrei Florea, Bucharest, Romania, Luis Felipe Macías-Ramos, Seville, Spain, David Orellana-Martín, Seville, Spain,

12 12 Structure of IMCS PR Committee Marian Gheorghe, Bradford, U.K. Chair, Petros Kefalas, Sheffield, U.K. (International Faculty, Greece), Savas Konur, Bradford, U.K., Maciej Koutny, Newcastle, U.K., Jianhua Xiao, Nankai, China, Publication Committee Linqiang Pan, Wuhan, China Chair, Marian Gheorghe, Bradford, U.K., Alberto Leporati, Milan, Italy, Gheorghe Păun, Bucharest, Romania, Mario Pérez-Jiménez, Seville, Spain, Gexiang Zhang, Chengdu, China, The main tasks of the Publication Committee are (1) to explore the possibility to initiate a series of MC monographs/collective volumes, (2) to establish a MC international journal, (3) to advise the organizers of CMC, ACMC, BWMC, MC workshops in what concerns the special issues of journals, (4) to help translating MC books in Chinese. The Publication Committee can become the Editorial Board of the MC series of books, but, of course, the journal should have a much larger Editorial Board. Conferences Committee Claudio Zandron, Milan, Italy Chair, zandron@disco.unimib.it Henry Adorna, Quezon City, Philippines Artiom Alhazov, Chişinău, Rep. of Moldova Bogdan Aman, Iaşi, Romania Matteo Cavaliere, Edinburgh, Scotland Erzsébet Csuhaj-Varjú, Budapest, Hungary Rudolf Freund, Wien, Austria Marian Gheorghe, Bradford, U.K. Honorary Member Thomas Hinze, Cottbus, Germany Florentin Ipate, Bucharest, Romania Shankara N. Krishna, Bombay, India Alberto Leporati, Milan, Italy Taishin Y. Nishida, Toyama, Japan

13 Linqiang Pan, Wuhan, China responsible of ACMC Gheorghe Păun, Bucharest, Romania Honorary Member Mario J. Pérez-Jiménez, Sevilla, Spain Agustín Riscos-Núñez, Sevilla, Spain Petr Sosík, Opava, Czech Republic K.G. Subramanian, Chennai, India György Vaszil, Debrecen, Hungary Sergey Verlan, Paris, France Gexiang Zhang, Chengdu, China Structure of IMCS 13 Awards Committee: Mario Pérez-Jiménez, Seville, Spain Chair, Marian Gheorghe, Bradford, U.K., Giancarlo Mauri, Milan, Italy, Gheorghe Păun, Bucharest, Romania, Linqiang Pan, Wuhan, China, Rules of functioning: 1. Prizes to be awarded annually: (1) The PhD Thesis of the Year, (2) The Theoretical Result of the Year, (3) The Application of the Year. 2. Each prize consists of diplomas for each co-author, one copy of the Hamangia thinker 1 and one voucher for a discount in the registration fee for the first of BWMC, CMC or ACMC to take place after the prize was voted; the discount will be fixed by the organizing committee of the meeting; in case of several authors, they will choose the one of them to benefit of the voucher. 3. Any registered member of IMCS can be nominated and can receive any of the three prizes. In cases (2) and (3), the prizes are awarded to the authors of a paper or of an application, with at least one of authors being a member of IMCS. The members of the Awards Committee cannot receive any prize, neither they can be coauthors of papers or applications which receive one of prizes (2) and (3). 4. If the Awards Committee considers necessary, each year at most one of the prizes can be awarded ex aequo, to two winners. 5. Any registered member of IMCS can propose a candidate for any prize, by sending to any member of the Awards Committee the relevant information (and any additional information requested by the Awards Committee). Implicitly, the Awards Committee can itself make nominations. 1 A Neolithic age clay sculpture, about 4000 years BC, found in Romania see the image at the next page.

14 14 Structure of IMCS 6. The nominations for the year Y should reach the Awards Chair before 20 of January of the year Y+1. The Awards Committee members decide the winners by the middle of February, and the prizes are awarded at the first edition of BWMC, CMC or ACMC where the winners participate in. 7. The members of the Awards Committee and the rules of functioning can be changed every year, after March 1, at the proposal of the Chair person or of any member of the IMCS Board, subject to a vote in the IMCS Board. The IMCS prizes are mainly meant to reward the excellency in MC research, equally focusing on theory and applications, and to encourage young researchers. The prizes are not subject to competitions, they do not identify the best PhD thesis or paper or application, they just point that a certain work/result is of a high value. This does not imply that other works/results are not so. We cannot rank scientific results like in sport, in a mathematical sense. We only want to call attention to certain works thus also calling attention to MC and to IMCS. The prestige of a prize will be given by the prestige of the winners, also on their evolution in time, during their careers. To reach these goals, we have to be conservative, exigent, transparent in our nominations and, especially, in selection. Nominations for the prizes for 2017 are waited for until January 20, 2018, and can be sent at any time, electronically, to any member of the Awards Committee (preferably with a CC to all members).

15 Constitution of the International Membrane Computing Society IMCS Article 1: Name (1.1.) The name of the Society shall be International Membrane Computing Society, abbreviated IMCS. (1.2.) The logo of IMCS is the one in the figure below. It should appear in all relevant places, such as IMCS web page, posters, calls, on the cover of the Bulletin of IMCS, etc. Article 2: Objects (2.1.) The society shall be a nonprofit academic organization, having as its goal to promote the development of membrane computing (MC), internationally, at all levels (theory, applications, software, implementations, connection with related areas, etc.).

16 16 Constitution of IMCS (2.2.) A special attention will be paid to the communication/cooperation inside MC community, to connections with other professional scientific organizations with similar aims, and to promoting MC to young researchers. (2.3.) IMCS will publish proceedings, journals or other materials, printed or electronically, as it sees fit. (2.4.) IMCS will organize yearly MC meetings, such as the Conference on Membrane Computing (CMC), the Asian Conference on Membrane Computing (ACMC), the Brainstorming Week on Membrane Computing (BWMC), as well as further workshops/meetings, as it sees fit. Article 3: Membership (3.1.) There are four categories of members: Honorary Members, Regular Members, Student Members, and Institution Members. (3.2.) The Honorary Members are elected by the IMCS Board ( voting, majority rule). Regular membership is open to all persons interested, on completing a membership form. (3.3.) Student Members can be undergraduate, master, and PhD students, and they are eligible for various facilities IMCS is planning for students. (3.4.) Institutions which want to join IMCS and support it can become Institution Members. Any support/sponsorship from an institution will be acknowledged in an appropriate way in IMCS publications. (3.5.) Any member, of any kind, is supposed to know and accept the Constitution of IMCS. Article 4: Structure (4.1.) The structure of IMCS and its governance are as specified in the next figure. The figure also specifies the ten Honorary Members with whom the Society starts (February 2016). (4.2.) Gheorghe Păun, the founder of MC, is appointed Honorary President of IMCS. (4.3.) The work of IMCS is organized and conducted by the Board of IMCS, consisting of the Executive Board and the Advisory Board. The Advisory Board should have between 10 and 20 members, hence in total the IMCS Board should contain between 20 and 30 members, (4.4.) The Executive Board consists of four individual positions: President, Vice President, Treasurer, and Secretary, and six Committees: Bulletin, Website, PR, Publication, Conferences, and Awards Committee. Each of these six Committees has a chair person. The Advisory Board also has a chair person. (4.5.) The four individual positions from the Executive Board, the six chair persons of the Committees, the members of the Advisory Board, and the chair of the Advisory Board are elected by the IMCS Board ( voting, majority rule). Each chair of a Committee appoints a number of Committee members as he/she sees fit.

17 Constitution of IMCS 17 (4.6.) All the elected positions are elected for one year. After one year, a change of an elected person can be proposed by the President or the Vice President of the IMCS Board, by the person itself (resignation), or by two thirds of members of the IMCS Board, and it is voted in the IMCS Board ( voting, majority rule). If there is no change proposal, then the person who occupies any position in the IMCS Board continues in the same position, for one further year. Article 5: Duties and competencies (5.1.) The IMCS Board President represents the Society in relation with any external entity, organizes/coordinates the activity of the IMCS Board, initiates voting in the IMCS Board, chairs any panel/meeting of the Society. (5.2.) The Vice President helps the President in all his/her activity, represents the President when he/she is not available (e.g., in chairing panels/meetings). Every year, the President and the Vice President present a common report

18 18 Constitution of IMCS about IMCS activity, first circulating it by in the IMCS Board and, after possible corrections, posting it in the IMCS web page. (5.3.) The Treasurer takes care of the income and expenses of IMCS, and each year presents a report in this respect to IMCS Board. This report is analyzed and voted in the IMCS Board ( voting, majority rule). (5.4.) The Secretary is responsible to keep a track record of the IMCS: memberships, reports, voting results, etc. (5.5.) The Bulletin Committee takes care of editing the Bulletin of IMCS, first accumulating information/materials in an electronic format and then printed, if needed/requested, with a periodicity to be decided by the IMCS Board. (5.6.) The Website Committee takes care of the IMCS web page, whose structure should be decided by the IMCS Board. (5.7.) The PR Committee is responsible with developing relationships with other similar organizations and promoting IMCS on various scientific forums, advertising its activity on specialised networks and at international events. (5.8.) The Publication Committee supervises the publication of proceedings, special issues of journals, collective volumes edited under the auspices of IMCS. Two particular goals of this Committee are to initiate a specialized journal, International Membrane Computing Journal, and a specialized series of monographs. (5.9.) The Conference Committee works as a steering committee for the two MC yearly conferences, CMC and ACMC, looking for venues, suggesting (in cooperation with the organizing committees) program committees and invited speakers, possible sponsors and publications. (5.10.) The Awards Committee collects nominations and decides the winners of three yearly IMCS Prizes: (1) The PhD Thesis of the Year, (2) The Theoretical Result of the Year, (3) The Application of the Year. The Awards Committee has its Rules of functioning, which are voted by the IMCS Board ( voting, majority rule). Article 6: Voting (6.1.) Each member of the IMCS Board (between 20 and 30 members) has one vote. (6.2.) A voting, on any subject, can be initiated by the President, the Vice President, or by two thirds of the IMCS Board members. (6.3.) The message proposing a vote should specify the issue to be decided in such a way that the alternatives YES and NO are clear. The message should be sent to all members of the IMCS Board, the voting messages of the members should also be sent to all members (full transparency). The voting should last 30 days. If a member is not replying in the first 15 days, the initiator of the vote should contact him/her once again. If a member is not replying even to the second message, then his/her vote is considered abstaining. (6.4.) Majority rule means that at least half of the IMCS Board have voted (YES, NO, or abstaining) and the decision is made according to the number of

19 Constitution of IMCS 19 YES and NO votes which is higher. In case of a draw, the vote of the President is decisive unless if the President does not decide to repeat the vote, maybe changing the object of the vote. (6.5.) All ambiguities and uncovered cases should be clarified by discussions in the IMCS Board and, if decided so, proposed as amendments to the Constitution. Article 7: Panels (7.1.) On the occasions of IMCS annual meetings, like BWMC, CMC, and ACMC, panels should be organized, chaired by the President, the Vice President or, in their absence, by another member of the IMCS Board designated by the President, to discuss current issues of the Society. Article 8: Finance (8.1.) Income: possible membership fees, as decided by IMCS Board, donations, sponsorhips, conference registration fees, participation to research projects. (8.2.) Expenses: IMCS prizes, students support, Bulletin of IMCS hardcopy, maintaining web pages, sponsoring MC conferences all these and anything else, under the control of the IMCS Board. Article 9: Amendments (9.1.) Amendments to IMCS Constitution can be proposed by any member of the IMCS Board, at any time. Any amendment should be discussed and voted in the IMCS Board ( voting, majority rule) and then, if accepted, published in the IMCS web page, thus being available to all members of IMCS. Article 10: Dissolution (10.1.) The dissolution of IMCS should be done in two steps: first, a vote in the IMCS Board is organized ( voting, two thirds majority), and, if the dissolution proposal passes, a general vote is organized, where all regular members participate ( voting, two thirds majority; in order the vote to be valid, at least half of the members should vote). (10.2.) If the Society decides to get dissolved, all remaining assets shall be donated to a similar organization, at the choice of the IMCS Board. Article 11. Provisory statement The present Constitution will get provisionally valid by being voted (by , majority rule), in March 2016, in the IMCS Board, as this Board was constituted by consensus during BWMC 2016 and soon after that. Then, it will be published in the IMCS web page and, as soon as possible, in 2016, it will be voted by all individual members of IMCS ( voting, majority rule). The vote will last one month and to voting will participate all individual members of IMCS, students or regular, registered until the last day of the previous month.

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21 IMCS Journal: Journal of Membrane Computing (JMEC) International Membrane Computing Society (IMCS) witnesses the processes of gestation and the birth of the new international journal, Journal of Membrane Computing (JMEC). The Editor-in-Chief of JMEC, Professor Linqiang Pan, signed the publishing agreement with Springer Nature Singapore Pte Ltd. on March 20, 2017, in Wuhan, China. JMEC is an international journal with four issues per volume (per year). The Journal Homepage is setting up and will be publicized soon. The first accepted papers are planned to be online in 2018 and thereafter will be published in the Spring of JMEC aims to foster the dissemination of new discoveries and novel technologies in the area of membrane computing and the related areas like bio-inspired computing and natural computing. The focus of this journal is to provide a publication and communication platform for researchers, professionals and industrial practitioners, covering the theoretical fundamentals and technological advances to various applications. JMEC solicits original, high-quality and previously unpublished research papers, survey and review articles, short communications, and tutorial papers.

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23 News from MC Research Groups Bioinformatics and Natural Computing at the University of Verona Vincenzo Manca The group of Bioinformatics and Natural computing, at the Department of Computer Science of the University of Verona, started in 2002, when Vincenzo Manca moved from the University of Pisa to the University of Verona, with the explicit request, from the Faculty of Science, of promoting activities in fields connecting Computer Science with Biology. In fact, at that time, the two main components of faculty were in those areas. V.M. started almost immediately to design and to realize the project of a bachelor curriculum in Bioinformatics, where programming, algorithms, and computer systems were taught jointly with chemistry (inorganic and organic) biochemistry, genetic, and molecular biology. In the end the structure of the resulting bachelor course had a kernel of basic mathematical and physical sciences, a computer science component and a bio-tech components, the last two almost equivalent as number of credits. This story is simple to tell now, but joining computer scientists and chemo-biologists in a common teaching project and with a common finality was really hard, of a difficulty comparable, maybe, to nuclear fusion. In the academic year 2006/2007 the first year of Bioinformatics started. The details of the structure of this Bachelor Course (3 years) and the contents of single courses can be found at the official site of the Department of Computer Science of the Verona University. A remarkable aspect of this course was the formal collocation within the schema of Italian scientific classes that labels the academic competences. Just for fixing the ideas, in Mathematics there are MAT01, MAT02, MAT08 including classical mathematical subjects (Analysis, Algebra,... ). In Computer Science there is only one class INF01, but there are a great number of classes CHEM01, BIO01,..., BIO18 for the chemo-bio scientific fields. Any teaching program has to be located in one class, in the case of Bioinformatics this class is INF01 (this is the only case in the whole academic programs on Bioinformatics active in Italy, usually hosted within biological classes). The above teaching project was supported by a big effort in establishing a group of bioinformaticians with different and complementary competences who could

24 24 V. Manca support the teaching activity especially in courses where the two competences meet. With the help of all faculty, in the years, a number of people was enrolled for filling the teaching and research needs of Bioinformatics. In academic year 2016/2017 (ten years after the bachelor course) a Program for a Master (two years) in Medical Bioinformatics was activated, and also this program is within the scientific class INF01. This premise wants just to explain the context where the research group of Bioinformatics and Natural computing is framed. In fact, the needs and the finalities of the courses required a corresponding research perspective very close to the applications. In the following we shortly present some research line of the group consisting of the following tenure track members. The members listed below cover 10 courses in both curricula of Bioinformatics mentioned above (a great number of PhD, graduate and undergraduate students has to be considered as an active part of the group). Manuele Bicego (Associate Professor) Vincenzo Bonnici (Post-doc) Alberto Castellini (Post-doc) Ferdinando Cicalese (Associate Professor) Giuditta Franco (Associate Researcher) Rosalba Giugno (Associate Professor) Zsuzsanna Liptak (Associate Researcher) Vincenzo Manca (Full Professor) Luca Marchetti (Lecturer at Verona Univ. and Researcher at COSBI of Trento) Informational Genomics (Bonnici, Franco, Giugno, Manca) Informational genomics is mainly based on the notion of genomic distribution. Any variable X defined on genomes and taking values in a (finite) set of values in correspondence to genome components (positions, substrings, segments, distances,... ) defines a genomic distribution where with each value x of X is associated the number of times this value occurs, and consequently its frequency of occurrence. In the second case we have a discrete probability distribution, therefore according to Shannon, a (genomic) information source (GIS). We can apply information theoretic concepts to GIS by means of which we can extract information useful to decipher the functional and evolutionary structures of genomes. A number of basic notions related to GIS were defined: dictionaries, that is, classes of substrings, called also words, or k-mers (with generic k equal to some specific length), indexes (max, min, average values of distributions), segmentations (partitions of a genome in contiguous portions), elongations (the set of all possible continuations of words occurring in a genome), recurrences (of specific words of classes of words), entropies (Shannon entropies for GIS), divergences (Kullback-Leibler informational divergences between GIS). Some classes of words have a great importance in the context of the concepts mentioned above: repeats, hapaxes, duplexes, creodes, nul-

25 Bioinformatics at the University of Verona 25 lomers, memers, double creodes, creode tails, tandems (their formal definitions, are based on specific patterns they must satisfy when they occur in a given genome). In this perspective genomes are long (from some thousand hundreds to some billions of characters), but they are long strings provided by nature for directing the functions and the evolutions of cells. These strings encode the information that cells use in their functioning. Therefore, this information has to be, in some way, related with the informational features emerging from their informational analyses. Starting from a number of basic distributions we computed all the notions indicated above, for a great number of real genomes from archea to H. sapiens (downloaded from public repositories). Another aspect, that is crucial for evaluating the data coming from the computations of distributional parameters, is the comparison with random genomes. We assume that a parameter of a genome G is informationally significant if it is different, in a statistic relevant manner, from the same parameter observed in a random genome R of the same length of G (R can be generated by using a pseudorandom generator). This assumption, which we call the Anti-randomness principle, resulted very useful in evaluating the genomic significance of some specific distributional parameters. The algorithms and the software needed in our computations must be very specific, otherwise their inefficiency would be prohibitive. For example, the entropy of the distribution of words of length 30 for H. sapiens would require to work with dictionaries near to billions of words. For this reason, data structures of suffix trees and suffix arrays were used for making affordable such a kind of computations (a specific software was implemented by V. Bonnici in this context). In a recent paper (Bonnici-Manca 2017) some laws were discovered that were validated by the measures of a number of entropic parameters over 70 genomes (from archea to primates). However, the main objective of these analyses is that of discovering basic principles helping in discriminating pathological cases with respect to heathy ones, for a deep comprehension of specific diseases at global genomic level. Recurrent bio-dynamics (Castellini, Franco, Manca, Marchetti) This research is aimed at discovering basic mechanisms underlying global functioning and internal relationships of complex systems, such as immune systems. It is mainly oriented to computational data-driven network models. In a more theoretical context, a biochemical network can be modeled by a multiset grammar and can be investigated from a dynamical viewpoint by linear recurrence systems. Recently, this interesting connection between multiset grammar and a (network) recurrent dynamics posed a minimization problem, which turned out to be NP-hard. Bio-data Analysis (Castellini, Franco) This research is aimed at inferring information and knowledge from real world data by employing standard techniques of statistical analysis, namely on immunological data, which are preprocessed and transformed in terms of either multivariate time series or cross-sectional data. Regarding analysis of cross-sectional data, a novel bi-clustering approach, which extends the affinity propagation clustering algorithm

26 26 V. Manca to the bi-dimensional case, was tested on synthetic microarray benchmarks, with encouraging results for real data. Regarding analysis of multivariate time series, an age-related correlation among human peripheral B lymphocyte subpopulations has been formalized by means of metabolic P systems, linear models, and probabilistic models. Ongoing research is improving heuristics for segmentation of multiple timeseries (by change point detection methods). The main goal of this research topic is to show different aspects of processes identified by different data partitions, which may have also non-synchronous evidence. Pattern Recognition (Bicego) This research is mainly focused on Pattern Recognition, namely on the study and development of automatic techniques and models able to extract information from real world data, typically in terms of classes or clusters. Recent activities are focused on probabilistic models: Hidden Markov Models, Mixtures, Topic Models - and on Kernel Machines - like Support Vector Machines. In these contexts novel models/methodologies are investigated (such as hybrid generative-discriminative methods), and alternative classification or clustering schemes for advanced model selection techniques. From an application point of view, the main focus is on Bioinformatics topics for designing efficient solutions to the analysis of counting data - namely data which express the level of presence of entities (e.g. gene expression data, or proteomics data). In this context, probabilistic graphical models are investigated, such as Topic Models, and Bag of Words. The ultimate goal is to devise highly interpretable solutions, the interpretability of methods and solutions being the most stringent need in nowadays bioinformatics research. Bio-algorithms (Cicalese, Liptak) This research concerns the development of efficient algorithms for problems in the area of computational biology, especially of a combinatorial type. In particular, algorithmic problems were extensively studied which occur in the area of sample identification from mass spectrometry data. One of the algorithmic problems arising from this and other applications (genome rearrangements, gene clusters), is that of Jumbled Pattern Matching, or abelian pattern matching. Here segments of the input sequence are sought which have the same multiplicity of characters as the query, but possibly in a different order. This question arises with mass spectrometry data, since the order of residues along the backbone has no influence on the molecular mass, which is measured in the experiment. Most recently, we studied a similar problem on trees, another data structure with multitudinous applications in bioinformatics. In addition, the research has given rise to theoretical studies of a new class of binary strings, which turn out to be relevant to abelian string problems, as well as to a novel data structure with the potential help to solve efficiently various abelian string problems. In another line of research, information theoretic measures are studied for the definition of similarity and dissimilarity functions among probability distributions. These mathematical tools are employed for both

27 Bioinformatics at the University of Verona 27 data clustering and for defining heuristics for inferring the right causality relationships among correlated phenomena, from samples on their joint distribution. Bio-networks (Bonnici, Giugno) The focus of this research is mainly on biological network modeling, perturbation and analysis and classification of phenotypes by coding and non-coding expression. Our research goes from Genome to Network analysis in order to understand biological systems under specific conditions. In particular, algorithms for searching in labeled directed and undirected graphs or multigraphs were developed. Up to date, ones of the fastest algorithms for subgraph database search have been designed, in the cases of matching with or without redundancy on the results. We have addressed the problem of finding significant unusually sub-networks, named motifs, by overcoming to the consuming step of network simulation and by providing analytical models. These algorithms are the basis to understand the topological proprieties of biological systems. In order to analyze their dynamics, signaling and metabolic pathways were designed by using languages, techniques, and tools well established in the context of electronic design automation. The underlying intuition is that several characteristics and issues to model biological systems are common to the electronics systems modeling, such as concurrency, reactivity, abstraction levels, as well as state space explosion during validation. Therefore, static and dynamic network characteristics were studied by associating to nodes profiles information from genome and trascriptome data. Pipelines for genomic variants detections and annotation were developed and customized both for bulk and single cell data, in the context of coding and non coding differential analysis. Models of network of interactions, constructed from data expression, were designed particularly for small, circular, circulant RNA interferences. Software and data were produced that are available for the open research development in appropriate platforms. Selected References Alaimo S, Giugno R, et al: Post-transcriptional knowledge in pathway analysis increases the accuracy of phenotypes classification. Oncotarget (2016) Bicego M, Lovato P: A bioinformatics approach to 2D shape classification. Computer Vision and Image Understanding, Vol. 145, pp (2016) Bollig-Fischer A, Marchetti L, Mitrea C, Wu J, Kruger A, Manca V, Draghici S: Modeling time-dependent transcription effects of HER2 oncogene and discovery of a role for E2F2 in breast cancer cell-matrix adhesion. Bioinformatics, Vol. 30, n. 21, pp (2014) Bonnici V, Giugno R: On the variable ordering in subgraph isomorphism algorithms. IEEE/ACM Transactions on Computational Biology and Bioinformatics, Vol. 14, n. 1, pp (2017) Bonnici V, Busato F, Micale G, Bombieri N, Pulvirenti A, Giugno R:

28 28 V. Manca APPAGATO: an APproximate PArallel and Stochastic GrAph Querying TOol for Biological Networks. Bioinformatics (2016) Bonnici V, Manca V: Informational laws of genome structures. Scientific Reports, Vol. 6, pp (2016) Bonnici V, Manca V: InfoGenomics Tools: a computational suite for informational analysis of genomes. Journal of Bioinformatics and Proteomics Review, Vol. 1, n. 1, pp (2015) Burcsi P, Fici G, Lipták Z, Ruskey F, Sawada J: On prefix normal words and prefix normal forms. Theoretical Computer Science, Vol. 659, pp (2016) Castellini A, Franco G, Manca V, Ortolani R, Vella A: Towards an MP model for B lymphocytes maturation. UCNC 2014, University Western Ontario, London, Ontario, Canada July 14-18, 2014, pp (2014) Castellini A, Manca V, Zucchelli M: An evolutionary procedure for inferring MP systems regulation functions of biological networks. Natural Computing, Vol. 14, n. 3, pp (2015) Denitto M, Farinelli A, Figueiredo M A T, Bicego M: A biclustering approach based on factor graphs and the max-sum algorithm. Engineering Applications of Artificial Intelligence, Vol. 59, pp (2015) Castellini A, Paltrinieri D, Manca V: MP-GeneticSynth: Inferring Biological Network Regulations from Time Series. Bioinformatics, Vol. 31, n. 5, pp (2015) Cicalese F et al: Diagnosis determination: decision trees optimizing simultaneously worst and expected testing cost. Proc. ICML 2014, pp (2014) Manca V: The principles of Informational Genomics. Theoretical Computer Science (2017) Manca V: Grammars for Discrete Dynamics. Machine Learning for Health Informatics LNAI (2016) Marchetti L, Manca V: MPTheory Java Library: a multi platform Java library for Systems Biology based on the Metabolic P theory. Bioinformatics, Vol. 31, n. 8, pp (2015)

29 Spiking Neural P Systems Research at Algorithms and Complexity Laboratory of the University of the Philippines Diliman Francis George C. Cabarle, Jym Paul Carandang, Ren Tristan dela Cruz, Kelvin C. Buño, John Matthew Villaflores, Miguel Ángel Martínez-del-Amor, Nestine Hope Hernandez, Richelle Ann B. Juayong, Henry N. Adorna Algorithms & Complexity Department of Computer Science University of the Philippines Diliman Diliman 1101 Quezon City, Philippines Main contact: hnadorna@up.edu.ph 1 Introduction The Algorithms and Complexity Laboratory (in short, ACLab) of Department of Computer Science, University of the Philippines Diliman, consists of a subgroup of nine members, listed as authors in this report, working mainly on membrane computing. The website for ACLab is at while the website of the subgroup for membrane computing is at com/site/aclabmcgroup/. The membrane computing subgroup of ACLab consists of professor Henry N. Adorna, three assistant professors: Francis George C. Cabarle, Kelvin C. Buño, and Nestine Hope Hernandez (working on other P system models), with Richelle Ann Juayong having recently finished her PhD dissertation on P systems with energy. Since 2009, ACLab has produced research on SN P systems (more details below). At present time, Kelvin Buño is in part working on dp Schemes which include distributed variants of P systems, which includes SN dp systems for his PhD work; for their masters theses, Jym Paul Carandang and John Matthew Villaflores are working on GPU simulators for SN P systems and their variants, while Ren Tristan dela Cruz is working on SN P systems with plasticity. Francis George C. Cabarle is doing postdoctoral research with Xiangxiang Zeng at Xiamen University (Xia- M.Á Martínez-del-Amor is mainly with the Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, Universidad de Sevilla, Avda. Reina Mercedes s/n, Sevilla, Spain

30 30 F.G.C. Cabarle et al. men, China). Henry N. Adorna is visting Linqiang Pan at Huazhong University of Science and Technology (Wuhan, China). Some of the main the research directions of ACLab for SN P systems (and their variants) include: their syntax and semantics for computing, applications, or modelling; their representations as vectors and matrices, in order to perform linear algebra operations in describing the system evolution; their simulation algorithms and software simulators (both sequential and parallel); their computing power and efficiency with respect to other P systems and models of computation. 2 Works on spiking neural P systems 2.1 Matrix representations and algorithms for GPU simulations The work on representing SN P systems as vectors and matrices started at BWMC2010 in [1], followed by a publication at CMC2010 in [2]. The earliest version of an SN P system simulator for graphics processing units (or GPUs) is in PCSC2011 in [3], followed by a journal version in [4]. Succeeding investigations by improving the simulation algorithm, software simulator, and GPUs include [5, 6]. Note that so far, these GPU simulators run on the CUDA hardware manufactured by NVIDIA corporation. The first and preliminary SN P system simulator using the open-standard software known as OpenCL is in [7]. Further improvements of the simulation and GPU simulator on NVIDIA CUDA hardware were afterwards referred to as CuSNP (short for CUDA for SN P systems) published as a preliminary work in [8], with following improvements reported at BWMC2016 in [9, 10, 11] with the most recent one in [12]. Lastly, since SN P systems in general are sparse graphs, we have started to work on simulators that make better use of GPUs with sparse matrix representations in [13]. 2.2 Variants of SN P systems Since SN P systems and most of their variants are static and directed graphs 2 some variants inspired by dynamic graphs focusing on edge-centric evolution (hence, synapse-centric evolution for SN P systems) were introduced. The first variant are SN P systems with structural plasticity introduced in [14], allowing neurons to use, besides the standard spiking rules, a second type of rules known as plasticity rules. Plasticity rules allowed neurons to create or delete their own synapses; see further works in [15, 16, 17, 18, 19] including a quick survey in [20]. The second variant for edge-centric dynamism are SN P systems with scheduled synapses introduced [21], where synapses can also (dis)appear in the system depending on a given schedule or duration. 2 A few variants have dynamism, e.g. neuron division and budding following dynamic graphs, but such variants are mainly focused on evolving the neurons instead of the synapse only.

31 SN P Systems Research at ACLab of UP Diliman 31 Lastly, works on SN dp systems where the entire input is divided in parts, so that the parts enter into different components of the system (each component is an SN P system) are given in [22, 23]. 2.3 More on SN P systems and their computations, formal methods Works comparing the structure and behaviour of SN P systems to other wellknown models for concurrency such as Petri nets and process algebra are given in [22, 24, 25, 26]. Other works on describing the computation of SN P systems with respect to their ingredients (e.g. rule types, delays), other invariant properties are addressed in[1, 2, 27, 28, 29], and their lower bound simulation of finite automata in [30, 31]. A quick survey of SN P systems including work from ACLab is provided in [20], as well as in a bibliography of SN P systems literature as of February 2016 in systemf. References 1. X. Zeng, H. N. Adorna, M. Á. Martínez del Amor, and L. Pan, When matrices meet brains, Proceedings of the Eighth Brainstorming Week on Membrane Computing, Sevilla, ETS de Ingeniería Informática, 1-5 de Febrero, 2010, X. Zeng, H. Adorna, M. Á. Martínez-del Amor, L. Pan, and M. J. Pérez-Jiménez, Matrix Representation of Spiking Neural P Systems, in Membrane Computing: 11th International Conference, CMC 2010, Jena, Germany, August 24-27, 2010 (M. Gheorghe, T. Hinze, G. Păun, G. Rozenberg, and A. Salomaa, eds.), pp , Berlin, Heidelberg: Springer Berlin Heidelberg, F. Cabarle, H. Adorna, and M. Martínez-del Amor, Spiking neural P system without delay simulator implementation using GPGPUs, in Proc. 11th Philippine Computing Science Congress, Naga city, Philippines, pp , F. Cabarle, H. Adorna, and M. A. Martinez-del Amor, Simulating Spiking Neural P systems without delays using GPUs, in Natural Computing for Simulation and Knowledge Discovery, pp , IGI Global, F. G. C. Cabarle, H. Adorna, and M. Martnez, A spiking neural P system simulator based on CUDA, Lecture Notes in Computer Science, vol LNCS, pp , F. G. C. Cabarle, H. Adorna, M. Martinez-Del-Amor, and M. Perez-Jimenez, Improving GPU simulations of spiking neural P systems, Romanian Journal of Information Science and Technology, vol. 15, no. 1, pp. 5 20, A. R. Lagunda, G. I. Palaganas, F. G. C. Cabarle, and H. Adorna, Spiking Neural P Systems GPU Simulation using OpenCL, Proc. 16th Philippine Computing Science Congress, March 2016, Puerto Princesa, Palawan, Philippines, pp , J. Carandang, J. Villaflores, F. G. C. Cabarle, and H. Adorna, CuSNP: Improvements on GPU Simulations of Spiking Neural P Systems in CUDA, Proc. 16th Philippine Computing Science Congress, March, 2016, Puerto Princesa, Palawan, Philippines, pp , 2016.

32 32 F.G.C. Cabarle et al. 9. J. P. Carandang, J. M. Villaflores, F. G. C. Cabarle, H. N. Adorna, and M. Á. Martínez del Amor, Improving Simulations of Spiking Neural P Systems in NVIDIA CUDA GPUs: CuSNP, Proc. 14th Brainstorming Week on Membrane Computing (BWMC2016), Sevilla, Spain, vol. 14, pp , J. Carandang, J. Villaflores, F. G. C. Cabarle, H. Adorna, and M. Á. Martínezdel Amor, CuSNP: Spiking Neural P Systems Simulators in CUDA, 5th Asian Conference on Membrane Computing 2016, J. P. Carandang, J. M. Villaflores, F. G. C. Cabarle, H. Adorna, and M. Á. Martínezdel Amor, CuSNP: Spiking Neural P Systems Simulators in CUDA, Romanian Journal of Information Science and Technology, vol. 20, no. 1, pp , J. P. A. Carandang, F. G. C. Cabarle, H. N. Adorna, N. H. S. Hernandez,, and M. A. Martinez-del Amor, Nondeterminism in Spiking Neural P Systems: Algorithms and Simulations, Pre-proc. 6th Asian Conference on Membrane Computing (ACMC2017), 21 to 25 September 2017, Xihua University, Chengdu, China, M. Á. M. del Amor, D. Orellana-Martín, F. G. C. Cabarle, M. J. Pérez-Jiménez, and H. N. Adorna, Sparse-matrix Representation of Spiking Neural P Systems for GPU, Fifteenth Brainstorming Week on Membrane Computing (BWMC2017), pp , F. G. C. Cabarle, H. N. Adorna, M. J. Pérez-Jiménez, and T. Song, Spiking neural p systems with structural plasticity, Neural Computing and Applications, vol. 26, no. 8, pp , F. G. C. Cabarle, N. H. S. Hernandez, and M. Á. Martínez-del Amor, Spiking Neural P Systems with Structural Plasticity: Attacking the Subset Sum Problem, in Membrane Computing: 16th International Conference, CMC 2015, Valencia, Spain, August 17-21, 2015 (G. Rozenberg, A. Salomaa, J. M. Sempere, and C. Zandron, eds.), pp , Springer International Publishing, F. G. C. Cabarle, H. N. Adorna, and M. J. Pérez-Jiménez, Sequential spiking neural P systems with structural plasticity based on max/min spike number, Neural Computing and Applications, vol. 27, no. 5, pp , F. G. C. Cabarle, H. N. Adorna, and M. J. Pérez-Jiménez, Asynchronous Spiking Neural P Systems with Structural Plasticity, in Unconventional Computation and Natural Computation (C. S. Calude and M. J. Dinneen, eds.), vol of LNCS, pp , Springer International Publishing, R. T. A. de la Cruz, F. G. C. Cabarle, and X. Zeng, On Languages Generated by Spiking Neural P System with Structural Plasticity, Pre-proc. 18th International Conference on Membrane Computing (CMC18), 24 to 28 July 2017, University of Bradford, U.K., pp , R. T. A. dela Cruz, F. G. C. Cabarle, and X. Zeng, Arithmetic and Memory Module using Spiking Neural P Systems with Structural Plasticity, Pre-proc. 6th Asian Conference on Membrane Computing (ACMC2017), 21 to 25 September 2017, Xihua University, Chengdu, China, H. N. Adorna, F. G. C. Cabarle, L. F. Macías-Ramos, L. Pan, M. J. Pérez-Jiménez, B. Song, T. Song, and L. Valencia-Cabrera, Taking the Pulse of SN P Systems: a Quick Survey, in Multidisciplinary Creativity: Homage to Gheorghe Păun on His 65th Birthday (M. Gheorghe, I. Petre, M. J. Perez-Jimenez, G. Rozenberg, and A. Salomaa, eds.), pp. 3 16, Spandugino, F. G. C. Cabarle, H. N. Adorna, M. Jiang, and X. Zeng, Spiking Neural P systems with Scheduled Synapses, IEEE Transactions on NanoBioscience (to appear), 2017.

33 SN P Systems Research at ACLab of UP Diliman F. G. C. Cabarle and H. N. Adorna, Theory and Practice of Computation: Workshop on Computation: Theory and Practice Quezon City, Philippines, September 2011 Proceedings, ch. Some Notes on Spiking Neural dp Systems and Petri Nets, pp Springer Japan, J. G. Q. Torres, K. C. Buño, and F. G. C. Cabarle, Some Notes on Spiking Neural dp Systems, Pre-proc. 6th Asian Conference on Membrane Computing (ACMC2017), 21 to 25 September 2017, Xihua University, Chengdu, China, F. G. C. Cabarle and H. N. Adorna, On Structures and Behaviors of Spiking Neural P Systems and Petri Nets, in Membrane Computing: 13th International Conference, CMC 2012, Budapest, Hungary, August 28-31, 2012 (E. Csuhaj-Varjú, M. Gheorghe, G. Rozenberg, A. Salomaa, and G. Vaszil, eds.), pp , Springer Berlin Heidelberg, R. A. B. Juayong, N. H. S. Hernandez, F. G. C. Cabarle, and H. N. Adorna, A Simulation of Transition P Systems in Weighted Spiking Neural P Systems, Proceedings of Workshop on Computation: Theory and Practice WCTP2013, ch. 2, pp , World Scientific, H. N. Adorna, K. C. Buño, and F. G. C. Cabarle, Notes in Delays and Bisimulations of spiking neural P systems using SNP Algebra, Proceedings of Workshop on Computation: Theory and Practice WCTP2013, ch. 2, pp , World Scientific, G. N. Ibo and H. N. Adorna, Characterizing Periodicity as a Dynamical Aspect of Generative SN P Systems, Pre-proc. 12th International Conference on Membrane Computing August, 2016, Fontainebleau, Paris, France, pp , F. G. C. Cabarle, K. C. Buño, and H. N. Adorna, Proceedings of Theory and Practice of Computation: 2nd Workshop on Computation: Theory and Practice, ch. Time after Time: Notes on Delays in Spiking Neural P Systems, pp Springer Japan, F. G. C. Cabarle, K. C. Buño, and H. N. Adorna, On the Delays in Spiking Neural P Systems, Philippine Computing Journal, vol. 7, no. 2, pp , F. G. C. Cabarle, H. N. Adorna, and M. J. Pérez-Jiménez, Notes on spiking neural p systems and finite automata, Thirteenth Brainstorming Week on Membrane Computing, pp , 02/ F. G. C. Cabarle, H. N. Adorna, and M. J. Pérez-Jiménez, Notes on spiking neural P systems and finite automata, Natural Computing, vol. 15, no. 4, pp , 2016.

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35 Book Presentations Computer der Natur (Computers Found in Nature) Author: Thomas Hinze Bookboon Publishing House, London, 118 pp. Second edition, 2017, ISBN Available for free as PDF for download at

36 36 T. Hinze Abstract Biological mechanisms of information processing turn out to be reliable, adaptive, and efficient. Commonly, they are mainly based on molecular interactions. Here, molecules act as a storage medium assuming the role of a data carrier affected and operated by fine-tuned biochemical reactions. This results in an alternative hardware at a nanometre scale, whose exploitation in engineering promises manifold applications beyond bionics able to advance many aspects of computer science. Beginning with foundations of chemical reactions and kinetic laws to capture the underlying temporal behaviour of substrate concentrations, numerous practical examples of chemical computers analogous as well as digital systems have been introduced and explained step by step, in a wide range from biological clock systems as chemical control circuits via molecular arithmetics up to cell signalling networks emulating finite automata and addressable register machines. Many figures illustrate the considered models of molecular computers. The German-speaking textbook gives a broad overview of the knowledge field dedicated to both beginners and advanced learners. Abstract in German Biologische Mechanismen der Informationsverarbeitung gelten als zuverlässig, anpassungsfähig und effizient. Sie beruhen größtenteils auf molekularen Interaktionen. Moleküle dienen hierbei als Speichermedium und übernehmen die Rolle des Datenträgers, auf dem feinabgestimmte biochemische Reaktionen operieren. Daraus resultiert eine alternative Hardware im Nanometermaßstab, deren ingenieurtechnische Erschließung über die Bionik hinaus mannigfaltige Anwendungen verspricht, die die Informatik in vielerlei Hinsicht bereichern und weiterentwickeln können. Aufbauend auf Grundlagen chemischer Reaktionen und kinetischen Gesetzmäßigkeiten ihres zeitlichen Ablaufs werden zahlreiche Beispiele chemischer Analog- und Digitalcomputer vorgestellt und leicht nachvollziehbar erklärt. Die Palette reicht dabei von biologischen Uhren als chemische Regelkreise über molekulare Arithmetik bis hin zu Zellsignalnetzwerken, die als endliche Automaten oder programmierbare Registermaschinen arbeiten. Zahlreiche Abbildungen veranschaulichen die einzelnen Molekularcomputermodelle. Das Lehrbuch gibt einen breiten Überblick über das Wissensgebiet und wendet sich gleichermaßen an Einsteiger wie Fortgeschrittene. Inhalt (Contents) Vorwort (Preface) 1. Chemisches Rechnen: Einfhrung und Motivation (Chemical Computing: Introduction and Motivation)

37 Books Announcements Grundlagen der Massenwirkungskinetik zur Beschreibung des Zeitverhaltens chemischer Reaktionen (Foundations of Mass-action Kinetics for Modelling the Temporal Behaviour of Chemical Reactions) 1. Systeme, Stoffe, Stoffkonzentrationen und Reaktionen (Systems, Substrates, Substrate Concentrations, and Reactions) 2. Reaktionskinetik (Reaction Kinetics) 3. Übungsaufgaben (Exercises) 3. Chemische Analogcomputermodelle (Models of Analogous Chemical Computers) 1. Addition (Addition) 2. Nichtnegative Subtraktion (Non-negative Subtraction) 3. Multiplikation (Multiplication) 4. Division (Division) 5. Quadratwurzel (Square Root) 6. Boolesche Operationen (Boolean Operators) 7. Diskussion der chemischen Arithmetik (Discussing Chemical Arithmetics) 8. Tiefpassfilter (Low-pass Filter) 9. Kaskadierter Integrator und Differentiator (Cascaded Integrator and Differentiator) 10. Brusselator zur Erzeugung spikefrmiger Oszillationen (Brusselator for Generation of Spike-shaped Oscillations) 11. Repressilator zur Erzeugung nahezu sinusfrmiger Oszillationen (Repressilator for Generation of Almost Sinusoidal Oscillations) 12. Goodwin-Modell als Prototyp steuerbarer chemischer Oszillatoren (Goodwin Model as a Prototype for Controllable Chemical Oscillations) 13. Chemischer Frequenzregelkreis nach dem Vorbild circadianer Uhren (Chemical Frequency Control Circuit as a Metamodel of Circadian Clocks) 14. Binrer Signalseparator (Binary Signal Separator) 15. Diskussion chemischer Analogcomputermodelle (Discussing Models of Analogous Chemical Computers) 16. Übungsaufgaben (Exercises) 4. Chemische Digitalcomputermodelle (Models of Digital Chemical Computers) 1. Datenkodierung und logische Gatter (Data Encoding and Logical Gates) 2. Ungetaktete Flip-Flops (Flip-flops without Clock) 3. Endliche Automaten am Beispiel eines binren Zhlers modulo 17 (Finite Automaton of a Binary Counter Machine modulo 17) 4. Chemische Registermaschine auf Basis der Massenwirkungskinetik (Chemical Register Machine Model Based on Mass-action Kinetics) 5. Übungsaufgaben (Exercises) 5. Literaturverzeichnis (References)

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39 Tutorials, Surveys Cooperative P Systems and the P Versus NP Problem Luis Valencia-Cabrera, Miguel Á. Martínez-del-Amor, D. Orellana-Martín, Ignacio Pérez-Hurtado, Mario J. Pérez-Jiménez Research Group on Natural Computing Department of Computer Science and Artificial Intelligence Universidad de Sevilla Avda. Reina Mercedes s/n, Sevilla, Spain {lvalencia, mdelamor, dorellana, perezh, marper}@us.es Summary. The P versus NP problem is undoubtedly the most important open question in computer science. Frontiers of tractability or efficiency expressed by means of syntactic or semantic ingredients in the framework of Membrane Computing, an unconventional computing paradigm, can bring a new approach to tackle P versus NP. In this context, the role of the cooperation of objects to trigger rewriting rules is analysed in order to obtain this kind of borderlines. Besides, a relationship among cooperative rewriting rules and instances of 2 SAT problem and 3 SAT problem is highlighted and their connections with results of computational complexity theory are described. Key words: Membrane Computing, P systems with active membranes, cooperative rules, the P versus NP problem, SAT problem. 1 Introduction The relevance of the P? = NP question is not only the inherent pleasure of solving a mathematical problem, since an answer to it could dramatically affect our everyday lives. On the one hand, a negative answer to this question would confirm that the majority of current cryptographic systems are secure from a practical point of view. On the other hand, a positive answer would not only show the uncertainty about the secureness of these systems, but also this kind of answer is expected to come together with a general procedure such that it will provide a deterministic algorithm solving any NP-complete problem in polynomial time. In an informal

40 40 L. Valencia-Cabrera et al. way, we can say that if P = NP then it would be possible to find solutions to search problems as easily as checking whether those solutions are correct, that is, almost all the algorithmic challenges that we face today could be solved in a practical way and computers could solve almost any mechanical task. The search of techniques other than the classical ones that allow us to tackle this problem becomes a very important challenge. In this context, frontiers of tractability expressed by means of syntactic or semantic ingredients in the framework of Membrane Computing can bring a new approach. Membrane computing is a flexible and versatile branch of natural computing, which arises as an abstraction of the compartmentalized structure of living cells, and the way biochemical substances are processed in (or moved between) membrane bounded regions [21]. Inspired by the structure of living cells, two main classes of membrane systems have been investigated: a hierarchical (cell-like) arrangement of membranes, inspired from the structure of the cell [21] and a net of membranes (placed in the nodes of a directed graph), inspired from the cellinterconnection in tissues [16] or inspired from the way that neurons communicate with each other by means of short electrical impulses (spikes), emitted at precise moments of time [10]. All classes of computing devices considered in the field of membrane computing are generally called P systems or membrane systems, which are parallel and distributed computational models based on processing multisets of objects in cell-like or tissue-like structures by means of rewriting rules. A P system is cooperative if it contains rules using cooperation, that is, rules that need more than one object to be triggered. This paper is devoted to study the role of cooperation of objects to trigger rewriting rules in order to obtain frontiers of efficiency by means of ingredients of membrane systems working in cell-like mode. For that, the use of cooperation in rewriting rules is combined with some syntactical ingredients and the behaviour of different cocktails are analysed. The paper is structured as follows. In next section, some general concepts are briefly described in order to make the work self-contained. Section 3 is devoted to present the different models of cell-like membrane systems that will be studied in this work, emphasizing the syntactic and semantic aspects of them. In the following section, frontiers of efficiency involving cooperation are presented in the different models considered through the paper. In Section 5, a relationship among cooperative rewriting rules and instances of 2 SAT and 3 SAT is highlighted from a complexity point of view. Finally, some conclusions are discussed. 2 Preliminaries An alphabet Γ is a non-empty set and their elements are called symbols. A string u over Γ is an ordered finite sequence of symbols, that is, a mapping from a natural number n N onto Γ. The number n is called the length of the string u and it is

41 Cooperative P Systems and the P Versus NP Problem 41 denoted by u. The empty string (with length 0) is denoted by λ. The set of all strings over an alphabet Γ is denoted by Γ. A language over Γ is a subset of Γ. A multiset over an alphabet Γ is an ordered pair (Γ, f) where f is a mapping from Γ onto the set of natural numbers N. The support of a multiset m = (Γ, f) is defined as supp(m) = {x Γ f(x) > 0}. A multiset is finite (resp. empty) if its support is a finite (resp. empty) set. We denote by the empty multiset and we denote by M(Γ ) the set of all multisets over Γ. Let m 1 = (Γ, f 1 ), m 2 = (Γ, f 2 ) be multisets over Γ, then the union of m 1 and m 2, denoted by m 1 + m 2, is the multiset (Γ, g), where g(x) = f 1 (x) + f 2 (x) for each x Γ. 2.1 Graphs and trees Let us recall some notions related with graph theory (see [6] for details). An undirected graph is an ordered pair (V, E) where V is a set whose elements are called nodes or vertices and E = {{x, y} x V, y V, x y} whose elements are called edges. A directed graph is an ordered pair (V, E) where V is a set whose elements are called nodes or vertices and E = {(x, y) x V, y V } whose elements are called arcs. A path of length k 1 from a node u to a node v in a graph (V, E) is a finite sequence (x 0, x 1,..., x k ) of nodes such that x 0 = u, x k = v and (x i, x i+1 ) E for 0 i k 1 (in the case of a directed graph or {x i, x i+1 } E in the case of an undirected graph. If k 2 and x 0 = x k then we say that the path is a cycle of the graph. A graph with no cycle is said to be acyclic. An undirected graph is connected if there exist paths between every pair of nodes. A free tree (tree, for short) is a connected, acyclic, undirected graph. A rooted tree is a tree in which one of the vertices (called the root of the tree) is distinguished from the others. In a rooted tree the concepts of ascendants and descendants are defined in a usual way. Given a node x (different from the root), if the last edge on the (unique) path from the root of the tree to the node x is {x, y} (in this case, x y), then y is the parent of node x and x is a child of node y. The root is the only node in the tree with no parent. A node with no children is called a leaf. 2.2 Decision problems Roughly speaking, a decision problem X is one whose solution/answer is either yes or no. This can be formally defined by an ordered pair (I X, θ X ), where I X is a language over a finite alphabet Σ X and θ X is a total boolean function over I X. The elements of I X are called instances of the problem X. Each decision problem X has associated a language L X over the alphabet Σ X as follows: L X = {u E X θ X (u) = 1}. Conversely, every language L over an alphabet Σ has associated a decision problem X L = (I XL, θ XL ) as follows: I XL = Σ and θ XL (u) = 1 if and only if u L. Then, given a decision problem X we have X LX = X, and given a language L over an alphabet Σ we have L XL = L.

42 42 L. Valencia-Cabrera et al. It is worth pointing out that any Turing machine M (with input alphabet Σ M ) has associated a decision problem X M = (I M, θ M ) defined as follows: I M = Σ M, and for every w Σ M, θ M (w) = 1 if and only if M accepts w. Obviously, the decision problem X M is solvable by the Turing machine M. The satisfiability problem The satisfiability problem (SAT problem) is described as follows: given a boolean formula in conjunctive normal form (CNF), to determine whether or not there exists an assignment to its variables, called truth assignment, on which it evaluates true. SAT is the first problem which was demonstrated that belongs to the class of NP-complete problems [7]. These problems are in the class NP and they verify an interesting property: its individual complexity can be extended to the entire class NP, that is, if SAT is a problem in P then all problems in NP also belongs to P. Different variants of the SAT problem were considered, in particular, for each k 1, k SAT problem is a special case of the SAT problem in which all clauses of the input formula have exactly k literals. It is well known that 2 SAT is in class P (in fact, it is an NL) and 3 SAT is an NP-complete problem [7]. The reachability problem The reachability problem is described as follows: given a directed graph G = (V, E) with two specified vertices s and t, determine whether or not there is a path from s to t. There are algorithms solving this problem, for instance, search algorithms based on breadth-first search or depth-first search. These algorithms determine whether two vertices are connected in O(max( V, E )) time. Moreover, they basically need to store at most V items, so these algorithms use O( V ) space. But this quantity of space can be reduced to O(log 2 V ) by using an algorithm that could be called middle-first search (see [20] for details, pp ). In particular, reachability problem is in class P. 3 Cell-like membrane systems In order to make this paper self-contained, the different models of membrane systems considered in this work are introduced detailing their syntax and semantics. 3.1 Basic transition P systems Next, the basic model introduced by Gh. Păun in its seminal paper [21] is presented by using a slightly different notation. Definition 1. A basic transition P system of degree q 1 is a tuple Π = (Γ, µ, M 1,..., M q, R, P R, i out ), where:

43 Cooperative P Systems and the P Versus NP Problem Γ is a finite alphabet. 2. µ is a rooted tree. 3. M 1,..., M q are multisets over Γ. 4. R is a finite set of rules of the following forms: a) [ u ] i v 1 [ v 2 [ v 3 ] j ] i, for i, j {1,..., q}, i j and u, v 1, v 2, v 3 M(Γ ). b) [ u ] i v 1 [ v 2 [ v 3 ] j ] i δ, for i, j {1,..., q}, i j and u, v 1, v 2, v 3 M(Γ ), and δ is a distinguished symbol such that δ / Γ. 5. P R R R is an strict partial order over R. 6. i out {0, 1,..., q}. A basic transition P system Π = (Γ, µ,, M 1,..., M q, R, P R, i out ), of degree q 1 can be viewed as a set of q membranes injectively labelled by 1,..., q, arranged in a hierarchical structure µ given by a rooted tree whose root is called the skin membrane of the system and with an environment labelled by 0 such that: (a) M 1,..., M q are multisets over the working alphabet Γ representing the objects initially placed in the q membranes of the system; (b) R is the set of rules that allows to evolve the system; (c) P R R R is a strict partial order relation over R providing priorities between rules, in such a manner that if (r 1, r 2 ) P R we say that rule r 1 has a higher priority than r 2 and we denote it by r 1 > r 2 ; and (c) i out {0, 1, 2,..., q} represents a distinguished zone which will encode the output of the system. We use the term zone i, 0 i q, to refer to membrane i in the case 1 i q and to refer to the environment in the case i = 0. An instantaneous description or a configuration at an instant t of a basic transition P system Π = (Γ, µ, M 1,..., M q, R, P R, i out ) is described by the membrane structure at instant t and all multisets of objects over Γ associated with all the membranes present in the system. The initial configuration of the system is (µ, M 1,, M q ). A rule r [ u ] i v 1 [ v 2 [ v 3 ] j ] i is applicable to a configuration C t at an instant t if the following holds: (a) membrane i is in C t ; (b) multiset u is contained in such membrane; (c) j is the label of a membrane immediately inside membrane i; and (d) there is no a rule r associated with membrane i applicable to C t such that r has a higher priority than r, that is, (r, r) P R. When applying such a rule, the objects specified by multiset u are consumed (multiset u is substracted from the multiset of membrane i), the objects specified by multiset v 1 will be moved to the zone immediately outside membrane i, the parent p(i) of that membrane (this zone is the environment in the case when i is the skin membrane: in this case, the objects leave the system and they never come back), the objects specified by multiset v 2 will be placed in the same membrane i, and the objects specified by multiset v 3 in the membrane with label j which must be a membrane immediately inside membrane i. A rule [ u ] i v 1 [ v 2 [ v 3 ] j ] i δ is applicable to a configuration C t at an instant t if the following holds: (a) the rule [ u ] i v 1 [ v 2 [ v 3 ] j ] i is applicable to C t ; (b) i i out ; and (c) i is not the label of the skin membrane. When applying the rule [ u ] i v 1 [ v 2 [ v 3 ] j ] i δ to a configuration C t, first the rule [ u ] i v 1 [ v 2 [ v 3 ] j ] i is

44 44 L. Valencia-Cabrera et al. applied to C t and then membrane i is dissolved. After dissolving a membrane, all objects and membranes previously present in it become elements of the contents of the immediately upper membrane which has not been dissolved. Given a basic transition P system Π, we say that configuration C t yields configuration C t+1 in one transition step, denoted by C t Π C t+1, if we can pass from C t to C t+1 by applying the rules from R synchronously, in a non-deterministic maximally parallel manner. This means the following: the objects to evolve in a transition step and the rules by which they evolve are chosen in a non-deterministic manner, but in such a way that in each membrane we have a maximally parallel application of rule (at each transition step a multiset of rules which is maximal is applied, no further applicable rule can be added). A computation of Π is a (finite or infinite) sequence of configurations such that: (a) the first term of the sequence is the initial configuration of the system; (b) each non-first term of the sequence is obtained from the previous configuration by applying rules of the system in a nondeterministic maximally parallel manner; and (c) if the sequence is finite (called halting computation) then the last term of the sequence is a halting configuration, that is, a configuration where no rule of the system is applicable to it. All computations start from an initial configuration and proceed as stated above; only halting computations give a result, which is encoded by the objects present in the output region i out in the halting configuration. Basic transition P systems have the ability to create an exponential workspace, expressed in terms of number of objects, in linear time (e.g. via evolution rules of the type [ a a 2 ] h ) 3.2 P systems with active membranes and electrical charges Cell-like P systems with active membranes having associated electrical charges with membranes were first introduced by Gh. Păun [22]. One of the main attractive of these membrane systems is the ability to create an exponential workspace, expressed in terms of number of objects and number of membranes, in linear time. Definition 2. A P system with active membranes of degree q 1 is a tuple Π = (Γ, µ, M 1,..., M q, R, i out ), where: 1. Γ is a finite alphabet. 2. µ is a rooted tree. 3. M 1,..., M q are multisets over Γ. 4. R is a finite set of rules (H denotes the set {1,..., q}) of the following forms: a) [ a u ] α h, for h H,α {+,, 0}, a Γ, u Γ (object-evolution rules). b) a [ ] α1 h [ b ]α2 h, for h H, α 1, α 2 {+,, 0}, a, b Γ (send in communication rules). c) [ a ] α1 h [ ] α2 h b, for h H, α 1, α 2 {+,, 0}, a, b Γ (send out communication rules).

45 Cooperative P Systems and the P Versus NP Problem 45 d) [ a ] α h b, for h H, α {+,, 0}, a, b Γ (dissolution rules). e) [ a ] α1 h [ b ]α2 h [ c ]α3 h, for h H, α 1, α 2, α 3 {+,, 0}, a, b, c Γ (division rules for elementary membranes). f) [ [ ] α1 h 1... [ ] α1 h k [ ] α2 h k+1... [ ] α2 h n ] α h [ [ ]α3 h 1... [ ] α3 h k ] β h [ [ ]α4 h k+1... [ ] α4 h n ] γ h, for k 1, n > k, h, h 1,..., h n H, α, β, γ, α 1,..., α 4 {+,, 0} and {α 1, α 2 } = {+, } (division rules for non elementary membranes). 5. i out H {0}. A P system with active membrane Π = (Γ, µ,, M 1,..., M q, R, i out ), of degree q 1 can be viewed as a set of q membranes injectively labelled by 1,..., q with electrical charges (positive +, negative or neutral 0) associated to them, arranged in a hierarchical structure µ given by a rooted tree whose root is called the skin membrane of the system and with an environment labelled by 0 such that: (a) M 1,..., M q are multisets over the working alphabet Γ representing the objects initially placed in the q membranes of the system; (b) R is the set of rules that allows to evolve the system; and (c) i out {0, 1, 2,..., q} represents a distinguished zone which will encode the output of the system (the term zone is used as above). Notice that P systems with active membranes have some important features: (a) they use three electrical charges; (b) the polarization of a membrane, but not the label, can be modified by the application of a rule; (c) the rules are noncooperative (the corresponding left-hand side consist of only one symbol), and (d) there are no priorities among rules. The concept of configuration at an instant t of a P system with active membranes Π = (Γ, µ, M 1,..., M q, R, i out ) is defined in a similar way to the one used in basic transition P systems. An object-evolution rule [ a u ] α h is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by h is in C t and its electrical charge is α; and (b) object a is contained in such membrane. When applying such a rule, object a is consumed and the objects specified by multiset u will be placed in that membrane h. A send-in rule a [ ] α1 h [ b ]α2 h is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by h, different from the skin membrane, is in C t and its electrical charge is α 1 ; and (b) object a is contained in the zone immediately outside the membrane h, the parent p(h) of that membrane. When applying such a rule, object a is consumed, object b will be placed in the same membrane h and the polarization of such membrane h will change to α 2. A send-out rule [ a ] α1 h b [ ]α2 h is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by h, different from the skin membrane, is in C t and its electrical charge is α 1 ; and (b) object a is contained in such membrane h. When applying such a rule, object a is consumed, object b will be placed in the zone immediately outside the membrane h, the parent p(h) of that membrane, and the polarization of such membrane h will change to α 2.

46 46 L. Valencia-Cabrera et al. A dissolution rule [ a ] α h b is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by h is in C t and its electrical charge is α; and (b) object a is contained in such membrane h. When applying such a rule, object a is consumed and the membrane is dissolved. After dissolving such membrane h, all objects and membranes previously present in it become elements of the contents of the immediately upper membrane which has not been dissolved, except object a triggering the rule that evolves to b. A division rule for elementary membranes [ a ] α1 h [ b ]α2 h [ c ]α3 h is applicable to a configuration C t at an instant t if the following holds: (a) an elementary membrane labelled by h, different from the skin membrane, is in C t and its electrical charge is α 1 ; (b) h i out ; and (c) object a is contained in such membrane h. When applying such a rule, object a is consumed and the membrane is divided into two membranes with the same label h, maybe of different polarizations α 2 and α 3 ; the object a specified in the rule is replaced in the two new membranes by possibly new objects b and c, respectively. A division rule for non-elementary membranes [ [ ] α1 h 1... [ ] α1 h k [ ] α2 h k+1... [ ] α2 h n ] α h [ [ ] α3 h 1... [ ] α3 h k ] β h [ [ ]α4 h k+1... [ ] α4 h n ] γ h, is applicable to a configuration C t at an instant t if the following holds: (a) a non-elementary membrane labelled by h, different from the skin membrane, is in C t and its electrical charge is α 1 ; (b) such membrane contains membranes with labels h 1,..., h n some of them (h 1,..., h k ) with electrical charge α 1 and the remaining (h k+1,..., h n ) with electrical charges α 2, being {α 1, α 2 } = {+, }; (c) if such membrane h 0 contains other membranes than those with labels h 1,..., h n then they must have neutral charges. When applying such a rule: (1) membrane h 0 is divided into two membranes with the same label h, maybe of different polarizations β and γ; (2) membranes with label h 1,..., h k contained in membrane h 0 are placed (with polarization α 3 ) inside of one of the new created membranes; (3) membranes with label h k+1,..., h n contained in membrane h 0 are placed (with polarization α 4 ) inside of the another new created membrane; and (4) if membrane h 0 contains other membranes than those with labels h 1,..., h n then they have neutral charges being duplicated in the new created membranes. P systems with active membranes differ from the basic transition P systems on the type of rules which are applied according to the following principles ([22]): All the rules are applied in parallel. In each transition step, one object of a membrane can be used by only one rule (chosen in a non deterministic way). If a membrane is dissolved, its content (multiset and internal membranes) is left free in the surrounding region. If at the same time a membrane labelled by h is divided by a rule of type (e)-(f) and there are objects in this membrane which evolve by means of rules of type (a), then we suppose that first the evolution rules of type (a) are used, and then the division is produced. Of course, this process takes only one step. The rules associated with membranes labelled by h are used for all copies of this membrane. At one step, a membrane can be the subject of only one rule

47 Cooperative P Systems and the P Versus NP Problem 47 of types (b)-(f), that is, these rules are applied in a sequential manner to each membrane. Given a P system with active membranes Π, we say that configuration C t yields configuration C t+1 in one transition step, denoted by C t Π C t+1, if we can pass from C t to C t+1 by applying the rules from R synchronously, in a non-deterministic maximally parallel manner according with the previous remarks. The concept of computation of Π is defined in a similar way that in the previous section. 3.3 Polarizationless P systems with active membranes P systems with active membranes and without electrical charges were initially studied in [2, 3]. In these systems, polarizations were replaced by the possibility to change the label of the membranes by means of some rules. However, in order to obtain polynomial-time solutions to computationally hard problems, two polarizations suffice (see [4] for details). In [28] bi-stable catalysts are used to compensate the loss of computational efficiency represented by avoiding polarizations. Definition 3. A polarizationless P system with active membranes of degree q 1 is a tuple Π = (Γ, µ, M 1,..., M q, R, i out ), where: 1. Γ is a finite alphabet. 2. µ is a rooted tree. 3. M 1,..., M q are multisets over Γ. 4. R is a finite set of rules (H denotes the set {1,..., q}), of the following forms: (a) [ a u ] h, for h H, a Γ, u Γ (object-evolution rules). (b) a [ ] h [ b ] h, for h H, a, b Γ (send in communication rules). (c) [ a ] h [ ] h b, for h H, a, b Γ (send out communication rules). (d) [ a ] h b, for h H, a, b Γ (dissolution rules). (e) [ a ] h [ b ] h [ c ] h, for h H, a, b, c Γ (division rules for elementary or weak division rules for non-elementary membranes). (f) [ [ ] h1... [ ] hk [ ] hk+1... [ ] hn ] h [ [ ] h1... [ ] hk ] h [ [ ] hk+1... [ ] hn ] h, where k 1, n > k, h, h 1,..., h n H (division rules for non-elementary membranes). 5. i out {0, 1,..., q}. The semantics of these rules are similar to the ones of P systems with active membranes and they are applied according to usual principles of these systems, described in the previous section. 3.4 P systems with symport/antiport rules In this section we introduce a kind of cell-like P systems that use communication rules capturing the biological phenomenon of trans-membrane transports of several chemical substances, in the same or in opposite directions. Specifically, two processes have been considered. The first one allows a multiset of chemical substances to pass through a membrane in the same direction. In the second one, two

48 48 L. Valencia-Cabrera et al. multisets of chemical substances (located in different biological membranes) only pass with the help of each other (an exchange of objects between both membranes). Next, we introduce an abstraction of these operation in the framework of P systems with symport/antiport rules following [23]. In these models, the membranes are not polarized. Definition 4. A P system with symport/antiport rules of degree q 1 is a tuple Π = (Γ, E, µ, M 1,..., M q, R, i out ), where: 1. Γ is a finite alphabet. 2. E Γ. 3. µ is a rooted tree. 4. M 1,..., M q are multisets over Γ. 5. R = R 1 R q, where R i is a finite set of rules associated with membrane i, of the following forms: Symport rules: (u, out) or (u, in), where u M(Γ ) such that u > 0; Antiport rules: (u, out; v, in), where u, v M(Γ ) such that u > 0 and v > 0; 6. i out {0, 1,..., q}. A P system with symport/antiport rules of degree q Π = (Γ, E, µ, M 1,..., M q, R, i out ) can be viewed as a set of q membranes, labelled by 1,..., q, arranged in a hierarchical structure µ given by a rooted tree whose root is called the skin membrane of the system, labelled by 1, and with an environment labelled by 0 such that: (a) M 1,..., M q are multisets over the working alphabet Γ representing the objects initially placed in the q membranes of the system; (b) E is the set of objects initially located in the environment of the system, all of them available in an arbitrary number of copies; (c) R = R 1 R q where R i is a finite set of communication rules over Γ associated with membrane i of µ; and (d) i out represents a distinguished zone which will encode the output of the system (the term zone is used as before). The length of rule (u, out) or (u, in) (resp. (u, out; v, in)) is defined as u (resp. u + v ). A P system with symport/antiport rules of degree q 1 Π = (Γ, E, µ, M 1,..., M q, R, i out ) where E =, is called a P system with symport/antiport rules and without environment. For each membrane i {2,..., q}, different from the skin membrane, we denote by p(i) the parent of membrane i in the rooted tree µ. We define p(1) = 0, that is, by convention the parent of the skin membrane is the environment. An instantaneous description or a configuration at an instant t of a P system with symport/antiport rules is described by the membrane structure at instant t,

49 Cooperative P Systems and the P Versus NP Problem 49 all multisets of objects over Γ associated with all the membranes present in the system, and the multiset of objects over Γ E associated with the environment at that moment. Recall that initially there are infinite copies of objects from E in the environment, and hence this set is not properly changed along the computation. The initial configuration of the system is (µ, M 1,, M q ; ). A symport rule (u, out) R i is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by i is in C t ; and (b) multiset u is contained in such membrane. When applying a rule (u, out) R i, the objects specified by multiset u are sent out of such membrane i into the region immediately outside, the parent p(i) of such membrane. This can be the environment in the case of the skin membrane. A symport rule (u, in) R i is applicable to a configuration C t at an instant t if the following holds:: (a) a membrane labelled by i is in C t ; and (b) multiset u is contained in the parent p(i) of such membrane i. When applying a rule (u, in) R i, the objects specified by multiset u goes out from the parent p(i) membrane of i and enters into the region defined by the membrane i. An antiport rule (u, out; v, in) R i is applicable to a configuration C t at an instant t if if the following holds: (a) a membrane labelled by i is in C t ; (b) multiset u is contained in such membrane; and (c) multiset v is contained in the parent p(i) of membrane i. When applying a rule (u, out; v, in) R i, the objects specified by multiset u are sent out of membrane i into the parent p(i) of i and, at the same time, bringing the objects specified by multiset v into such membrane i. The rules of a P system with symport/antiport rules are applied in a nondeterministic maximally parallel manner: at each step we apply a multiset of rules which is maximal, so no further applicable rule can be added. Given a P system with symport/antiport rules Π, we say that configuration C t yields configuration C t+1 in one transition step, denoted by C t Π C t+1, if we can pass from C t to C t+1 by applying the rules from R 1 R q following the previous remarks. The concept of computation of Π is defined in a similar way that in the previous section. P systems with symport/antiport rules and membrane division or membrane separation In this section, we introduce new types of rules (membrane division and membrane separation) inspired by the mitosis and the membrane fission processes, in the framework of P systems with symport/antiport rules. These rules provide a mechanism to construct an exponential workspace (expressed in terms of number of objects and number of membranes) in linear time. Definition 5. A P system with symport/antiport rules and membrane division of degree q 1 is a tuple Π = (Γ, E, µ, M 1,..., M q, R, i out ), where: 1. Π = (Γ, E, µ, M 1,..., M q, R, i out ) is a P system with symport/antiport rules.

50 50 L. Valencia-Cabrera et al. 2. R = R 1 R q, where R i is a finite set of symport/antiport rules, associated with membrane i, which can also contain rules of the following form: [a] i [b] i [c] i, where i / {1, i out } and a, b, c Γ (division rules for elementary membranes). A division rule [a] i [b] i [c] i R i is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by i, different from the skin membrane, is in C t ; (b) i i out ; and (c) object a is contained in such membrane. When applying a division rule [a] i [b] i [c] i, under the influence of object a, the membrane with label i is divided into two membranes with the same label; in the first copy, object a is replaced by object b, in the second one, object a is replaced by object c; all the other objects residing in such membrane i are replicated and copies of them are placed in the two new membranes. Definition 6. A P system with symport/antiport rules and membrane separation of degree q 1 is a tuple where: Π = (Γ, Γ 0, Γ 1, E, µ, M 1,..., M q, R, i out ) 1. Π = (Γ, E, µ, M 1,..., M q, R, i out ) is a P system with symport/antiport rules. 2. {Γ 0, Γ 1 } is a partition of Γ, that is, Γ = Γ 0 Γ 1, Γ 0, Γ 1, Γ 0 Γ 1 = ; 3. R = R 1 R q where R i is a finite set, associated with membrane i, of rules symport/antiport rules which can also contain rules of the following form: [a] i [Γ 0 ] i [Γ 1 ] i, where i / {1, i out } and a Γ (separation rules). A separation rule [a] i [Γ 0 ] i [Γ 1 ] i R i is applicable to a configuration C t at an instant t if the following holds: (a) a membrane labelled by i, different from the skin membrane, is in C t ; (b) i i out ; and (c) object a is contained in such membrane. When applying a separation rule [a] i [Γ 0 ] i [Γ 1 ] i R i, in reaction with an object a, the membrane i is separated into two membranes with the same label; at the same time, object a is consumed; the objects from Γ 0 are placed in the first membrane, those from Γ 1 are placed in the second membrane. With respect to the semantics of these variants, the rules of such P systems are applied in a non-deterministic maximally parallel manner, with the following important remark: when a membrane i is divided (resp. separated), the division rule (resp. separation rule) is the only one from R i which is applied for that membrane at that step (however, some rules can be applied in a daughter membrane). The new membranes resulting from division (resp. separation) could participate in the interaction with other membranes or the environment by means of communication rules at the next step providing that they are not divided (resp. separated) once again. The label of a membrane precisely identify the rules which can be applied to it.

51 3.5 Recognizer membrane systems Cooperative P Systems and the P Versus NP Problem 51 Let us recall that solving a decision problem can be expressed in terms of recognizing the language associated with it. Recognizer P systems were introduced in [32] and they provide a natural framework to solve decision problems. In this section, the term membrane system is used to refer any cell-like P systems introduced at the previous sections. An arbitrary membrane system of the order q, q 1, will be described by a tuple (Γ, Γ 0, Γ 1, E, µ, M 1,..., M q, R, P R, i out ) where we can think that Γ 0 = Γ 1 = for membrane systems without separation rules, E = for basic transition P systems or P systems with active membranes, and P R only for basic transition P systems. Next, we introduce the concept of recognizer associated with the membrane systems defined in the previous section. Definition 7. A recognizer membrane system Π = (Γ, Γ 0, Γ 1, E, µ, M 1,..., M q, R, i out ) is a membrane system verifying the following: The working alphabet Γ has two distinguished objects yes and no, with at least one copy of them present in some initial multisets, but none of them initially present in E; there exists an additional alphabet Σ (the input alphabet) strictly contained in Γ such that E Γ \ Σ; M 1,..., M q are multisets over Γ \ Σ; i in {1,..., q} is the label of the input membrane; the output zone i out is the environment; all computations halt; if C is a computation of Π, then either object yes or object no (but not both) must have been released into the environment, and only at the last step of the computation. For each multiset M over the input alphabet Σ, a computation of Π with input multiset M starts from the configuration of the form (µ, M 1,..., M iin + M,..., M q, ), where the input multiset M has been added to the content of the input membrane i in. That is, we have an initial configuration associated with each input multiset M over Σ in recognizer membrane systems. We denote by Π + M the P system Π with input multiset M. 3.6 Polynomial complexity classes of recognizer membrane systems Next, according to [27], we define what solving a decision problem by a family of recognizer P systems with symport/antiport rules, in a uniform way, means.

52 52 L. Valencia-Cabrera et al. Definition 8. A decision problem X = (I X, θ X ) is solvable in polynomial time by a family Π = {Π(n) n N} of recognizer membrane systems (in a uniform way) if the following conditions hold: the family Π is polynomially uniform by Turing machines, that is, there exists a deterministic Turing machine working in polynomial time which constructs the system Π(n) from n N (n expressed in unary); there exists a pair (cod, s) of polynomial-time computable functions over I X such that: for each instance u I X, s(u) is a natural number and cod(u) is an input multiset of the system Π(s(u)); for each n N, s 1 (n) is a finite set; the family Π is polynomially bounded with regard to (X, cod, s), that is, there exists a polynomial function p, such that for each u I X every computation of Π(s(u)) + cod(u) is halting and it performs at most p( u ) steps; the family Π is sound with regard to (X, cod, s), that is, for each u I X, if there exists an accepting computation of Π(s(u)) + cod(u), then θ X (u) = 1; the family Π is complete with regard to (X, cod, s), that is, for each u I X, if θ X (u) = 1, then every computation of Π(s(u)) + cod(u) is an accepting one. According to Definition 8, we say that for each u I X, the recognizer membrane system Π(s(u)) + cod(u) is confluent, in the sense that all possible computations of the system must give the same answer. If R is a class of recognizer membrane systems, then we denote by PMC R the set of all decision problems which can be solved in polynomial time (and in a uniform way) by means of systems from R. The class PMC R is closed under complement and polynomial-time reductions (see [27] for details). 4 Frontiers of tractability in membrane systems We say that a class of recognizer membrane systems F is presumably efficient if there exists an NP-complete problem that can be solved in polynomial time by a family of systems from F. From the properties of the NP-completeness, we deduce that any NP-complete problem can be solved in polynomial time by families of a presumably efficient class of recognizer membrane systems. Because class PMC F is closed under complement and polynomial-time reductions (see [27] for details), if the class F is presumably efficient then NP co-np PMC F. We say that a class of recognizer membrane systems F is feasible if only tractable problems can be solved in polynomial time by a family of systems from F, that is, if PMC F = P. According to these definitions, if P = NP then a class F is feasible if and only if it is presumably efficient. Besides, if P NP then each feasible class is not presumably efficient. Nevertheless, under that hypothesis a non-feasible class could be non-presumably efficient (as a consequence of the

53 Cooperative P Systems and the P Versus NP Problem 53 Ladner theorem by which if P NP then there exist NP-intermediate problems, that is, problems which are neither in the class P nor in the class of NP-complete problems, see [11] for details). Let F 1 and F 2 be two models from a computing paradigm such that F 1 is an extension of F 2, in other words, F 1 is obtained from F 2 by adding some syntactic or semantic ingredients (called additional ingredients). In this case, each solution of a decision problem in model F 2 is also a solution in model F 1. In this context, if F 2 is a feasible model and F 1 is a presumably efficient model, then we say that the additional ingredients provide a frontier of tractability between tractability and NP-hardness. Feasible Presumably Efficient F F 2 1 Let us consider two models F 1 and F 2 of recognizer membrane systems such that F 2 is feasible, F 1 is presumably efficient and model F 1 is an extension of model F 2. On the one hand, translating an efficient solution of an NP-complete problem by a family of systems in F 1, into an efficient solution by a family of systems in F 2 amounts to proving P=NP. On the other hand, proving that without the additional ingredients in F 1 it is not possible to solve an NP-complete problem in polynomial-time, then the result P NP follows. Hence, each frontier of tractability provide a tool to tackle the P versus NP problem. 4.1 Basic transition P systems Let us recall that the decision problem associated with a Turing machine M with input alphabet Σ M is the problem X M = (I M, θ M ), where I M = ΣM, and for every w ΣM, θ M (w) = 1 if and only if M accepts w. Then we say that a Turing machine M is simulated in polynomial time by a family of recognizer membrane systems from R if X M PMC R. In [9] an efficient simulation of deterministic Turing machines by recognizer basic transition P systems was given. Proposition 1. (Sevilla theorem) Every deterministic Turing machine working in polynomial time can be simulated in polynomial time by a family of recognizer basic transition P systems.

54 54 L. Valencia-Cabrera et al. Also, in [9] was shown that each confluent basic transition P system can be (efficiently) simulated by a deterministic Turing machine. As a consequence, if a decision problem is solvable in polynomial time by a family of recognizer basic transition P systems, then there exists a deterministic Turing machine solving it in polynomial time. Then, we have the following result: P = PMC T, being T the class of all recognizer basic transition P systems. Thus, the use of cooperative rules in basic transition P systems and their ability to create exponential workspace (in terms of number of objects) in linear time is not enough to efficiently solve NPcomplete problems (assuming that P NP). 4.2 P systems with active membranes Let us denote by DAM the class of all recognizer P systems with active membranes and let N AM be the class of recognizer P systems with active membranes which do not make use of division rules. In the framework of cell-like membrane systems, confluent systems from N AM can be efficiently simulated by a deterministic Turing machine [43]. Proposition 2. (Milano theorem) A deterministic P system with active membranes but without membrane division can be simulated by a deterministic Turing machine with a polynomial slowdown. As a consequence of the Milano theorem, we have PMC N AM P. Bearing in mind that the reverse implication is easily deduced from Definition 8, we have PMC N AM = P. By using membrane systems from DAM which do not make use of dissolution rules, different efficient solutions to strongly NP-complete problems (SAT [27], Clique [3], Bin Packing [29], Common Algorithmic Problem [30]) have been given. Since the class PMC DAM is closed under complement and polynomial-time reductions, we deduce that NP co-np PMC DAM. Remark 1: In the framework of P systems with active membranes, the use or not of the division rules provides a borderline for the tractability of decision problems, assuming that P NP. Thus, by using division rules we can solve NP-complete problems in polynomial time, but without division rules only problems in P can be solved in an efficient way. Then, in this framework, cooperative rules are not necessary to obtain frontiers of tractability. 4.3 Polarizationless P systems with active membranes Let us denote by DAM 0 (α, β, γ, η) the class of all recognizer polarizationless P systems with active membranes which make use of division rules such that: If α = +e (α = e, resp.) then object-evolution rules are permitted (forbidden, resp.);

55 Cooperative P Systems and the P Versus NP Problem 55 if β = +c (β = c, resp.) then communication rules are permitted (forbidden, resp.); if γ = +d (α = d, resp.) then dissolution rules are permitted (forbidden, resp.); and if η = +n (η = n, resp.) then division rules for elementary and nonelementary membranes are permitted (only division rules for non-elementary membranes are permitted). In the same way, we denote by SAM 0 the corresponding class when separation rules are considered instead of division rules. The so-called Păun s conjecture can be formally formulated in terms of membrane computing complexity classes as follows: P = PMC DAM 0 (+e,+c,+d, n). Currently, this is a relevant open problem. However, several partial solution have been given. Let Π be a recognizer polarizationless P system with active membranes which does not make use of dissolution rules. A directed graph (called dependency graph) can be associated with Π verifying the following property: every accepting computation of Π is characterized by the existence of a path in the graph between two specific nodes. Based in this concept and by using the fact that reachability problem is in class P, the following result has been provided [8]: P = PMC DAM 0 (+e,+c, d, n) = PMC DAM 0 (+e,+c, d,+n) Thus, polarizationless P systems with active membranes which do not make use of dissolution rules cannot solve NP-complete problems in polynomial time (unless P=NP). This result can be considered as a partial affirmative answer to the Păun s conjecture. Let us now consider polarizationless P systems with active membranes making use of dissolution rules. Will it be possible to solve NP-complete problems in that framework? Several authors [1, 8] gave a positive answer when division for nonelementary membranes are allowed. The mentioned papers provide solutions in linear time to SAT problem and Subset Sum problem, respectively. Hence, we have NP co-np PMC DAM 0 (+e,+c,+d,+n). Therefore, a partial negative answer to Păun s conjecture is given: assuming that P NP and making use of dissolution rules and division rules for elementary and non-elementary membranes, computationally hard problems can be efficiently solved avoiding polarizations. The answer is partial because efficient solvability of NP-complete problems by means of families from DAM 0 (+e, +c, +d, n) is unknown. Remark 2: In the framework of polarizationless P systems with active membranes, the use or not of dissolution rules provides a borderline for the tractability of decision problems, assuming that P NP, that is, by using dissolution rules we can solve NP-complete problems in polynomial time, but without dissolution rules only problems in P can be solved in an efficient way.

56 56 L. Valencia-Cabrera et al. 4.4 Cooperation in polarizationless P systems with active membranes The role of dissolution rules in the framework of DAM 0 is crucial in order to provide polynomial-time solutions to computationally hard problems (assuming that P NP). In this section we prove that by using (very restrictive) cooperative rules in polarizationless P systems with active membranes, it is possible to solve NP-complete problems in an efficient way. Next, several types of minimal cooperation in object-evolution rules are considered in the framework of polarizationless P systems with active membranes. The term minimal cooperation is used in the following sense: the left-hand side of such rules consists of two symbols. Minimal cooperation (mc): object-evolution rules are of the form [ u v ] h, where u, v M(Γ ) such that 1 u 2. Primary minimal cooperation (pmc): object- evolution rules are of the form [ u v ] h, where u, v M(Γ ) and 1 u, v 2. Bounded minimal cooperation (bmc): object- evolution rules are of the form [ u v ] h, where u, v M(Γ ) and 1 v u 2. Minimal cooperation and minimal production (mcmp): object- evolution rules are of the forms [ a b ] h or [ a b c ] h, where a, b, c Γ. In polarizationless P systems with active membranes and minimal cooperation in object-evolution rules, the remaining rules (send-in communication rules, send-out communication, dissolution and division) are non-cooperative rules. Besides, the rules are applied according to the same principles than in classical P systems with active membranes. In the expression DAM 0 (α, β, γ, δ), parameter α associated with objectevolution rules is extended as follows: if α = mc then minimal cooperation in object- evolution rules are permitted. if α = pmc then primary minimal cooperation in object- evolution rules are permitted. if α = bmc then bounded minimal cooperation in object- evolution rules are permitted. if α = mcmp then minimal cooperation and minimal production in objectevolution rules are permitted. Next, we summarize some interesting results. 1. Families of systems from DAM 0 (+e, +c, +d, +n) can solve PSPACE-complete problems in polynomial time, that is, PSPACE PMC DAM 0 (+e,+c,+d,+n) [5]. In fact, PSPACE = PMC DAM 0 (+e,+c,+d,+n) (see [36] and [37] for details). 2. Families of systems from DAM 0 (+e, +c, d, +n) can efficiently solve only problems in class P, that is, PMC DAM 0 (+e,+c, d,+n) = P (see [8] for details).

57 Cooperative P Systems and the P Versus NP Problem Families of systems from SAM 0 (+e, +c, d, +n) can efficiently solve only problems in class P, that is, PMC SAM 0 (+e,+c, d,+n) = P (see [42] for details). 4. Families of systems from DAM 0 (bmc, +c, d, n) can solve NP-complete problems in polynomial time, i.e., NP co-np PMC DAM 0 (bmc,+c, d, n) (see [38] for details). 5. Families of systems from SAM 0 (bmc, +c, +d, +n) can efficiently solve only problems in class P, that is, PMC SAM 0 (bmc,+c,+d,+n) = P (see [40] for details). 6. Families of systems from SAM 0 (pmc, +c, d, n) can solve NP-complete problems in polynomial time, i.e., NP co-np PMC SAM 0 (pmc,+c, d, n) (see [38] for details). 7. Families of systems from DAM 0 (mcmp, +c, d, n) can solve NP-complete problems in polynomial time, i.e., NP co-np PMC DAM 0 (mcmp,+c, d, n) (see [41] for details). 8. Families of systems from SAM 0 (mcmp, +c, +d, +n) can efficiently solve only problems in class P, that is, P = PMC SAM 0 (mcmp+c,+d,+n) (see [41] for details). From these results, the following frontiers of tractability are obtained. In the framework DAM 0 (, +c, d, n): passing from non-cooperative objectevolution rules to bounded minimal cooperation in object-evolution rules. In the framework SAM 0 (, +c, d, n): passing from non-cooperative objectevolution rules to primary minimal cooperation in object-evolution rules. In the framework AM 0 (bmc, +c, d, n): passing from separation rules to division rules. In the framework AM 0 (mcmp, +c, d, n): passing from separation rules to division rules. Non Efficiency Efficiency Frontiers (Feasible) (Presumably Efficient) DAM 0 (+e, +c, d, n) DAM 0 (mcmp, +c, d, n) minimal cooperation and minimal production SAM 0 (+e, +c, d, n) SAM 0 (pmc, +c, d, n) primary minimal cooperation SAM 0 (bmc, +c, d, n) DAM 0 (bmc, +c, d, n) separation vs division SAM 0 (mcmp, +c, d, n) DAM 0 (mcmp, +c, d, n) separation vs division

58 58 L. Valencia-Cabrera et al. 4.5 P systems with symport/antiport rules The class of all recognizer P systems with symport/antiport rules and with membrane division (resp. membrane separation) will be denoted by CDC (resp. CSC). For each natural number k 1, we denote by CDC(k) (resp. CSC(k)) the class of all recognizer P systems with membrane division (resp. membrane separation) and with symport/antiport rules of length at most k. In the case of P systems without environment, we denote by ĈDC, ĈSC, ĈDC(k) and ĈSC(k), respectively, the corresponding class. Obviously, recognizer P systems from CDC(1), CSC(1), ĈDC(1) and ĈSC(1), are non-cooperative systems, and the remaining recognizer P systems are cooperative systems. Next, we summarize some interesting results. Families of non-cooperative P systems with symport/antiport rules can only efficiently solve problems in class P, that is, P = PMC CDC(1) = PMC CSC(1) (see [14] for details). In [39] a family of P systems with division rules and symport/antiport rules using minimal cooperation solving in polynomial-time the HAM-CYCLE problem, a well known NP-complete problem [7], has been given. Thus, NP co-np PMC CDC(2). Families of P systems with separation rules and symport/antiport rules using minimal cooperation can only efficiently solve problems in class P, that is, P = PMC CSC(2) (see [13] for details). However, in the cited paper, a family of P systems with separation rules and using symport/antiport rules with length at most three solving in polynomial-time the SAT problem, has been given, that is, NP co-np PMC CSC(3). The role of the environment is irrelevant when we try to provide polynomialtime solutions to NP-complete problems by means of families of P systems with symport/antiport rules and membrane division. Specifically, for each k N we have PMC CDC(k+1) = PMCĈDC(k+1) (see [15] for details). Families of P systems with symport/antiport rules and separation rules but without environment, can only efficiently solve problems in class P, that is, P = PMCĈSC (see [12] for details). Hence, the role of the environment is relevant when we try to provide polynomial-time solutions to NP-complete problems by means of families of P systems with symport/antiport rules and membrane separation From these results, the following frontiers of tractability are obtained.

59 Cooperative P Systems and the P Versus NP Problem 59 Non Efficiency Efficiency Frontiers (Feasible) (Presumably Efficient) CDC(1) CDC(2) (length) CSC(2) CSC(3) (length) CSC(2) CDC(2) (separation vs division) CSC(2) CDC(2) (separation vs division) ĈSC CSC (environment) 5 Satisfiability problems and cooperation in rewriting rules In this section, a relationship among cooperative rewriting rules and instances of 2 SAT problem and 3 SAT problem is highlighted. 5.1 Implication graph associated with instances of 2 SAT Let ϕ = C 1 C p be a boolean formula consisting of p (p 2) clauses C j = lj 1 l2 j, 1 j p, that is, ϕ is an instance of 2 SAT. The implication graph G ϕ = (V ϕ, E ϕ ) is the directed graph defined as follows: V ϕ is the set of all literals associated with the set V ar(ϕ) of variables of ϕ. E ϕ V ϕ V ϕ is the following set of arcs: (x, y) V ϕ V ϕ if and only if there exists a clause C = l 1 l 2 of ϕ such that x = l 1 y = l 2 or x = l 2 y = l 1. According with the previous definition, each clause C j = lj 1 l2 j of ϕ has associated two arcs (lj 1, l2 j ) and (l2 j, l1 j ). In some sense, these arcs capture the boolean values of logical implications associated with each clause of ϕ in a natural way. It is worth pointing out that the implication graphs associated with instances of 2 SAT verify the following result (see [20] for details): Theorem 1. Let ϕ = C 1 C p be an instance of 2 SAT with p (p 2) clauses. Then the following assertions are equivalent: ϕ is unsatisfiable. There exists a variable x V ar ϕ such that there are paths in G ϕ from x to x and from x to x. From this theorem, by using the tractability of the reachability problem, it is easy to follow that 2 SAT is a problem in class P.

60 60 L. Valencia-Cabrera et al. 5.2 Rewriting rules associated with instances of k SAT Let k 2 and ϕ = C 1 C p be a boolean formula consisting of p (p 2) clauses C j = lj 1... lk j, 1 j p, that is, ϕ is an instance of k SAT. Let us recall that each clause C j is logically equivalent to the boolean formula lj 1 lk 1 j lj k. If σ is a truth assignment which makes true the formula ϕ and σ(lj 1 lk 1 j ) = 1 then we deduce that σ(lj k) = 1. In this context, the expression lj 1 lk 1 j lj k is associated with clause C j and it can be viewed as a rewriting rule in the following sense: the rule is applicable for a truth assignment σ which makes true the formula ϕ if and only if σ(lj 1 lk 1 j ) = 1. The application of that rule produces σ(lj k ) = 1 as some kind of information. Following the same idea and bearing in mind the properties of logical equivalence of boolean formulas, different rewriting rules could be associated with formula ϕ. Specifically, for each t, 1 t k, rewriting rules of the type l 1 j l t j lk j lt j are associated to clause C j, where the expression lj 1 l j t lk j means that literal lj t does not appear in the left-hand side. In this paper, the particular case t = k has been considered. Next, we show that any instance of 2 SAT problem has associated noncooperative rewriting rules (their left-hand side contain only one object) and any instance of 3 SAT problem has associated cooperative rewriting rules (their lefthand side contain at least two objects). Instances of 2 SAT Let ϕ = C 1 C p be a boolean formula consisting of p (p 2) clauses C j = lj 1 l2 j, 1 j p, that is, ϕ is an instance of 2 SAT. Then, clause l1 j l2 j has associated the non-cooperative rewriting rule lj 1 l2 j. As usual, the term non-cooperative refers to the propery that its left-hand side contains only one object (in this case, one literal). We say that the non-cooperative rewriting rule lj 1 l2 j is applicable for a truth assignment σ associated with the set of variables V ar ϕ of ϕ if σ(lj 1 ) = 1; that is, in order to determine if such a rule is applicable for σ only is necessary to know the truth value of one literal. When applying a rule lj 1 l2 j for a truth assignment σ, we deduce the following information: σ(lj 2) = 1. Instances of 3 SAT Let ϕ = C 1 C p be a boolean formula consisting of p (p 2) clauses C j = lj 1 l2 j l3 j, 1 j p, that is, ϕ is an instance of 3 SAT. We associate with clause lj 1 l2 j l3 j the cooperative rewriting rule l1 j l2 j l3 j. As usual, the term

61 Cooperative P Systems and the P Versus NP Problem 61 cooperative refers to the propery that its left-hand side contains more than one object (more than one literal). We say that cooperative rewriting rule lj 1 l2 j l3 j is applicable for a truth assignment σ associated with the set of variables V ar ϕ of ϕ if σ(lj 1) = 1 and σ(lj 2 ) = 1; that is, in order to determine if such a rule is applicable for σ is necessary to know the truth value of two literals. So, these literals must cooperate in order to apply the rewriting rule. When applying a rule lj 1 l2 j l3 j for a truth assignment σ, we deduce the following information: σ(lj 3) = 1. Cooperation as a new frontier of tractability It is well known that 2 SAT P and 3 SAT is an NP-complete problem. On the one hand, non-cooperative rewriting rules have been associated with instances of 2 SAT and cooperative rewriting rules have been associated with instances of 3 SAT. On the other hand, passing from 2 SAT to 3 SAT can be interpreted as passing from tractability to (the presumable) intractability (assuming that P NP). In this context we can consider that passing from non-cooperative rewriting rules to cooperative ones amounts to passing from tractability to (the presumable) intractability (assuming that P NP). 6 Conclusions The quest for tools that provide new approaches to address the problem P versus NP, is a major challenge in computer science due to the relevance of the above mentioned problem. This paper focuses on tools related to frontiers of tractability expressed in terms of syntactic or semantic ingredients associated with models in a computing paradigm. The role of cooperation of objects to trigger rewriting rules is analysed in order to obtain this kind of borderlines in the framework of Membrane Computing. Specifically, some cell-like membrane systems have been considered: Basic transition P systems. P systems with active membranes -with/without electrical charges and with/ without environment- and with membrane division or membrane separation. P systems with symport/antiport rules -with or without environment- and with membrane division or membrane separation. References 1. A. Alhazov, L. Pan and Gh. Păun. Trading polarizations for labels in P systems with active membranes. Acta Informaticae, 41, 2-3(2004),

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65 From Super-cells to Robotic Swarms: Two Decades of Evolution in the Simulation of P Systems Luis Valencia-Cabrera, David Orellana-Martín, Miguel Ángel Martínez-del-Amor, Mario J. Pérez-Jiménez Research Group on Natural Computing Department of Computer Science and Artificial Intelligence University of Sevilla, Spain Avda. Reina Mercedes s/n, Sevilla, Spain {lvalencia,dorellana,mdelamor}@us.es Summary. Membrane Computing provides machine-oriented models of computation, with types and variants including different elements inspired from living cells. Proven computationally complete from their inception, they also showed their ability to solve computationally hard problems. Thus, it is crucial accompanying the theoretical studies with the practical materialization of these devices. While their real implementation presents serious limitations, a more affordable goal is the simulation of these machines, both to aid in the understanding of the theoretical models, and to provide tools to solve problems in different areas (Biology, Economy, robot control, etc.) by P systems-based models. Several works have analysed the history and state of the art of the simulation tools for P systems, the last one in Therefore, instead of repeating this information, we decided to provide a brief summary, along with an interactive tool to visualize on-line the evolution of these simulators, intended to stay updated as new simulation tools keep appearing on stage. 1 Introduction Natural computing is characterised by the fact that it provides conceptual and practical tools inspired in living nature, observing phenomena that can be somehow viewed as calculation procedures. Some branches within this area are inspired by evolutionary processes (as genetic, evolutionary algorithms and others) or certain types of animal behaviours (ant/bee colonies, etc.). These types of approaches have been mostly applied to optimization problems through the provision of approximate solutions. Others, instead, focus in lower levels of organization, as it might be the case of DNA computing or Membrane computing, among others. Since its appearance in November 1998, as introduced in [12, 13], membrane computing has supplied a number of computational devices (P systems, initially

66 66 L. Valencia-Cabrera et al. called super-cell systems), of different types (cell-like, tissue-like, neuron-like, multienvironment, P colonies, kernel P systems, MP systems, etc.), with many variants, each one providing different elements making them sound to attack certain relevant problems from a theoretical or a practical point of view. While many of these types and variants of P systems have been used to solve computationally hard problems, and a number of them have provided subtle tools to get thinner and thinner boundaries between P and NP classes (assuming P NP), other set of types and variants have been useful in the computational modelling of certain real-life phenomena, both at a micro and a macro level. The design and verification of solutions are tough tasks, especially when dealing with NP-complete problems solved with minimalistic variants of P systems trying to reduce the number of ingredients involved in the design of these machines (types and length of rules, structures, etc.) In addition, the computational modelling of complex systems, usually involving a huge number of elements and interactions among them, represents another challenge in terms of the difficulty of understanding how the system is evolving taking into account all the processes involved in such an evolution. Given this scenario, it is crucial to have as much help as possible to handle these P systems, and software tools here play an essential role. They can aid in many related tasks, which can differ depending on the specific needs. These assistants can be used to define or specify the syntax and semantics of the solutions for hard problems or the computational models for complex systems. Besides, some tools can be used to help in the formal verification of these solutions. In addition, the simulation of the computational devices supports researchers in the task of exploring the implications of the addition or removal of certain elements on the systems under study. Bearing in mind the needs just introduced, a variety of software applications and tools have been provided along the years. Not surprisingly, there have been a number of publications explaining those developments. Besides, several works have been presented reviewing the state of the art in the simulation, since the early days of membrane computing up to now. Consequently, it will not be the goal of this survey to provide a new analysis of the same tools. Instead, we will go very briefly over the main facts on the evolution of the simulators, recalling the main sources where any reader can find deep explanations about every simulator, and will present some on-line tool where all the main simulation tools are placed along a time line, starting in 1998 to the present time, and aiming to continue growing in the future. The rest of this work is consequently devoted to this brief outline of the milestones the discipline of Membrane computing has gone through. Section 2 summarizes the initial steps in the simulation of P systems, from 1998 to After that, Section 3 describes the main facts related with the period between 2005 and Then, Section 4 enumerates the most recent developments introduced in this decade, from 2011 to In Section 5, an interactive tool to show the evolution in the simulation of simulator of P systems is presented. Finally, Section 6 analyses

67 Two Decades of Simulation in Membrane Computing 67 the present and some prospects about the future of the simulation in Membrane computing. 2 Early days in Membrane Computing Almost two decades ago, the appearance of Membrane computing in 1998 [12] opened a fascinating branch of Natural computing bringing elements from its cellular inspiration, formal languages theory, Turing and the essence of computability models. The super-cell systems introduced at that moment (re-named definitely as P systems in 2000 [13]) provided characterisations of recursively enumerable languages. The computational power of these systems was not studied in that first work, but while the computational completeness was obtained without the need of a parallel synchronized evolution of objects and membranes, it was highlighted there the intuition that the theoretical model proposed, taking advantage of the inherent parallelism present in biological cells, could provide tools to explore an exponential search space working non-deterministically on a number of processors. Later works would prove the actual ability of some variants, as P systems with active membranes, to solve hard problems efficiently, providing a deep analysis in terms of computational complexity and the power of these devices. Besides this intuition, since this very first foundational technical report, the remarks section pointed towards the possibility of a practical implementation of super-cell systems, in biological or electronic media. At that respect, the document emphasized the importance of looking for answers to many open problems by directing the theoretical studies to the most promising and practically relevant path. The need of implementing or at least simulating these theoretical devices was therefore always there from the inception of the discipline of Membrane computing. Consequently, the first simulators for P systems were developed soon after the appearance of the foundational work [12], being submitted in 1999, before the first regular paper [13] presenting the discipline (see [15, 17]). As pointed out in a first survey work about simulators [6] (included in [2]), the focus in this first stage was to try to understand better the recently created computing model, with both pedagogical and research purpose, acting as assistants to study different computations of P systems, helping in tasks as design and verification of these cellular solutions, possibly suggesting invariants that could help in proofs. As also highlighted in that survey of software available up to [6], some restrictions were present when simulating P systems: on the one hand, the non-determinism of these devices cannot be in essence captured, but somehow simulated by the generation of pseudo-random numbers; on the other hand, a crucial aspect in Membrane computing gets definitely constrained or significantly reduced when simulating in digital computers: the massive parallelism inherent at different levels in P systems (objects inside each region evolving simultaneously - not to mention the evolution of different parts of a string in the case of string-based systems -, and every compartment - membrane, cell or neuron - possibly evolving at

68 68 L. Valencia-Cabrera et al. the same time). Despite those practical limitations, especially present in the early days, the first generation of simulators for cellular systems could provide interesting tools for teaching and mostly research, aiding to a better understanding, valuable support for theoretical research and assistant for verification. For those interested in the details of these initial works, we strongly recommend the reading of [6]. They were extensively analysed in that book chapter, and it would make no major sense to repeat the work conducted there, so we will simply enumerate the main contributions of those years in each direction. First of all, in those early days appeared the first P system simulators for transition P systems: in [15] they used LISP to build a simulator aimed to address real-life problems (as the Brusselator model or initial models in ecological systems, see [16]). and [17], in Prolog. They were followed closely by [18] (in Scheme, based on the formalization presented in [19]) and [20] (in Haskell, based on previous works analysing algorithmic aspects, the framework and the data structures used, see [21, 22, 23, 24, 25]). After these ones, a new simulator for transition P systems using Java was presented in [26]. Along with these sequential programs, the first two parallel simulators were presented, with the aim of partially exploiting the capabilities of the theoretical models in terms of efficiency: a first one [27, 28] was implemented in C++, making use of MPI to communicate threads, while the other one [29], implemented in Java, made use of RMI for the communication among processes in different computers. After the first two simulators mentioned [15, 17], simultaneously to the contributions just outlined, some developments were made for additional types of cell devices. That was the case of P systems with active membranes (as defined in [1]), with a first simulator [30] (also see [14]), in Visual C++, that also worked with catalytic cell systems. After that two new simulators were presented, one using CLIPS in [31] (to solve problems as Bin-packing [32] or CAP [33]), and another contribution using Prolog, in [34], to solve other NP-complete problems (see [35, 36, 37]), and also including a tool to analyse the descriptive complexity of P systems, based on the so-called Sevilla carpets (see [38, 39, 40]). A relevant contribution was also made within other types of P systems, as it would be the case of the implementation of catalytic P systems presented in [41], and the simulator for maximally parallel multiset-rewriting systems with promoters/inhibitors in [42], used to prove theorems presented in [43, 44]. Other simulator developed during those early years in Membrane computing was SubLP-Studio, for both L and P systems (see [45], available at [138]). It included some graphical components showing the growth of plants [46, 47]. Besides complete simulators, other interesting tools appeared during that period, as a library (in C language) to ease the representation and simulation of P systems [48]. Finally, in that period some interesting contributions appeared trying to get closer to the implementation of P systems in silico through hardware components, as the FPGA based implementation described in [49], were they cited as possible precursor [50].

69 Two Decades of Simulation in Membrane Computing 69 A much broader explanation of the main milestones of this period and details over the simulators can be found at [6]. Besides, an online tool has been provided to follow the evolution in an interactive way, also providing further details and references, in [139]. 3 A second stage: from particular to general The previous section outlined some facts related with the initial steps in Membrane computing. It implied an exciting period where the emergence of the discipline led to the growth of the research community, the appearance of tools to support the theoretical findings, and some case studies related with NP-complete problems and with modelling of biological phenomena started to shed some light about future lines for the practical use of P systems tools in short term. The theories, algorithms and software developed laid crucial foundations for the new stages, clarifying the challenges to face when facing the simulation of the types of machines designed within Membrane computing, sound structures to handle these devices and technical constraints imposed by the state of the technology. During the following few years, from until 2010, we witnessed an explosion in the number of types of variants of P systems, from classical cell-like to tissue-like and neuron-like P systems, along with new stochastic and probabilistic approaches, involving environments and other elements enriching the conceptual tool-kit of frameworks to solve problems by means of cellular systems. This burst came together with a proportional increase in the number of simulation tools to help managing the different problems addressed during this period. As the number of possible domains to apply new variants of P systems was growing, the choice of developing new software tools for each specific problem or sub-typology started to be at least questionable, given the significant effort for a limited profit. It was therefore the moment of changing the focus of the simulators from specific to general purpose tools, able to deal not with a P system designed for a specific problem, but software to manage as many machines as possible within a certain framework inside Membrane computing, or even providing general functionality applicable to devices as different as cell-like, neuron-like or multi-environment P systems. In most cases, during the previous stage, ad-hoc software tools were developed for checking specific models. This idea was progressively turning into more general views and tools. Besides general purpose tools, a significant feature characterised several simulators in this generation: the application to real-world problems in biology. For a detailed analysis of the software for P systems until 2010 we recommend [3], providing an exhaustive survey within the Oxford Handbook on Membrane Computing [3], most complete collective volume to date. In addition, due to the permanent evolution of the discipline, it would be also advisable to check the P systems web site [137] for updated information about software simulators and tools. As mentioned in previous sections and detailed in Section 5, we have tried to present in this bulletin a collection of the main references in web format, includ-

70 70 L. Valencia-Cabrera et al. ing the main achievements in each period. We strongly recommend the reader to contact the author if you detect any simulation tool within Membrane computing missing in this site that you consider convenient to include, so that we can maintain it as complete as possible, as a valuable resource for the community and those others interested in Membrane computing and especially those focused in the simulation of its devices, P systems. In what follows, we will briefly over a non-complete list of significant contributions to this period, but a more exhaustive list can be found in the sources cited and the interactive tool provided in [139]. During the last few years of the first generation, some works had started exploring new paths in the application of Membrane computing. Along with all these simulators presented in that phase for classical P systems, where maximal parallelism was generally applied as the execution strategy, it took place the appearance of the first tool [51] for the modelling of biological processes [53] through a variant of P systems (from [52]) including probabilistic elements applied for the execution of the rules. A different approach (unless with a development inspired in [15, 17, 30]) emerged in those years by the University of Verona, leading to the development of PSim (available since 2004 in [138], and described years later in [54, 55]). Along the year 2006, different initiatives led to the development of several simulators from different places of a growing community. Thus, in a joint effort between Sevilla and Sheffield two implementations of multi-compartmental Gillespie s algorithm to simulate stochastic P systems (in C and Scilab, respectively) was delivered [143], along with a tool [57] (written in Java) for the manipulation of SBML files from CellDesigner. The framework designed was applied to several relevant papers in the modelling of biological phenomena at a micro level [58, 59, 60, 60]. Other significant contributions of this generation were made by the ad-hoc simulators (available at [138]) developed by Cazzaniga-Pescini (written in C, with parallelism given by MPI) to simulate certain stochastic processes in biology (see [62, 64, 63, 65, 66]). Further developments were made during those years applied to the modelling of biological phenomena, as the simulator of stochastic processes Cyto-sim [67, 145], based on the formal model presented in [68]. Another important field related with P systems appeared between 2004 and 2006, membrane algorithms [70, 71, 72, 69], providing new approaches to attack NP-complete optimization problems using ingredients from Membrane computing. In [69] some solutions for TSP were presented, along with a software tool (in Java), to simulate several variants of these devices. Along with the previous tools for cell-like P systems, many others emerged during this period for other types of devices. Thus, in [73] it was presented a simulator for tissue-like P systems [74], formalism presenting a graph structure (instead of the tree-based one of cell-like P systems), used to solve NP-complete problems as in [75]. Similarly, in 2008 a tool for the simulation and verification of Spiking neural P systems [76] was developed [77]. Additionally, a third novel type

71 Two Decades of Simulation in Membrane Computing 71 of P systems, the conformon P systems [79, 80], received a special attention, hence leading to the appearance of a first simulator [78] to handle them, being applied to the study of biochemical processes with implications in medicine [81]. Apart from pure simulators, other tools were also developed during thee period. Thus, concerning the graphical representation of P systems in relation with plant modelling, several works [82, 83, 84, 85] deepened into the works conducted by Geourgiou et al. in the previous generation. Other works were devoted to the simulation of P systems by other computing models, for instance by means of multi/agent systems [86], aiming to exploit the inherent parallelism of membrane computing solutions through a distributed application. Another great example of the trend within this period, as a paradigmatic case passing from particular to general, it is worth highlighting the open source project Xholon [146], general-purpose software for modelling, transformation, and simulation not only for P systems but for a broader range of computing models including finite state machines, cellular automata and agent based systems. Based on XML and Java, it supports the Unified Modeling Language (UML) 2.1, systems biology modelling including SBML, and other models and tools. And we could not finish our overview of this period without mentioning another ambitious open source project within the area of Membrane computing: P-lingua [87, 89]. It aimed to become a standard for the specification/definition of P systems, providing a whole framework including from a programming/specification language for different types and variants of P systems, to a number of simulators for those variants, with different tools for parsing/debugging or export of P systems in different formats to increase interoperability with other software. To end up with this stage, it is worth mentioning a remarkable fact, while not representative from the period and not software related, given its possible significance for the future: a first in vitro implementation of a kind of membrane computing device, using test tubes as membranes and DNA molecules as objects, evolving under the control of enzymes. A more precise introductory explanation can be found at [7], and the details in [90, 91]. Despite some difficulties and limitations, there is no doubt about the relevance of the achievements reached by that project. 4 A third step: the era of applications As explained in the previous section, the second period in the simulation of P system implied the development of general purpose simulators and tools, aimed to provide the P systems researchers and developers software that they could adapt to analyse their specific P systems under study, instead of being developed for a specific system. The next step in the natural evolution of these devices is, not surprisingly, crossing the barrier of the assistants for the Membrane computing community, so that they can act as useful tools for people working in other disciplines, from researchers to managers in companies or institutions, that can take

72 72 L. Valencia-Cabrera et al. advantage of the solutions based on P systems to solve their problems, not focused in the internal details provided by these tools but receiving the software delivered as black boxes that they can use as practical aids in their decision-making processes. It is what we could somehow call the era of applications, putting the results of our research and findings at disposal of a much broader community of researchers and, in general, at disposal of the society. It is unknown in Membrane computing community if we will see a real implementation of P systems, beyond the attempts commented in previous stages, with their actual power and efficiency in the new few decades. We do not know if we will have those machines able to solve NP-complete problems in polynomial time, in many cases even linear time, but does that not necessarily mean we will have to wait until that moment in biochemical technology to find some relevant use of P systems. As Babbage kept working on his ideas, not simply waiting until the precise moment when Turing, Von Neumann and their contemporaries witnessed the first electronic computers based on similar principles, membrane computing must keep moving, finding new ways to provide a step further. Researchers keep looking for frontiers of the practical tractability of very relevant problems through different variants of P systems, and explore ways for their practical implementation. But additionally we can say today that we have provided some conceptual and practical tools that can help solving real problems, as the computational modelling of certain phenomena that have been shown to be approachable with the techniques and software tools developed. Novel modelling approaches and conceptual and software tools have been delivered, some of them showing desirable features more suitable to address certain modelling problems that would be significantly harder to handle with more traditional tools. In the last decade we have seen different examples in this sense both at micro and macro levels. Last year, 2016, a new survey was published analysing the state of the art in the simulation of P systems [8], providing a rich list of sources up to 2016, so it would not be worth detailing the same study here, but we will briefly outline some facts of the most recent progress in this topic. Additionally, as the branch of Membrane computing and in particular the simulation of P systems seems to be maintaining a good health, we will add some more recent advances appeared within the last year. Some of the main results within this third stage were, not surprisingly, consequence of the achievements in the previous stage, to develop the conceptual and software tools deepening into the study of different modelling approaches. A good example of this transition from previous works was MetaPlab [142], a computational framework for MP systems, moving several steps forward wit respect to the first simulator [54] based on the metabolic algorithm introduced in [56], developed in the previous stage. Many successful applications of MP systems can be found at [141, 142], with special attention at biomedical studies at a micro level. Another exponent of this trend was the development of a family of software applications (EcoSim) to model and simulate the population dynamics of different ecosystems, by using the simulation engines provided by P-Lingua framework

73 Two Decades of Simulation in Membrane Computing 73 [87, 89, 147]. The identification of common needs for the simulation of those different phenomena, along with any other P systems covered by the framework, led to the appearance of MeCoSim [92, 148], allowing the customisation, by simple configuration, of apps adapted to each particular problem, having all the infrastructure of P-Lingua framework available for parsing, debugging, simulation, verification, and providing a GUI with custom input and output tables, charts, graphs, visualization of structures and connectors with external tools as invariants detectors or model checkers. The disposal of this generic infrastructure boosted the study of different real problems, mostly using probabilistic P systems (known as PDP systems) applied to ecosystems ([94, 95, 96], among many others) and other fields ([97, 98, 99]), once the researchers could focus on the specificities of their particular problems and not in the software implementation of the tools needed. Additionally, a protocol was designed [100] to standardise the modelling process by PDP systems, from the problem to the software solution customised for the specific scenario under study. Another work [101] focused on the calibration of the parameters in ecosystems modelling based on P systems. A recent book chapter [11] summarised the main facts and case studies in PDP systems modelling. And if at a macro level the simulation of ecosystems through P systems following a probabilistic approach experienced a significant growth with major achievements, at a micro level the success of the studies within systems and synthetic biology using P systems following a stochastic approach was remarkable. These significant contributions were supported by solid tools developed with Infobiotics Workbench [102, 149]. This software, aimed to facilitate the incremental modelling and rapid prototyping of multi-compartment systems, accepting two languages to describe the biological models: an extension of SBML (that could be generated from CellDesigner) and a domain specific language (DSL), implementing lattice population P systems [93]. As the authors mention, the reactions described (using the DSL language) in these models can be organized in modules (parametrisable sets of reactions), which promote (sub)model reuse and hence facilitate debugging of model entities capturing biological functions. Some other works appeared during this period, focused on the simulation of numerical P systems, as it was the case in [103] or [104]. It was applied to the simulation of robots in [105], and later simulators were developed in sequential [151] and parallel [106] variants (this one based on GPU cards). A later contribution, by 2014, to the practical application of membrane computing in biology, making use of P-Lingua framework, was MeCoGUI [150], incorporating probabilistic guarded P systems[107]. This application was used recently for the simulation of the ecosystem of Pieris napi oleracea in eastern North America[108, 152]. Additionally, another interesting application related with the study of regenerative processes using these tools was recently published [109]. The following year, 2015, other biological applications at a micro level appeared. Thus, a new simulator of P systems was developed in [110] to study the evolution of the resistance of certain organisms/parasites against antibiotics. Simultaneously, in [111] the authors studied new techniques for solving a Mitogen Ac-

74 74 L. Valencia-Cabrera et al. tivated Protein Kinases (MAPK) cascade by means of P systems, using P-Lingua and MeCoSim. Other applications emerged in 2016 and 2017 [112, 113, 114, 115] and will continue appearing. Additionally, new approaches for the simulation of P systems keep coming up, as it is the case of an interesting agent-based simulator for kernel P systems with division rules [116], from the same team that also developed kpworkbench [117, 153], providing another interesting set of tools for the simulation and verification of kernel P systems. To continue with a fair outline of the achievements on this era, we cannot reduce our mentions to biological applications, because a big amount of efforts and remarkable results were made within a complementary research line: the development of simulation tools taking advantage of the inherent parallelism of P systems by their simulation on parallel architectures. An extensive survey on this topic was presented in 2015 [9], summarising the simulation tools making use of parallel technologies, more specifically those based on GPU devices. Among other works detailed there, we can recall the line followed along PMCGPU project [154]. Please look for the details about parallel simulators in [9], including among others [118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128]. Some more recent results appear after that surveys, including [10, 129]. Apart from those novel technologies applied to the simulation of different types of P systems, the research line started with P-Lingua framework kept including new variants along the years, as tissue P systems with cell division [130], spiking neural P systems [131], tissue P systems with cell separation [132], cell-like P systems with symport/antiport rules [133], and more recently cell-like SN P systems [134]. And if we started this survey talking about super-cells, the original name of P systems that opening the title of the paper, we think it would be fair closing this overview of simulators in membrane computing paying a special attention to one of the most recent promising research lines within the area: the control of robots and robotic swarms. Over the last few years this path is being intensively explored in different groups from Romania to China. Thus appeared in 2015 Lulu [135], an open-source simulator for P colonies, XP colonies and colonies of XP colonies (P swarms), applied to the control of single and multiple robots. See [155] for additional information about the software and its use, that can be downloaded from [156]. In 2016 an application of XP colonies with robotic swarms appeared [136], and this year, 2017, they presented a book [4] collecting the most relevant information from their works about distributed control of robotic swarms using membrane computing. In the last few years, other works have analysed the different simulation tools appeared in membrane computing over the last two decades. As far as we are concerned, the last survey is [8] in 2016, as commented at the beginning of this section. That work provides an excellent source, listing the main contributions, describing the simulators and classifying them in categories depending degree of generality, applications addressed, etc. As the authors highlight, their paper is an attempt to provide the details of the tools that are available for membrane

75 Two Decades of Simulation in Membrane Computing 75 computing... On one hand we have tools that are being used for specific type of P Systems or the tools which have a specific application. On the other hand there are tools which are comparatively generic in nature. Further this paper lists the tools that have been designed and developed to be used for the biological applications of P Systems. In fact, the idea of deploying in this work a web tool aiming to stay updated came after reading this extensive analysis in that survey, which also included a useful timeline from 2000 to We observed the convenience of such a line to visualize the evolution in the simulators, and at the same time we saw it was practically impossible to provide all the tools in a static picture, both due to its inherently dynamic nature (given the number of tools still under construction and to be developed), and because of the space limitation. Thus, we considered an interactive tool aiming to keep updated could fill that gap for members of membrane computing community and for people approaching the discipline for the first time and aiming to see what happened in the past and what is going on up to now. 5 An on-line tool to explore the timeline Along this work we have recalled the main periods in the evolution of Membrane computing, in relation with the development of software tools for the modelling, simulation and verification of P systems, as well as their use for the practical solution of relevant problems. The main references within this field has been outlined, pointing towards collective volumes and survey works have discussed in more depth each of the periods since the inception of the discipline in As made significantly evident in previous sections, the last two decades have been very prolific in terms of the number of tools for the modelling and simulation within the area of membrane computing. A good measure is the number of works outlined throughout this paper, and the presence of several survey papers every few years to try to summarise the new contributions taking place and highlight the main achievements within each period. Thus, it was not the goal of this work in the bulletin to repeat those deeper works in the analysis of the simulation tools, but simply putting all of them together, trying to show the whole picture. And as this whole picture is dynamic and will continue growing, we thought a suitable tool to handle this evolution in the simulation tools would be an interactive tool that we could keep updated. Thus, a web resource has been prepared and put at disposal of the community, in order to have an updated source with the main results in the simulation of P systems in an interactive way. Thus, the tool shared online is as shown in Fig. 1, presented as a separate web. As we can see, it presents a line that we can follow, move, zoom in/out, etc. to explore the evolution of the different contributions over the years. Additionally, the central part shows the details of a specific publication or site, some labels depending on the type of resource, and possible links to external resources. We

76 76 L. Valencia-Cabrera et al. can click on each separate publication in th timeline, or start navigating to the left or right to go through each single register sorted by publication date. Fig. 1. Timeline - isolated Additionally, the inclusion of this timeline within any other web resource is straightforward, and we not only allow its sharing through other sites but also encourages the community to do it. Please do not hesitate to contact the authors to share the minimal details needed to include this resource in your site if considered convenient (of course, the styles to present the same tools in other sites can be changed through css). In Fig. 2 we can see the timeline included in the site of the Research Group on Natural Computing. Fig. 2. Timeline - embedded This timeline tool has been deployed in the website of RGNC at [157].

77 Two Decades of Simulation in Membrane Computing 77 6 Conclusion The crucial role played by the tools to aid in the tasks of modelling, simulation, verification, etc. of models within the area of membrane computing has been made clear along the last two decades. The evolution of the tools from pedagogical helpers to the current frameworks for the actual solution of practical problems, real applications, reinforces the relevance of these tools to complement the theoretical studies essential for the robustness of the achievements in the discipline. As highlighted in the previous sections, the aim of this survey is to serve as an updated source for members inside and outside the community of membrane computing interested in the simulation of P systems, so we encourage again the reader to contact the authors if some errors or missing resources are detected and you consider they should be included in this text, considering it is within the scope of this work. 7 References In what follows, the main contributions to this work are listed under different headings (collective volumes, surveys and extensive bibliography), depending on their category. In these three lists the publications are sorted in order of appearance within the text, which is mostly chronological inside each category of publication. References Books and collective volumes 1. Păun, Gh. Membrane Computing. An Introduction. Springer, Ciobanu, G., Păun, Gh., Pérez-Jiménez, M.J. (eds.) Applications of Membrane Computing. Springer, Păun, Gh., Rozenberg, G., Salomaa, A. (eds.) The Oxford Handbook of Membrane Computing. Oxford University Press, New York, Florea, A.G., Buiu, C. Membrane Computing for Distributed Control of Robotic Swarms: Emerging Research and Opportunities, Hershey, PA: IGI Global, Zhang, G., Pérez-Jiménez, M.J., Gheorghe, M. Real-life Applications with Membrane Computing. Series: Emergence, Complexity and Computation, 25, Springer, Surveys 6. Gutiérrez-Naranjo, M.A., Pérez-Jiménez, M.J., Riscos-Núñez, A. Available membrane computing software. In Ciobanu, G., Păun, Gh., Pérez-Jiménez, M.J. (eds.) Applications of Membrane Computing, Springer, Heidelberg (2006),

78 78 L. Valencia-Cabrera et al. 7. Díaz-Pernil, D., Graciani-Díaz, C., Gutiérrez-Naranjo, M.A., Pérez-Hurtado, I., Pérez-Jiménez, M.J. Software for P systems. In Păun, Gh., Rozenberg, G., Salomaa, A. (eds.) The Oxford Handbook of Membrane Computing, Oxford University Press, 2010, Raghavan, S., Chandrasekaran, K. Tools and Simulators for Membrane Computing - A Literature Review. In: Gong M., Pan L., Song T., Zhang G. (eds.) Bio-inspired Computing Theories and Applications. BIC-TA Communications in Computer and Information Science, 681, Springer, Singapore, Martínez-del-Amor, M.A., García-Quismondo, M., Macías-Ramos, L.F., Valencia- Cabrera, L., Riscos-Núñez, A., Pérez-Jiménez, M.J. Simulating P Systems on GPU Devices: A Survey. Fundamenta Informaticae, IOS Press, 136 (2015), Martínez-del-Amor, M.A., Macías-Ramos, L.F., Valencia-Cabrera, L., Pérez- Jiménez, M.J. Parallel simulation of Population Dynamics P systems: updates and roadmap. Natural Computing, 15, 4 (2016), Zhang, G., Pérez-Jiménez, M.J., Gheorghe, M. Data Modeling with Membrane Systems: Applications to Real Ecosystems. Real-life Applications with Membrane Computing, Springer International Publishing, 2017, Extensive bibliography 12. Păun, Gh. Computing with membranes. Turku Center for Computer Science, TUCS Technical report 208 (1998), Păun, Gh. Computing with membranes. Journal of Computer and System Sciences, 61, 1 (2000), Păun, Gh. P systems with active membranes: Attacking NP-complete problems. Journal of Automata, Languages and Combinatorics, 6 (2001), Suzuki, Y., Tanaka, H. On a LISP Implementation of a Class of P Systems. Romanian Journal of Information Science and Technology, 3, 2 (2000), Suzuki, Y., Fujiwara, Y., Tanaka, H., Takabayashi, J. Artificial life applications of a class of P systems: Abstract rewriting systems on multisets. In Calude, C.S., Păun, Gh., Rozenberg, G., Salomaa, A. (eds.) Multiset Processing. Mathematical, Computer Science, and Molecular Computing Points of View. Lecture Notes in Computer Science, 2235, Springer, 2001, Maliţa, M. Membrane computing in Prolog. In: Pre-Proceedings of the Workshop on Multiset Processing, Curtea de Arges, Romania, TR 140, CDMTCS, University of Auckland, (2000), Balbontín-Noval, D., Pérez-Jiménez, M.J., Sancho-Caparrini, F. A MzScheme implementation of transition P systems. In: Păun, G., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) Lecture Notes in Computer Science, Springer Heidelberg, 2597 (2003), Pérez-Jiménez, M.J., Sancho-Caparrini, F. A formalization of transition P systems. Fundamenta Informaticae, 49 (2002), Arroyo, F., Luengo, C., Baranda, A.V., Mingo, L. A software simulation of transition P systems in Haskell. In: Păun, G., Rozenberg, G., Salomaa, A., Zandron, C. (eds.), Lecture Notes in Computer Science, Springer Heidelberg, 2597 (2003), Baranda, A.V., Castellanos, J., Arroyo, F., Gonzalo, R. Data structures for implementing transition P systems in silico. In: Pre-Proceedings of the Workshop on

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87 Two Decades of Simulation in Membrane Computing García-Quismondo, M. A Java-Based P-Lingua Simulator for Enzymatic Numerical P Systems software tools/java simulator enps/ 152. Pieris oleracea model website. PGSP systems: Pieris oleracea 153. kpworkbench website PMCGPU project Florea, A.G., Buiu, C. Lulu - a software simulator for P colonies. Use case scenarios and demonstration videos. Zenodo. (2015) Florea, A.G., Buiu, C. Lulu - an open-source software simulator of P colonies and P swarms. pcol sim (2016) Timeline in RGNC.

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89 P Systems Attacking Hard Problems Beyond NP: A Survey Petr Sosík Research Institute of the IT4Innovations Centre of Excellence, Faculty of Philosophy and Science, Silesian University in Opava Opava, Czech Republic, petr.sosik@fpf.slu.cz Summary. In the field of membrane computing, great attention is traditionally paid to the results demonstrating a theoretical possibility to solve NP-complete problems of polynomial time by means of various models of P systems. A bit less common is the fact that almost all models of P systems with this capability are actually stronger: some of them are able to solve PSPACE-complete problems in polynomial time, while others have been recently shown to characterize the class P #P (which is conjectured to be strictly included in PSPACE). A large part of these results has appeared in the last few years. In this paper we focus on strong models of membrane systems which have potential to solve hard problems belonging to classes containing NP (although it has not been proved yet whether this inclusion is strict or not). These include P systems with active membranes, P systems with proteins on membranes and tissue P systems, as well as P systems with symport/antiport and membrane division. We provide a survey of computational complexity results of these membrane models, pointing out some features providing them with their computational strength. 1 Introduction P system is a bio-inspired computing model of rather high level of abstraction which is motivated by regulatory functions of biological membranes in living cells. The key ingredient of P system is an abstract membrane which lets pass only certain objects, only in certain directions or only under some conditions. The membranes can be embedded, hence the name membrane system. Objects can also react and produce other objects. Gradually, many variants of membrane system have been proposed, enriched with further operations as membrane division, membrane separation (denoted together as membrane fission), membrane creation, tissue P systems, P systems with membrane proteins etc. It soon became evident that, allowing operations of membrane fission (and ignoring any physical space/time constraints since these are abstracted in the definition of P systems), one can apply such P systems to solve NP-hard problems in polynomial time [47]. The question of the upper bound of the class of problems

90 90 P. Sosík solvable in polynomial time remained open for several years. Finally, the class PSPACE was shown to be the upper bound for several models of confluent P systems [53, 54, 56], the question remaining open in the other cases. This upper bound was shown to be tight in some cases, while some other models have been recently reported to characterize the class P #P, defined by means of oracles for counting problems [22, 24]. The complexity class PSPACE plays an important role in the theory of parallel computing. The parallel computation thesis [9], formulated already in 1976, states that the time used by a (reasonable) parallel machine is polynomially related to the space used by a (reasonable) sequential machine. Many seminal models models of parallel computation, including PRAM and the alternating Turing machine, are known to meet this condition. Consequently, the class of machines characterizing by their polynomially time-bounded computations the class PSPACE is sometimes called the second machine class (as opposed to the first machine class characterizing the class PTIME). The results summarized in this survey confirm that also some models of confluent membrane systems, as P systems with active membranes and P systems with proteins on membranes, are second class machines, while others being on the candidate list (as the P systems with symport/antiport and membrane division, or maybe spiking neural P systems [15]). In the remaining sections we shall study the influence of several abstract operations in membrane systems, as membrane fission, membrane dissolution, membrane polarization, symport/antiport of objects, regulation by proteins on membranes etc., on their computational power. Interesting results have been also recently reported for the case of time-free P systems. Our aim is to give a survey of the known results arranged in a comparative way. Where possible, we omit technical details and refer the reader to more technical papers. We also omit extensive explanation of definitions and we limit the number of examples which can be also found in the sources referred to. 2 Preliminaries A multiset M over an underlying set A is a pair (A, f) where f : A N is a mapping. If M = (A, f) is a multiset then its support is defined as supp(m) = {x A f(x) > 0}. The total number of elements in a multiset, including repeated memberships, is the cardinality of the multiset. A multiset is empty (resp. finite) if its support is the empty set (resp. a finite set). If M 1 = (A, f 1 ), M 2 = (A, f 2 ) are multisets over A, then we define the union of M 1 and M 2 as M 1 +M 2 = (A, g), where g = f 1 + f 2. In the rest of the paper we often refer to standard computational complexity classes P, NP, co-np and PSPACE. We also denote by PP the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 0.5 for all instances. Finally, P #P denotes the class of problems solved by polynomial-time Turing machines with oracles for counting

91 P Systems Attacking Hard Problems Beyond NP 91 problems. It is known that this class coincides with P PP. We refer the reader to, e.g., [19] for more details. Definition 1. A P system of degree m 1 is a construct Π = (O, H, µ, w 1,..., w m, R, i out ) where: 1. O is the alphabet of objects; 2. H is the set of labels of membranes; 3. µ is a membrane structure of degree m with membranes labeled with elements of H; 4. w 1,..., w m O are the multisets of objects initially present in the m regions of µ; 5. R is a finite sets of evolution rules (associated with labels) which can change contents of membranes and eventually also structure of the system; types of rules are specified in further sections; 6. i out H indicates the output region of Π. The membrane structure and the multisets represented by w i, 1 i m, in Π constitute the initial configuration of the system. A transition between configurations means applying a maximal multiset of evolution rules in parallel. Other execution modes as sequential or minimal parallelism are possible. The computation stops when there is no rule which can be applied to objects and membranes in the last configuration.the result of computation is then defined by the content of the output membrane. In this paper we study the accepting (or recognizer) variant of P systems. A recognizer P system solving a decision problem has a specific input membrane i in which initially contains a multiset of objects encoding an instance of the problem. Alternatively, if the system solves only one instance, the instance may be encoded within the structure of the system; then we speak about P systems without input membrane. A recognizer P system must furthermore comply with the following requirements: (a) the alphabet O contains two distinguished elements yes and no; (b) all computations halt; and (c) exactly one of the object yes (accepting computation) or no (rejecting computation) must be sent to the output membrane of the system, and only at the last step of each computation. 3 Complexity Classes of Membrane Systems Consider a decision problem X = (I X, θ X ) where elements of I X are called instances and θ X is a total boolean function over I X. In a family of recognizer systems without input membrane, denoted by Π = {Π(w) : w I X }, an instance w of a problem X is encoded into the structure of a P system Π(w). The system Π(w) is supposed to solve the instance w. If we use recognizer P systems with input membrane, then such a family is denoted by Π = {Π(n) : n N}. A member

92 92 P. Sosík Π(n) of the family solves all the instances of the problem X of size n, properly encoded as its input. (Let us denote by w the size of an instance w I X.) Definition 2 ([36]). A family of recognizer membrane systems is polynomially uniform by Turing machines if there exists a deterministic Turing machine which constructs each member Π of the family in polynomial time with respect to the size of the instance(s) solved by Π. In the sequel we will for short denote such a family just as uniform. Formally, [36] defines the conditions of soundness and completeness of Π with respect to X. A conjunction of these two conditions ensures that for every w I X, if θ X (w) = 1, then every computation of Π(w) is accepting, and if θ X (w) = 0, then every computation of Π(w) is rejecting. Note that the P system Π(w) can be generally nondeterministic, i.e, it may have different possible computations, but with the same result. Such a P system is also called confluent. Definition 3 ([36]). A decision problem X is solvable in polynomial time by a family Π = {Π(w) : w I X } of recognizer P systems without input membrane if the following holds: The family Π is polynomially uniform by Turing machines. The family Π is polynomially bounded; that is, there exists a polynomial p such that for each instance w I X, every computation of Π(w) performs at most p( w ) steps. The family Π is sound and complete with respect to X. The family Π is said to provide a semi-uniform solution to the problem X. Analogously one could define a family Π = {Π(n) : n N} of recognizer P systems with input membrane which provide a uniform solution to the problem X. We refer to [36] for more details. Let C be a class of recognizer P systems. We denote by PMC C the set of all decision problems which can be solved in a uniform way and polynomial time by means of families of systems from C. We denote by PMC C the set of all decision problems which can be solved by such families in a semi-uniform way. By definition, for any class C we obtain PMC C PMC C. 4 P systems with Active Membranes The term active membranes denotes the fact that, in this model, an operation inspired by division of live cells is introduced: membranes can divide into two, copying their contents (including eventual embedded membranes) into both descendants. Some objects, however, can be modified, imitating mutations. Besides, the membranes contain objects which can evolve and pass between membranes under pre-defined conditions. The following definition is given without any broad

93 P Systems Attacking Hard Problems Beyond NP 93 explanation and examples; for further details please see, e.g., [47] or [46]. A P system with active membranes [47], abbreviated here as AM system, is a construct Π = (O, H, µ, w 1,..., w m, R, i out ), where O, H, w 1,..., w m and i out are as in Definition 1, µ is a membrane structure of m membranes with possible polarizations {+,, 0}, and R is a finite set of developmental rules of the following forms: (a) [ h a v] α, for h H, α {+,, 0}, a V, v V h (object evolution rules); (b) a[ h ] α1 h [ α2 hb] h, for h H, α 1, α 2 {+,, 0}, a, b V (communication rules); (c) [ h a ] α 1 h [ h ] α 2 h b, for h H, α 1, α 2 {+,, 0}, a, b V (communication rules); (d) [ h a ] α b, for h H, α {+,, 0}, a, b V h (dissolving rules); (e) [ h a ] α 1 h [ h b ] α 2 h [ h c ] α 3 h, for h H, α 1, α 2, α 3 {+,, 0}, a, b, c V (division rules for elementary membranes); (f) [ h0 [ h1 ] + h 1... [ hk ] + h k [ hk+1 ] h k+1... [ hn ] h n ] α2 h 0 [ h0 [ h1 ] α 3 h 1... [ hk ] α 3 h k ] α 5 h 0 [ h0 [ hk+1 ] α 4 h k+1... [ hn ] α 4 h n ] α 6 h 0, for n > k 1, h i H, 0 i n, and α 2,..., α 6 {+,, 0}; (division of non-elementary membranes). All the above rules are applied in parallel, but at one step, an object a can be subject to only one rule of type (a) (e) and a membrane h can be subject to only one rule of type (b) (f). In the case of type (f) rules, this means that none of the membranes h 0,..., h n listed in the rule can be simultaneously subject to another rule of type (b) (f). However, this restriction do not apply to membranes with neutral charge contained in h 0. The system Π starts its computation in the initial configuration (µ, w 1,..., w m ) when all the membranes are neutral (polarization 0), and it continues the computation until no rule can be applied. We summarize basic known results in Table 1, describing the influence of various combinations of operations on the computational power of AM systems. Each column in the table corresponds to a specific combination of operations described in the above definition of AM systems. The last row compares the computational power of polynomially uniform families of AM systems using these operations with known computational complexity classes. Most of these characterizations hold for both uniform and non-uniform solutions; some, however, are known to hold for only one of these variants. The reader is referred to, e.g., [20, 55] for more details on some of these results. Table 1 summarizes results reported in [1, 22, 39, 37, 38, 54, 57]. The results in Table 1 refer to P systems with active membranes with no restriction of the depth of their membrane structure. When the non-elementary

94 94 P. Sosík Evolution rules (a) * X X Communication rules (b),(c) X X X Membrane dissolution (d) * * * Division of elementary membranes (e) X * Division of non-elementary membranes (f) X Class of problems solved = = = in polynomial time P P #P PSPACE Table 1. The computational power of uniform families of recognizer AM systems with polarization. X denotes used operations, * denotes operations not affecting the computational power. membrane division is allowed but the depth of the membrane structure is bounded, the resulting families of P systems can solve intermediate problems in the counting hierarchy, located between P #P and PSPACE. Specifically, P systems with active membranes and membrane structure of a constant depth d > 1 are able to solve problems in the complexity class P C d P, where C d P is the d-th level of the counting hierarchy [23]. 4.1 Polarizationless Active Membranes In AM systems without polarization, denoted by AM 0 systems, all membranes are always polarized neutrally. In such a case the condition that the membranes h 1,..., h k and h k+1,..., h n in rules of type (f) have opposite polarization is relaxed. We start with rather surprising result: if, in addition to polarizations, we remove also the rules (d) of membrane dissolution, the system looses almost all its computational power [32]. Actually, such a system can be replaced by a computationally equivalent AM 0 system with a single membrane and with evolution rules only, which immediately transform certain input objects to yes and others to no. In other words, the behaviour of such a system is trivial. It is interesting that even the non-elementary membrane division does not increase the power of these systems. Whenever the the membrane dissolution is allowed, the resulting computational power corresponds to that of conventional computers, even only with object evolution rules (a). The power of these families remains unchanged if we add communication rules of type (b) a (c), and also division rules (e) restricted to the form [ h a ] h [ h b ] h [ h b ] h (symmetric division of elementary membranes). Interestingly, if unrestricted elementary membrane division is allowed, only an upper bound of the resulting computational power is known [22] (second column in Table 2. The so-called Păun s conjecture claims that the computational power is still P. With non-elementary membrane division and only the rules of types (d), (e), (f) are allowed, then semi-uniform families of AM 0 systems solve in polynomial time problems in NP co-np [58].

95 P Systems Attacking Hard Problems Beyond NP 95 Finally, with all possible operations (a) (f) including the non-elementary membrane division we again get the power of the second class computers, i.e., the capability to solve PSPACE-complete problems in polynomial time. The results based on publications [2, 21, 32, 31, 39, 54, 58] are summarized in Table 2. Evolution rules (a) X X X Communication rules (b),(c) * X X Membrane dissolution (d) X X X X Division of elementary membranes (e) X X X Div. of non-elementary membranes (f) X X Class of problems solved = = in polynomial time P P #P NP co-np PSPACE Table 2. The computational power of uniform families of recognizer AM systems without polarization. X denotes used operations, * denotes operations which do not affect the computational power. Open Problem 1 Find a relevant example of P systems with active membranes where the classes of problems solvable by uniform and semi-uniform way are different. Open Problem 2 Is the lower bound NP co-np or the upper bound PSPACE for the power of families of AM 0 systems with rules of types (d), (e), (f) optimal? What is their relation to the class P #P? 4.2 Time-Free P Systems with Active Membranes The results presented so far in this section rely often on the assumption that each rule of a P system is executed in exactly one time step. This is rather strong condition and efforts have been made to relax it. A timing function has been defined, assigning to each rule a number of steps needed to execute it. Time-free P systems were defined in [8], in the sense that the result of computation does not depend on the choice of the timing function. The need for time-free solutions to computationally hard problems was formulated in [7]. The first time-free solution to such a problem (particularly, the SAT problem) by a family of P systems with active membranes was presented in [51]. The convention was adopted that only those steps when at most one rule starts its execution are counted to the final time of the computation. Since then, also other types of membrane systems (with proteins on membranes, tissue P systems) were shown to provide time-free solutions to hard problems including PSPACE-complete ones, see the next sections for details. However, an exact characterization of the class of problems solvable by families of P systems with active membranes in a time-free manner remains open.

96 96 P. Sosík 4.3 Non-confluent P Systems with Active Membranes Until now we have only assumed confluent P systems, i.e., such that a given P system (with a given input) has only rejecting or only accepting computations, even if the system itself can be non-deterministic, i.e., it can reach the same result by various computational paths. In this subsection we consider non-confluent P systems with active membranes, where the same P system can produce both accepting and rejecting computations. The input is accepted if there is at least one accepting computation, as in the case of non-deterministic Turing machine. A few results are known about the power of families of non-confluent P system with active membranes or non-confluent P systems in general). The paper [40] confirms the rather intuitive idea that non-confluent P systems should be able to solve NP-complete problems in polynomial time even without membrane division. Indeed, the paper also demonstrates that the class of problems solvable by this kind of non-confluent P system without membrane division in polynomial time is exactly the class NP. Recently, authors of [25] demonstrated that polynomially uniform families of non-confluent P systems with active membranes of depth 1, using only object evolution, send-out communication and elementary membrane division rules (hence without dissolution and non-elementary division rules) can solve in polynomial time all problems in PSPACE. Exact characterization the computational power of such non-confluent families of P systems with active membranes remains open. Since the space used by such a P system during its computation is at most exponential, the survey paper [30] indicates the upper bound EXPSPACE on their power. Open Problem 3 Does the class PSPACE characterize the power of uniform families of non-confluent P systems with active membranes (eventually, under some restrictions)? 5 P systems with Symport/Antiport Rules and Membrane Division Membrane systems with symport/antiport rules were introduced in [44] as an alternative to P systems manipulating and changing objects. Their basic philosophy is to allow just a regulated transport of objects through membranes, but not their change. Unlike rules of types (b), (c) in P systems with active membranes, there can be more than one object participating at each rule. Formally, symport rules are of the form (u, in) or (u, out), and move objects of multiset u through a membrane; antiport rules are of the form (u, out; v, in), and move the objects of multiset u outside a membrane while moving the objects of multiset v inside. After demonstrating the computational universality of P systems with symport/antiport in [44], a sequence of papers appeared, studying effects of various

97 P Systems Attacking Hard Problems Beyond NP 97 restrictions or generalizations to this kind of P systems. A membrane computing model combining symport/antiport with membrane division was introduced in [48]. The authors demonstrated a (uniform) linear time solution to the NPcomplete problem, subset sum, by using division rules for elementary membranes and communication rules of length at most 3. Furthermore, when division rules for non-elementary membranes are allowed, these P systems can efficiently solve the PSPACE-complete problem QSAT in a uniform way. Hence, this type of P systems also aspires to the status of second class computers and we conjecture that PSPACE is also an upper bound on their computational efficiency in polynomial time. It remains another open problem whether these P systems with only elementary membrane division characterize the class P #P, similarly as in the case of P systems with active membranes. 6 P systems with Proteins on Membranes P systems using with active objects (proteins) on membranes were studied in several different types (see [] for an overview). The type most relevant for our survey on solutions to hard computational problems is the model developed at Louisiana Tech. University (so-called Ruston model), combining membrane systems and brane calculi [6]. The model was introduced in [42] and [43] where insoluble membrane proteins have been modelled in the broad area of membrane computing. Besides the crucial role of membrane proteins in cells, further research motivation is the fact that maximally parallel processing of different species of molecules in membrane systems was not realistic. The Ruston model limits this parallelism through the use of a limited number of trans-membrane proteins (protein channels). Definition 4. A P system with proteins on membranes and membrane division is a tuple Π = (O, P, µ, w 1 /z 1,..., w m /z m, E, R 1,..., R m, i o ), where m is the degree of the system (the number of membranes), O is the set of objects, P is the set of proteins (with O P = ), µ is the membrane structure (a rooted tree) with membranes labelled uniquely 1,..., m w 1,..., w m are (strings representing the) multisets of objects present in the m regions of the membrane structure µ, z 1,..., z m are multisets of proteins present on the m membranes of µ, E O is the set of objects present in the environment (in an arbitrarily large number of copies each), R 1,..., R m are finite sets of rules associated with the m membranes of µ, and i o is the label of the output membrane. Both proteins and objects can be manipulated via rules associated with membranes. In all of these rules, a, b, c, d are objects, p, p are proteins, and i is a label of a membrane.

98 98 P. Sosík Type Rule Effect 1cp [ i p a] i [ i p b] i modify an object, a[ i p ] i b[ i p ] i but not move 2cp [ i p a] i a[ i p ] i move an object, a[ i p ] i [ i p a] i but not modify 3cp [ i p a] i b[ i p ] i modify and move a[ i p ] i [ i p b] i one object 4cp a[ i p b] i b[ i p a] i interchange two objects interchange and modify 5cp a[ i p b] i c[ i p d] i two objects The abbreviation cp means change protein. If p p in every rule, then we denote them ncpp, n = 1, 2, 3, 4, 5 (pure change-protein rules). If, on the other hand, p = p in every rule, then we denote the rules by nres, n = 1, 2, 3, 4, 5, where res stands for restricted. An intermediate case is to allow at most two states for each protein, p, p, and each rule must change from p to p and back (like in the case of bistable catalysts). The rules are used in the non-deterministic maximally parallel way. However, at each step each object and each protein can be involved in application of at most one rule. The membranes are not considered as involved in the rule applications hence the same membrane can appear in any number of rules at the same time. P systems with proteins on membranes have been shown to be computationally universal (in the Turing sense) for a broad range of parameters including the number of membranes, number of proteins on each membrane and types of applied rules [17, 34, 42]. To divide a membrane, we use the following type of rule (referred to as type 6 ), where p, p, p are proteins (possibly equal): [ i p ] i [ i p ] i [ i p ] i Membrane i can be non-elementary. The rule doesn t change the membrane label i and instead of one membrane, at next step, we will have two membranes with the same label i and the same contents (except for p and p ) replicated from the original membrane. The first solution to the PSPACE-complete problem QSAT by non-uniform families of these P systems was shown in [56]. The same paper also establishes PSPACE as an upper bound of the polynomial-time computations of this kind of P systems, confirming their status as second class computers. Interestingly enough, an analogous result holds also for time-free mode (see Section 4.2 for definition). The authors of the strongest recently known result [49] construct a uniform family of P systems with proteins on membranes solving the problem QSAT in polynomial time in a time-free manner. The paper [49] also provides a useful list of references to the broader area of P systems with proteins on membranes. Open Problem 4 Under what restrictions on the form of rules would families of P systems with proteins on membranes still be able to solve QSAT in polynomial time? Particularly, what if only rules of type res are allowed?

99 P Systems Attacking Hard Problems Beyond NP 99 Open Problem 5 What is the computational power of families of these P systems without membrane division? Do they characterize the class P, and what happens under various restrictions on the form of rules? Open Problem 6 What if only elementary membrane division is allowed? 7 Tissue P systems The basic idea of tissue P systems is the principle of symport and antiport [44]. Symport rules move objects across a membrane together in one direction, whereas antiport rules move objects across a membrane in opposite directions. In tissue P systems these two variants were unified as a unique type of rule manipulating a certain number of objects. From the original definitions of tissue P systems [28, 29], several research lines have been developed and other variants have arisen (see, for example, [3, 5, 13, 16, 18]). Definition 5. A tissue P system of degree q 1 is a tuple where: Π = (Γ, E, M 1,..., M q, R, i out ), 1. Γ is a finite alphabet whose elements are called objects; 2. E Γ is a finite alphabet of objects initially in the environment of the system in inexhaustibly many copies each, and 0 is the label of the environment; 3. M 1,..., M q are strings over Γ, representing the finite multisets of objects placed in the q cells of the system at the beginning of the computation; 4. R is a finite set of communication rules of the form (i, u/v, j), for i, j {0, 1, 2,..., q}, i j, u, v Γ, and the length of the rule is uv > 0; 5. i out {0, 1, 2,..., q} is the output cell. When applying a rule (i, u/v, j), the objects of the multiset represented by u are sent from region i to region j and, simultaneously, the objects of the multiset v are sent from region j to region i. A communication rule (i, u/v, j) is called a symport rule if u = λ or v = λ. A symport rule (i, u/λ, j), with i 0, j 0, provides a virtual arc from cell i to cell j. A communication rule (i, u/v, j) is called an antiport rule if u λ and v λ. An antiport rule (i, u/v, j), with i 0, j 0, provides two arcs: one from cell i to cell j and another one from cell j to cell i. Thus, every tissue P systems has an underlying directed graph whose nodes are the cells of the system and the arcs are obtained from communication rules. Recognizer variant of tissue P system is defined analogously as in the previous sections. We denote the class of recognizer tissue P systems by T C. The following result was shown by simulation of basic transitional P systems in [11]: P = PMC T C.

100 100 P. Sosík 7.1 Tissue P Systems with Cell Division Tissue P system with cell division is based on the cell-like model of P systems with active membranes [47]. The biological inspiration is the following: alive tissues are not static network of cells but new cells are produced by membrane division in a natural way. In these models, the cells are not polarized; the two cells obtained by division have the same labels as the original cell, and if a cell is divided, its interaction with other cells or with the environment is blocked during the division process. Division rules: [a] i [b] i [c] i, where i {1, 2,..., q} and a, b, c Γ, and i i out. In reaction with an object a, the cell i is divided into two cells with the same label; in the first cell the object a is replaced by b; in the second cell the object a is replaced by c; the output cell i out cannot be divided. For each natural number k 1, we denote by TDC(k) the class of recognizer tissue P systems with cell division and communication rules of length at most k. We denote by TDC the class of recognizer tissue P systems with cell division and without restriction on the length of communication rules. Obviously, TDC(k) TDC for all k 1. The following result in [14] states that only problems in P can be solved by families of recognizer tissue P systems with the rules of length 1, on one hand: P = PMC T DC(1). (1) On the other hand, [41] places a tight borderline between efficiency and nonefficiency in the sense of the length of rules: NP co-np PMC T DC(2). (2) Finally, the exact characterization of the power of these P systems with unlimited length of rules is given in [24]: PMC T DC (4) = PMC T DC = P #P. (3) Recall that P #P denotes the class of problems solved by polynomial-time Turing machines with oracles for counting problems. This result remains unchanged when one considers either deterministic or confluent P systems, and also regardless whether the family of tissue P systems is uniform or semi-uniform. Finally, let us mention that several time free solutions to NP-complete problems by tissue P systems with cell division have been published. As an example we cite [26, 50] demonstrating time-free solutions to Maximum Clique Problem, Hamilton Path Problem and Subset Sum. It remains open whether tissue P systems with cell division can solve problems from P #P in a time-free manner, or whether analogous time-free solutions are possible with the operation of cell separation described in the next section.

101 7.2 Tissue P Systems with Cell Separation P Systems Attacking Hard Problems Beyond NP 101 The operation of membrane separation was introduced in [33]. It is motivated by the fact that during a cell division, its content is split between the two descendants. To imitate this behaviour within the framework of P systems, we partition the working alphabet Γ into two non-empty parts, that is, Γ = Γ 1 Γ 2, Γ 1, Γ 2, Γ 1 Γ 2 =. Separation rules: [a] i [Γ 1 ] i [Γ 2 ] i, where i {1, 2,..., q} and a Γ, and i i out. In reaction with an object a, the cell i is separated into two cells with the same label; at the same time, object a is consumed; the objects from Γ 1 are placed in the first cell, those from Γ 2 are placed in the second cell; the output cell i out cannot be separated. As in the previous section, we introduce the notation TSC(k) or TSC for the class of recognizer tissue P systems with cell separation and communication rules of length at most k, or without restriction, respectively. The known results are summarized as follows: P = PMC T SC(2) (characterization of P, [35]); NP co-np PMC T SC(3) ((in)tractability borderline, [35]); PMC T SC = P #P (upper bound on the computing power, [24]). It is interesting that, although the frameworks of tissue P systems with cell division and cell separation is rather similar, the borderline between tractability and intractability is placed differently. Further variants of tissue P systems with cell division/separation and their computational power were studied, e.g., in [27]. Open Problem 7 What happens if only symport (respectively, only antiport) rules are allowed in tissue P systems with cell division or cell separation? The results in [24] improve previously known upper bound PSPACE on the power of tissue P systems reported in [52, 53]. The authors of [24] conjecture that the class PSPACE can still be reached when one assumes non-deterministic non-confluent tissue P systems, that is, allowing the same tissue P system to have both accepting and rejecting computations, with the final result determined by the existence of at least one accepting computation. A similar result has been already shown for the case of non-confluent active membrane systems of depth 1, see Sec. 4.3 for details. 8 Conclusion We have addressed a sequence of problems and results connected with (uniform families of) powerful membrane system able to solve in polynomial time problems

102 102 P. Sosík belonging to classes beyond NP co-np. The models in this survey include P systems with active membranes, P systems with proteins on membranes, and tissue P systems, all of them manipulating objects from a certain alphabet. A different category of P systems capable of solving hard problems with pre-computed exponential resources are spiking neural P systems using anonymous spikes as their data medium [15]. We focused on polynomially uniform families of recognizer P systems working in polynomial time and often using the strategy of trading space for time. Details of the cited results are mostly omitted, and rather their synopsis is given which would allow to compare the power of various features used in the studied P system models. Given that new variants of P systems with novel operations continuously emerge and are studied, the reader can find some of those not included in this survey in the cited references. To briefly summarize, it is now generally accepted that the class of problems decided by polynomially uniform families of confluent P systems with the operation of non-elementary membrane division in polynomial time equals PSPACE. When only elementary membrane division is allowed, the resulting class of problems (or its upper bound) is often P #P. Although an unlimited number of cells can be produced by cell division, separation or creation, they cannot be accessed individually unless nondeterminism is involved as in [4]. Intermediate classes can be obtained with a limited depth of the membrane structure. Many results in this area still remain open and we have listed some of them. We focused mostly on the (families of) P systems with restrictions on their computational time. Several results are known also when comparing computational space used by P systems and Turing machines. For instance, it is known that P systems with active membranes working in polynomial space characterize exactly the class PSPACE. We refer the reader to [30] for more details and a recent survey on computational complexity approaches to P systems. Acknowledgements The author is grateful to Alberto Leporati, Enrico Porreca and Luca Manzoni for their comments on several results cited in this paper. This work was supported by the Ministry of Education, Youth and Sports Of the Czech Republic from the National Programme of Sustainability (NPU II) project IT4Innovations Excellence in Science - LQ1602, and by the Silesian University in Opava under the Student Funding Scheme, project SGS/13/2016. References 1. Alhazov, A., Martín-Vide, C., Pan, L.: Solving a PSPACE-complete problem by P systems with restricted active membranes. Fundamenta Informaticae 58(2), (2003)

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107 Open Problems A Dozen of (Meta/Mega?) Research Topics Gheorghe Păun Institute of Mathematics of the Romanian Academy PO Box 1-764, Bucureşti, Romania, and Department of Computer Science and Artificial Intelligence University of Sevilla Avda. Reina Mercedes s/n, Sevilla, Spain gpaun@us.es, curteadelaarges@gmail.com Summary. This note considers three basic research directions in membrane computing characterizations of the computing power of Turing machines, computing more than Turing machines, efficiency (solving computationally hard problems in a feasible time) by basic classes of P systems (cell and tissue multiset rewriting systems, symport/antiport systems, spiking neural P systems, numerical P systems). (Types of) Results reported in the literature are briefly mentioned, several unsolved cases are pointed out, and directions of further research are proposed. 1 A Synthesis Table Let us start abruptly with the table below, a map (and summary) of the whole discussion. Here is a dictionary of abbreviations: CS = context-sensitivity features, λ = erasing possibilities, acc = acceleration, 2 x space = exponential space (used in a

108 108 Gh. Păun time-space trade-off), precomp = using pre-computed resources, mcre = membrane creation, div = membrane division, sep = membrane separation, p = promoters, i = inhibitors, we = what else?, bud = budding, E = important role of the regular expression in rules of SN P systems, expl = explanations needed, +, = using only these operations. 2 A Dozen of Problems Only one dozen, in order to get a nice title... otherwise the list can be much longer. Moreover, we do not have well formulated, crisp open problems, but, in general, research directions which need to be explored, starting with the technical formulation of ideas and of basic open problems. 2.1 Equivalence with Turing Machines The column Power RE refers to characterizations of the Turing computing power. In formal language theory it is well-known the fact that context-sensitivity plus erasing ensure universality (for instance, each RE language is the projection of a context-sensitive language). That is why two sub-columns are mentioned, marked with CS and λ. In basic membrane computing, that is, dealing with multiset processing in cell-like, tissue-like or spiking neural P systems, we have a large number of universality results (strictly speaking, computational completeness results). We have mentioned explicitly symport/antiport systems, because, for other issues, this class should be considered separately. For all these types of P systems, contextsensitivity and erasing are provided for free by the definition (by the biochemistry of the cell erasing can be obtained by sending objects into the environment or by making them unable to evolve/react further). However, also the numerical P systems are known to be universal, and here the explanation should be different, as we do not deal with multiset rewriting. This is the first question, Q1 in the table: where lies the power of numerical P systems (with polynomials with integer coefficients in the programs) and how this depends on the production functions (enzymatic/non-enzymatic, using only some of the four operations +,,,, etc.) Note the crucial importance of operations from the efficiency point of view, see [6]. 2.2 Passing Beyond The Turing Barrier The second column has the header Power > RE, with the meaning that we deal here with hypercomputing ideas and results (in spite of the fact that Martin Davis considers hypercomputing a myth, see, e.g., [3]). Two sub-columns were indicated: acc and others. The first one refers to the idea of acceleration, explored in [2] as a

109 A Dozen of Research Topics 109 way to compute the uncomputable in terms of cell-like P systems endowed with rules for membrane creation. Ways to accelerate other classes of P systems were not proposed and examined yet, and the cases of symport/antiport, spiking neural and numerical P systems are associated with questions Q2, Q3, Q4, respectively. Problem Q5 is a generic notation for the general question of exploring for P systems other ideas known in the hypercomputation area some surveys of such ideas can be found, e.g., in [10], [11], [16]. Actually, besides acceleration, another idea was also explored in membrane computing as a way to get hypercomputations, namely evolutionary lineages of P systems, see [15] and its references; I would include also this idea in others. In particular, it would be interesting/intriguing to prove hypercomputing results for numerical P systems, without using tricks (a term of Martin Davis), for instance, without using real numbers in the functions involved in the systems. 2.3 Efficiency Basically, this is about the possibility of solving computationally hard problems, typically, NP-complete problems, in a feasible time, typically, in a polynomial time. Plenty of results of this type are known in membrane computing, most of them based on trading-off time for space: a way to create an exponential working space during the computation is considered, and this space is then used, in a massively parallel manner, to solve (decidability, but also numerical) problems in a polynomial time. Two ideas were mainly used: dividing membranes (We include here also separation of membranes and budding of neurons, creating membranes, even the replication of string objects) and using pre-computed resources. The latter idea was considered only for spiking neural P systems what about other classes of P systems (problem Q7 in the table)? These two ideas were not considered yet for numerical P systems (hence the problems Q6 and Q8), with an important detail: these systems were proved already to be efficient, in the case of using all the four operations +,,,, [6], for enzymatic numerical P systems, even without using a space-time trade-off. This is really interesting: numerical P systems have an intrinsic efficiency, they do not need to use an exponential working space. Which is the explanation of this fact, where this intrinsic efficiency is placed? The answer seems to be immediate: in each step, one performs complex computations: evaluating production functions, computing the parts of the current production which has to be distributed, the distribution itself to the target variables. All these are done in no time, without any cost. Then, of course, questions Q6 and Q8 should be adequately formulated for classes of numerical P systems which are not covered by the results in [6]: nonenzymatic, using only a subset of the four operations, etc. The intrinsic efficiency of numerical P systems suggests a fruitful research direction: do also other classes of P systems possess a similar intrinsic efficiency?

110 110 Gh. Păun Otherwise formulated: do they perform complex computations during the steps which are, by definition, supposed to last only one time unit, irrespective how complex they are? If this is not the case, can we find such features and hide them in the definition in order to get efficiency without using the basic trick of trading-off time for space? For basic multiset rewriting cell-like P systems and also for other types of P systems, the Milano theorem tells us that this is not possible, hence we have to consider more sophisticated P systems. One candidate are spiking neural P systems, where in each step we check the membership of multisets to sets of multisets described by regular expressions (of arbitrary complexity), but, for the deterministic case, again a Milano theorem was proved, see [8]. Expected enough, in the nondeterministic case the efficiency is obtained, [9]. What one can add to (rules of) P systems in order to make them efficient in an intrinsic way? the question is denoted by Q9 in the table, for all classes of P systems. Promoters and inhibitors make the computation steps more complex, as we have to check, in no time, whether or not a given multiset is a submultiset of the current multiset of objects present in a compartment. Is this enough? (Probably not.) What else to check, what other predicate(s) can we associate with the rules? There appears here also a more general problem, which is not mentioned in the table, problem Q10. Up to now we have discussed about efficiency (in [13] I call it fypercomputing, following the model of hypercomputing, replacing the initial h with f, from fast). However, we can distinguish three levels of efficiency: strong fypercomputing = solving NP-complete problems in (deterministic) polynomial time, very strong fypercomputing = solving PSPACE problems in polynomial time, and weak efficiency = solving problems in P in a faster way than by any known sequential algorithm. The last class might contain problems of a practical interest, for which the best sequential algorithm is polynomial, but with large parameters (coefficients, degrees, number of variables). For instance, it would be nice to have examples of mappings with a practical significance/usefulness which could be computed in a faster way (in terms of parameters involved in the polynomial estimating the time) by means of numerical P systems. Due to the intrinsic parallelism of these systems, it is plausible that such mappings exist. The last sub-column of the efficiency column proposes (Q11) to look for other ideas, different from those mentioned above, able to speed-up computations in P systems. A large space for imagination, for the proverbial creativity of PhD students... Similarly global, the last row of the table, marked with Q12, asks about extending all the previous results and questions to other classes of P systems, for instance, to kernel P systems (see the survey [5] and the references therein), to polymorphic P systems (see the survey [1] and the references therein), to conformon P systems, [4], etc.

111 A Dozen of Research Topics Final Remarks Of course, we have supposed here that the reader is familiar with membrane computing basic notions and results, as well as with basic references, including the handbook [14], the bibliography provided in the membrane computing web page, [17], the bibliographies presented in the Bulletin of IMCS, [18]. It is also understood that this note is only a challenge to carry out a more systematic and analytic study of the power and efficiency of P systems, starting from the question why are P systems powerful/efficient, what is present there by definition, used or not yet used in proofs, what we can add in a natural, minimalistic and bio-realistic way in order to increase the power and/or the efficiency, with the important details that power refers to characterizing the power of Turing machines but also of passing beyond that, while efficiency can also be classified in at least three levels, as suggested above (strong, very strong, and weak). There still exists plenty of work to be done in membrane computing... References 1. A. Alhazov, R. Freund, S. Ivanov: Polymorphic P systems. A survey. Bulletin of IMCS, 2 (December 2016), C. Calude, Gh. Păun: Bio-steps beyond Turing. CDMTCS Research Report 226, Univ. of Auckland (November 2003), and BioSystems, 77 (2004), M. Davis: The myth of hypercomputation. In C. Teuscher, ed., Alan Turing: Life and Legacy of a Great Thinker, Springer, 2004, P. Frisco: Computing with Cells. Advances in Membrane Computing. Oxford Univ. Press, M. Gheorghe: A survey of kernel P systems. Bulletin of IMCS, 3 (June 2017), A. Leporati, G. Mauri, A.E. Porreca, C. Zandron: Enzymatic numerical P systems using elementary arithmetic operations. Pre-proc. 14th Intern. Conf. on Membrane Computing, Chişinău, 2013, A. Leporati, A.E. Porreca, C. Zandron, G. Mauri: Improving universality results on parallel enzymatic numerical P systems. Proc. 11th Brainstorming Week on Membrane Computing, Sevilla, 2013 (L. Valencia-Cabrera et al., eds.), Fenix Editora, Sevilla, 2013, A. Leporati, C. Zandron, C. Ferretti, G. Mauri: On the computational power of spiking neural P systems. Fifth Brainstorming Week on Membrane Computing, Sevilla, 2007 (M.A. Gutiérrez-Naranjo et al., eds.), Fenix Editora, Sevilla, 2007, A. Leporati, C. Zandron, C. Ferretti, G. Mauri: Solving numerical NP-complete problems with spiking neural P systems. Membrane Computing. 8th Intern. Workshop, Thessaloniki, Greece, June 2007 (G. Eleftherakis et al., eds.), LNCS 4860, Springer, Berlin, 2007, T. Ord: Hypercomputation: Computing More Than the Turing Machine. Honours Thesis, Dept. of CS, Univ. of Melbourne, September T. Ord: The many forms of hypercomputation. Applied Mathematics and Computation, 178 (2006),

112 112 Gh. Păun 12. Gh. Păun: Computing with membranes. J. Comput. Syst. Sci., 61 (2000), (see also TUCS Report 208, November 1998, Gh. Păun: Towards fypercomputations (in membrane computing), Languages Alive. Essays Dedicated to Jürgen Dassow on the Occasion of His 65 Birthday (H. Bordihn, M. Kutrib, B. Truthe, eds.), LNCS 7300, Springer, Berlin, 2012, Gh. Păun, G. Rozenberg, A. Salomaa, eds.: The Oxford Handbook of Membrane Computing. Oxford University Press, P. Sosík, O, Valík: On evolutionary lineages of membrane systems. Membrane Computing. Proc. of WMC 2005, Vienna, Austria, July 2005 (R. Freund et al., eds.), LNCS 3850, Springer, Berlin, 2006, A. Syropoulos: Hypercomputation: Computing Beyond the Church-Turing Barrier. Springer, Berlin, The P Systems Website: Bulletin of International Membrane Computing Society: computing.net/imcsbulletin/index.php.

113 Dualistic Open Problems in Membrane Computing Luca Manzoni and Antonio E. Porreca Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi di Milano-Bicocca Viale Sarca 336/14, Milano, Italy {luca.manzoni,porreca}@disco.unimib.it Summary. We present some high-level open problems in the complexity theory of membrane systems, related to the actual computing power of confluence vs determinism, semiuniformity vs uniformity, deep vs shallow membrane structures, membrane division vs internal evolution of membranes. 1 Confluence vs determinism P systems solving decision problems (recogniser P systems [14]) are usually required to be confluent [14] rather than strictly deterministic. That is, they are allowed to have multiple computations, as long as all of them agree on the final result, acceptance or rejection. This sometimes simplifies the presentation of some algorithms. For instance, a classic membrane computing technique [12, 19] consists in generating all 2 n truth assignments of n variables by using membrane division rules of the form [x i ] h [t i ] h [f i ] h, with 1 i n. The membrane division is triggered separately in each membrane with label h by one of the objects x i, nondeterministically chosen at each computation step. Irrespective of all such nondeterministic choices, the end result is invariably a set of 2 n membranes, each containing a different truth assignment. Notice, however, that this kind of nondeterminism can be completely avoided by serialising the generation of truth assignments for each variable: first all instances of x 1 trigger the division, then all instances of x 2, and so on. This can be achieved by adding an extra subscript to each object, which counts down to zero and only then starts the division process. This work was partially supported by Fondo d Ateneo 2016 of Università degli Studi di Milano-Bicocca, project 2016-ATE-0492 Sistemi a membrane: classi di complessità spaziale e temporale.

114 114 L. Manzoni, A.E. Porreca It is often the case that confluent nondeterminism can be avoided in a similar way, although this is usually proved by exhibiting a deterministic algorithm, rather than showing how to remove the nondeterminism from existing algorithms. It is then natural to ask whether this is indeed always the case, or if there exists a variant of P system where confluent nondeterminism is strictly stronger than determinism. For powerful enough P systems (e.g., able to efficiently simulate deterministic Turing machines, or stronger than that) we feel that the existence of such a variant would be very surprising, although there do exist confluent nondeterministic algorithms with no known deterministic version. For instance, the currently known proof of efficient universality (i.e., the ability to simulate any Turing machine with a polynomial slowdown) of P systems with active membranes using elementary membrane division [1] relies on a massive amount of nondeterministic choices performed at each simulated step; these are due to the fact that send-in communication rules cannot differentiate among membranes having the same label and electrical charges. 2 Semi-uniformity vs uniformity Recogniser P systems usually appear in families Π = {Π x : x Σ }, where each member of the family is associated to a string x and accepts if and only if x belongs to a given language. A family of P systems is usually required to be at least semi-uniform, that it, to have an associated Turing machine M with some suitable resource bound (usually, polynomial time) such that M on input x outputs a suitable encoding of Π x [14, 9]. A more restrictive condition on families of P systems is full-fledged uniformity [14, 9]: there exist two Turing machines F and E (again, usually with polynomial runtime) such that F on input n = x constructs a P system skeleton Π n, valid for all strings of length n, and E on input x produces a multiset w encoding x, which is then placed inside the input region of Π n, giving the P system Π x that computes the answer. It is known [10] that, for restrictive enough resource bounds, uniformity is weaker than semi-uniformity. However, when polynomial-time semi-uniform solutions to problems sometimes appear in the literature first, polynomial-time uniform solutions usually follow. We conjecture that polynomial-time uniformity and semi-uniformity do indeed coincide for powerful enough P systems, such as standard P systems with active membranes [12]. The idea here is that a semi-uniform family could be made uniform by simulating the semi-uniform portion of the construction, depending on the actual symbols of x Σ n, with the P system constructed for all strings of length n.

115 Dualistic Open Problems in Membrane Computing Membrane division vs internal evolution The computing power of a single membrane (for cell-like P systems) or cell (for tissue-like P systems) working in polynomial time usually has a P upper bound, as already proved by the Milano theorem [19]; the only way to exceed this bound would be to include really overly powerful rules (e.g., rules able to perform an NPcomplete task in a single step). The P upper bound can actually be achieved by having cooperative rewriting rules (even minimal cooperation [18, 17] suffices) or rules able to simulate them indirectly (e.g., active membrane rules with membrane charges [8]). Several techniques for simulating polynomial-time Turing machines using a single membrane are known [7]. Any additional power beyond P of models presented in the literature is due to membrane division, first exploited in order to solve NP-complete problems in polynomial time [12]. Membrane division enables us to create exponentially many processing units working in parallel; by using communication rules, these can synchronise and exchange information (this is the famous space-for-time trade-off in membrane computing). It is reasonable to expect that P system variants where the power of a single membrane working in polynomial item coincides with P can be standardised in a Turing machine normal form : each membrane performs a Turing machine simulation 2, and the communication and division rules implement a network, whose shape can be exploited to simulate nondeterminism, alternation, or oracle queries [7]. Notice that what previously described does not necessarily carry over to variants of P systems with weaker rules internal to the membranes, such as P conjecture systems [13, Problem F] (active membranes without charges), which do not seem able to simulate cooperation [2], or with communication restricted to a single direction, either send-out [5, 6, 16] or send-in only [15]. 4 Deep vs shallow membrane structures Let us now consider cell-like P systems with membrane division, for instance P systems with active membranes [12]. It has already been shown that the nesting depth of membranes (more specifically, the nesting depth of membranes with associated division rules, which we might call division depth) is one of the most influential variables when establishing the efficiency of these P systems. Indeed, P systems without membrane division (i.e., with division depth 0) are known to characterise the complexity class P in polynomial time [19]. At the other end of the spectrum, we have P systems with active membranes with elementary and non-elementary division rules (i.e., with polynomial division depth), which characterise PSPACE in polynomial time. 2 This can be trivially implemented by having each membrane simulate a Turing machine which, in turn, simulates the original membrane via a Milano theorem.

116 116 L. Manzoni, A.E. Porreca When only elementary membrane division is allowed (i.e., division depth 1), then the intermediate complexity class P #P is characterised in polynomial time [3, 6]. This class contains all decision problems solved by deterministic polynomialtime Turing machines with oracles for counting problems in the class #P [11]. It has been proved that moving from any constant division depth d to division depth d + 1 allows the P systems to simulate Turing machines with more powerful oracles [4]. We conjecture that this is in fact a proper hierarchy. This result would require proving the upper bounds corresponding to the known lower bounds. It also remains open to characterise the computing power of polynomial-time P systems with other division depths, such as O(log n). References 1. Alhazov, A., Leporati, A., Mauri, G., Porreca, A.E., Zandron, C.: Space complexity equivalence of P systems with active membranes and Turing machines. Theoretical Computer Science 529, (2014) 2. Gutiérrez-Naranjo, M.A., Pérez-Jiménez, M.J., Riscos-Núñez, A., Romero-Campero, F.J.: On the power of dissolution in P systems with active membranes. In: Freund, R., Păun, Gh., Rozenberg, G., Salomaa, A. (eds.) Membrane Computing, 6th International Workshop, WMC Lecture Notes in Computer Science, vol. 3850, pp Springer (2006) 3. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Simulating elementary active membranes, with an application to the P conjecture. In: Gheorghe, M., Rozenberg, G., Sosík, P., Zandron, C. (eds.) Membrane Computing, 15th International Conference, CMC Lecture Notes in Computer Science, vol. 8961, pp Springer (2014) 4. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Membrane division, oracles, and the counting hierarchy. Fundamenta Informaticae 138(1 2), (2015) 5. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Monodirectional P systems. Natural Computing 15(4), (2016) 6. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: The counting power of P systems with antimatter. Theoretical Computer Science (2017), in press 7. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Subroutines in P systems and closure properties of their complexity classes. In: Graciani, C., Păun, Gh., Riscos-Núñez, A., Valencia-Cabrera, L. (eds.) 15th Brainstorming Week on Membrane Computing, pp No. 1/2017 in RGNC Reports, Fénix Editora (2017) 8. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: A toolbox for simpler active membrane algorithms. Theoretical Computer Science 673, (2017) 9. Murphy, N., Woods, D.: The computational power of membrane systems under tight uniformity conditions. Natural Computing 10(1), (2011) 10. Murphy, N., Woods, D.: Uniformity is weaker than semi-uniformity for some membrane systems. Foundamenta Informaticae 134(1 2), (2014) 11. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1993) 12. Păun, Gh.: P systems with active membranes: Attacking NP-complete problems. Journal of Automata, Languages and Combinatorics 6(1), (2001)

117 Dualistic Open Problems in Membrane Computing Păun, Gh.: Further twenty six open problems in membrane computing. In: Gutíerrez- Naranjo, M.A., Riscos-Nuñez, A., Romero-Campero, F.J., Sburlan, D. (eds.) Proceedings of the Third Brainstorming Week on Membrane Computing. pp Fénix Editora (2005) 14. Pérez-Jiménez, M.J., Romero-Jiménez, A., Sancho-Caparrini, F.: Complexity classes in models of cellular computing with membranes. Natural Computing 2(3), (2003) 15. Valencia-Cabrera, L., Orellana-Martín, D., Martínez-del Amor, M.A., Riscos-Núñez, A., Pérez-Jiménez, M.J.: Restricted polarizationless P systems with active membranes: Minimal cooperation only inwards. In: Graciani, C., Păun, Gh., Riscos-Núñez, A., Valencia-Cabrera, L. (eds.) 15th Brainstorming Week on Membrane Computing, pp No. 1/2017 in RGNC Reports, Fénix Editora (2017) 16. Valencia-Cabrera, L., Orellana-Martín, D., Martínez-del Amor, M.A., Riscos-Núñez, A., Pérez-Jiménez, M.J.: Restricted polarizationless P systems with active membranes: Minimal cooperation only outwards. In: Graciani, C., Păun, Gh., Riscos- Núñez, A., Valencia-Cabrera, L. (eds.) 15th Brainstorming Week on Membrane Computing, pp No. 1/2017 in RGNC Reports, Fénix Editora (2017) 17. Valencia-Cabrera, L., Orellana-Martín, D., Martínez-del-Amor, M.A., Riscos-Núñez, A., Pérez-Jiménez, M.J.: Computational efficiency of minimal cooperation and distribution in polarizationless P systems with active membranes. Fundamenta Informaticae 153(1 2), (2017) 18. Valencia-Cabrera, L., Orellana-Martín, D., Martínez-del-Amor, M.A., Riscos-Núñez, A., Pérez-Jiménez, M.J.: Reaching efficiency through collaboration in membrane systems: Dissolution, polarization and cooperation. Theoretical Computer Science (2017), in press 19. Zandron, C., Ferretti, C., Mauri, G.: Solving NP-complete problems using P systems with active membranes. In: Antoniou, I., Calude, C.S., Dinneen, M.J. (eds.) Unconventional Models of Computation, UMC 2K, Proceedings of the Second International Conference, pp Springer (2001)

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119 With Theorem Provers and P Systems: How to Build a New Computational Universe Mike Stannett The University of Sheffield, Department of Computer Science Regent Court, 211 Portobello, Sheffield S1 4DP United Kingdom m.stannett@sheffield.ac.uk Summary. What will future generations be able to compute? Will they still be bound by the Church-Turing Thesis, or will they have developed devices more powerful than Turing machines? We recall the long-standing argument that the development of super-turing machines is a real possibility in the context of relativistic hypercomputation. However, it is difficult to resolve the matter one way or the other, due to a lack of fully formalised models of the proposed computing systems. We discuss one particular attempt to provide such a formalisation, with particular emphasis on the need to use interactive theorem provers, and the role that P systems will play in this process. 1 Introduction Standard computation theory tells us that effective computation is faithfully modeled by Turing machine behaviours, but the problem with this thesis is obvious: what one considers to be effective depends on the resources that are available. If the only tools you have are straightedge and compass, you might reasonably conclude that numbers like π are not effectively computable, but add even a basic calculator and you can square the circle with ease. For mathematicians in the 1930s, the practical resources available for computation were thought to be understood, but we need to remember the wider historical context. All practical devices are physical objects, so we need to consider how physics itself was understood when the pioneers of computer science first developed their models in the 1930s and 40s [Kle50]. The two main theories underpinning modern physics, quantum theory and general relativity, were of course already established. But Hubble had only recently shown that (1925) other galaxies exist beyond the Milky Way, and (1929) that the universe is expanding [Hub26, Hub29]. The idea that the universe might have started with a Big Bang wasn t widely accepted until the discovery of cosmic microwave background radiation in 1964 [PW65], and the prediction of black holes was considered an oddity until the discovery of neutron stars

120 120 M. Stannett in the mid-60s suggested that even-more-extreme gravitationally collapsed compact objects might really exist [HO65, Shk67, HBP + 68]. It is unsurprising, therefore, that the idea of using spacetime singularities to boost computational power only emerged in the 1990s. The story with regard to quantum theory is similar. It was only in the late 1920s that the wave nature of the electron was confirmed [Nob65], and experimental verification of quantum entanglement had to wait until 1982 [ADR82], roughly the same time that quantum computing was first being mooted [Ben80, Man80, Fey82]. Since the inception of the field nearly 40 years ago, quantum computing has become one of the hottest research areas in computation theory, and its significance is already well known. In this paper, I ll focus instead on a long-running project looking at how and why cosmological factors also matter: in normal circumstances here on Earth, computation may well be what we think it is, but with access to more (and still physically feasible) resources, we should be able to do much better in future. 2 Proving the relativity of computational power Cosmology may seem a bit unrelated to computation theory, but it really does matter. Complexity theory, for example, takes it for granted that there s always time to perform one more step in a computation if you need to. If you can t do that, then asymptotic complexity means nothing, because asymptotic behaviour only makes sense in a world where computations can run for exactly as long as they need to. But recent experiments have shown that the expansion of the universe is accelerating, and this has led to various theories as to how the universe will eventually end [Woo16]. In the Big Rip scenario, for example, all matter will have been ripped apart in around 22 billion years time. In the Big Freeze, energy eventually becomes so thinly distributed that nothing can ever happen. But the key point in these and other scenarios is the central agreement that the universe as we know it will eventually end. If these physicists are correct, every computation will eventually stop, because it will no longer have a universe to support it in which case, asymptotic complexity has no physical meaning, it is simply a mathematical abstraction. Fortunately for complexity theorists, other aspects of general relativity (GR) stand ready to come to their rescue. As long ago as 1992, Hogarth suggested that some models of GR allow an observer to view an eternity in finite time [Hog92], which implies their ability to observe the outcome of a supertask (a task involving infinitely many steps). This theme, the computational power of systems in general relativistic spacetimes, has been studied by myself and others [Sta13, Sta15]; Andréka, Németi and their colleagues [EN02, NA06, ANS12] have investigated concrete schemes that exploit relativistic effects to perform hard computations (or rather, hypercomputations effective computations that go beyond what is possible using Turing machines [Sta06]).

121 How to Build a New Computational Universe 121 In my opinion, these studies provide compelling evidence that future civilisations may well be able to develop devices that outperform Turing machines, not just in terms of execution times, but in absolute terms. They will allow effective computation of functions that are currently considered uncomputable. Unfortunately, compelling evidence isn t quite the same as irrefutable mathematical proof, and it is this which we hope eventually to provide. 3 Theorem provers and relativistic hypercomputation The problem with existing studies into relativistic computation is the same problem that affects physics proofs in general; they are only as good as the humans who developed and reviewed them. Given that many proofs published in good faith have subsequently been found to contain errors, we need stronger guarantees that our own proofs, in particular, are sound. An added complication comes from the way physics is conducted and reported. A widespread lack of formal logical foundations means that physicists rely on shared common knowledge when reporting and interpreting results, and the actual assumptions underpinning the logical deductions made in these reports are rarely, if ever, stated in a form that is both explicit and formal. This raises the twin spectres of ambiguity and inconsistency. If assumptions aren t made explicit, how can we know whether two different authors have made compatible or conflicting assumptions? If we combine findings from their papers, how do we know we re not implicitly relying on embedded sets of assumptions that are actually inconsistent with one another? In order to overcome this problem, we have been looking at ways to axiomatize the physical systems and procedures involved in executing these proposed hypercomputations. Our goal, initiated in joint work with Németi [SN14, SN16], is to demonstrate that results in relativity theory can be deduced formally using an interactive theorem prover (in our case, Isabelle/HOL). The use of such theorem provers is, I believe, central to the main project, because it forces us to make every assumption explicit. One cannot rely on common knowledge in a machine proof, you have to cross every t and dot every i. An unexpected complication but one that offers scope for many open problems is that even subtle changes in ones axioms can change the expected behaviours of objects and observers in relativistic spacetimes. This has long been a topic of interest to the Andréka-Németi group, and I have been fortunate to work with them from time to time (in particular, I d like to acknowledge Judit X. Madarász and Gergely Székely). We have found not only that long-treasured beliefs, like if you could travel faster than light you would be able to time-travel, are false [AMN + 14], but that even the foundational principles of relativity theory are intrinsically ambiguous [MSS17].

122 122 M. Stannett 4 Modelling mobility through spacetime using P systems A common feature of relativistic hypercomputation schemes is the separation of observer and (standard) computer. This allows gravitational time-dilation effects to be exploited, whereby a program runs forever as experienced by the computer, but the observer sees everything run to completion in finite time. Central to capturing this scenario formally is the ability to model motion through spacetime along a given trajectory. Axiomatizing this capability is not straightforward, since in addition to axiomatizing what we mean by observer and computation, we need to capture the idea that computation is localised to the computer, but at the same time allow the results of computation to be influenced by the global structure of spacetime. One approach might be to use a grid of cellular automata, and describe the computer s motion in terms of computations jumping from one cell to the next along the required trajectory. However, this isn t satisfactory, because the presence of the computer (a massive object) will distort the underlying grid, and might even change its global structure. Modelling these interactions may be possible, but we envisage it being quite a complex task. Following a suggestion by Marian Gheorghe (personal communication) it seems a better approach would be to use P systems. The strategy is quite straightforward: the computation is modelled in terms of P system computation, with the P system itself moving as required through spacetime. This ensures that computation is localised within known compartments, without any requirement to specify any kind of global grid structure. The only complication is that we still need to decide how to model the P system s changing location. To overcome this problem, we need to clarify what we mean by location. We cannot use a coordinate system, because we cannot guarantee a priori whether the spacetimes we are interested in will be coordinatizable. This suggests we need to employ a mathematical structure that captures the notion of proximity without requiring an actual metric to be defined, and one approach is to use general topological spaces. We have made a small move in this direction (joint work with Csuhaj-Varjú, Gheorghe and Vaszil) by considering how topologies can be used to localise interactions [CGSV15] and control P system behaviours [CGS12], but a great deal still remains to be done. 5 Summary In summary, there is no obvious reason why relativistic phenomena, and in particular, spacetime singularities, could not potentially be harnessed by future generations. Possible candidates, like the supermassive black hole at the centre of our own galaxy, seem to be widely available in the night sky, so why shouldn t future scientists be able to travel to these singularities and incorporate them as structural components within experimental devices? Of course, they would need a good reason to do so, and one approach is to provide a cast-iron proof that harnessing the power of a singularity can lead to a step-change in computational

123 How to Build a New Computational Universe 123 power. Generating this proof will be time consuming and difficult, and require dedication. In particular, we need some way of modelling the controlled motion of localised computations through a spacetime which actively responds to those computations presence by changing its global structure. We can and should build a comprehensive axiomatic framework for machine-aided reasoning about the components involved in these systems, which combines key aspects of general topology, relativity theory, gravitational interactions, P systems and more. The time has come to reconsider, in computational high-definition, the new vistas that open up when we look to the limitless possibilities of the universe around us. References [ADR82] A. Aspect, J. Dalibard, and G. Roger. Experimental Test of Bell s Inequalities Using Time-Varying Analyzers. Phys. Rev. Lett., 49(25): , [AMN + 14] H. Andréka, J. X Madarász, I. Németi, M. Stannett, and G. Székely. Faster than light motion does not imply time travel. Classical and Quantum Gravity, 31(0):095005, [ANS12] H. Andréka, I. Németi, and G. Székely. Closed Timelike Curves in Relativistic Computation. Parallel Process. Lett., 22: , [Ben80] P. Benioff. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. Journal of Statistical Physics, 22(5): , [CGS12] E. Csuhaj-Varjú, M. Gheorghe, and M. Stannett. P systems controlled by general topologies. In J. Durand-Lose and N. Jonoska, editors, Unconventional Computation and Natural Computation (UCNC 2012), volume 7445 of LNCS, pages Springer, [CGSV15] E. Csuhaj-Varjú, M. Gheorghe, M. Stannett, and Gy. Vaszil. Spatially localised membrane systems. Fundamenta Informaticae, 138(1 2): , [EN02] G. Etesi and I. Nemeti. Non-Turing computations via Malament-Hogarth space-times. Int. J. Theor. Phys., 41: , [Fey82] R. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21(6): , [HBP + 68] A. Hewish, S. J. Bell, J. D. H. Pilkington, P. F. Scott, and R. A. Collins. Observation of a Rapidly Pulsating Radio Source. Nature, 217(5130):709, [HO65] A. Hewish and S. E. Okoye. Evidence of an unusual source of high radio brightness temperature in the Crab Nebula. Nature, 207(4992):59 60, [Hog92] M. Hogarth. Does General Relativity Allow an Observer to View an Eternity in a Finite Time? Foundations of Physics Letters, 5: , [Hub26] E. Hubble. Extragalactic nebulae. Astrophysical Journal, 64(64): , December [Hub29] E. Hubble. A relation between distance and radial velocity among extragalactic nebulae. PNAS, 15(3): , [Kle50] S. C. Kleene. Introduction to Metamathematics. van Nostrand, 1950.

124 124 M. Stannett [Man80] Yu. I. Manin. Vychislimoe i nevychislimoe [Computable and Noncomputable]. Sov. Radio, (in Russian). [MSS17] J. X. Madarász, G. Székely, and M. Stannett. Three Different Formalisations of Einstein s Relativity Principle. The Review of Symbolic Logic, 10(3): , September [NA06] I. Németi and H. Andréka. Can General Relativistic Computers Break the Turing Barrier? In Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30- July 5, 2006, volume 3988 of LNCS. Springer, [Nob65] Nobel Foundation. Clinton Joseph Davisson: The Nobel Prize in Physics In Nobel Lectures, Physics Elsevier, se/ physics/laureates/1937/davisson-bio.html. Accessed 23/11/2017. [PW65] A. A. Penzias and R. W. Wilson. A Measurement of Excess Antenna Temperature at 4080 Mc/s. Astrophysical Journal, 142: , [Shk67] I. S. Shklovsky. On the Nature of the Source of X-Ray Emission of SCO XR-1. Astrophysical Journal, 148(1):L1 L4, April [SN14] M. Stannett and I. Németi. Using Isabelle/HOL to verify first-order relativity theory. Journal of Automated Reasoning, 52(4): , [SN16] M. Stannett and I. Németi. No faster-than-light observers. In Archive of Formal Proofs. April [Sta06] M. Stannett. The case for hypercomputation. Applied Mathematics and Computation, 178(1):8 24, July [Sta13] M. Stannett. Computation and spacetime structure. International Journal of Unconventional Computing, 9(1 2): , [Sta15] M. Stannett. Towards Formal Verification of Computations and Hypercomputations in Relativistic Physics. In J. Durand-Lose and B. Nagy, editors, International Conference on Machines, Computations, and Universality (MCU 2015), volume 9288 of LNCS, pages Springer, September [Woo16] V. Woollaston. A Big Freeze, Rip or Crunch: how will the Universe end? WIRED, 10 October article/ how-will-universe-end. Accessed 23/11/2017.

125 Clock-freeness and Related Concepts in P Systems: from Definitions to Open Problems Artiom Alhazov 1,2, Rudolf Freund 3, Sergiu Ivanov 4,5, Linqiang Pan 2,6, and Bosheng Song 2 1 Institute of Mathematics and Computer Science Academy of Sciences of Moldova Academiei 5, Chişinău, MD-2028, Moldova artiom@math.md 2 Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan , China boshengsong@hust.edu.cn, lqpan@mail.hust.edu.cn 3 Faculty of Informatics, TU Wien Favoritenstraße 9 11, 1040 Vienna, Austria rudi@emcc.at 4 LACL, Université Paris Est Créteil Val de Marne 61, av. Général de Gaulle, 94010, Créteil, France sergiu.ivanov@u-pec.fr 5 TIMC-IMAG/DyCTiM, Faculty of Medicine of Grenoble, 5 avenue du Grand Sablon, 38700, La Tronche, France sergiu.ivanov@univ-grenoble-alpes.fr 6 School of Electric and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou , China P systems are a class of distributed parallel computing devices inspired by some basic features of biological membranes, which have the restriction that each rule is executed in exactly one time unit (a global clock is assumed for timing systems). However, a more realistic working mode of P systems, from biological point of view, is to consider systems without a time restriction on each rule since the execution time of biochemical reactions is very sensitive to environmental factors that could affect it in an unpredictable way. Here we first present the definitions of timed, time-free and clock-free P systems, and then some possible research directions are proposed. This note mainly consists of thoughts and comments on the notions and results elaborated in [2] followed by [1], also highlighting some questions still remaining open.

126 126 A. Alhazov et al. Timed P systems were introduced in [3], where an integer that represents the execution time is associated with each rule. Note that when a rule is started, then the occurrences of symbol-objects used by this rule are not available anymore for other rules. A particular class of timed P systems, called time-free P systems, was also defined in [3], where such P systems are shown to generate (or accept) the same family of vectors of natural numbers, independently of the value assigned to the execution time of each rule. For time-free P systems, we have the following facts [1]: 1. If a rule is simultaneously applied multiple times, then all instances finish simultaneously. 2. If a rule is simultaneously applied in different membranes with the same label, then all rules finish simultaneously. 3. If a rule is applied at different steps of a computation, then all instances last for the same amount of time. 4. If a rule is applied in different non-deterministic branches of a computation, then all instances last for the same amount of time. Time-free P systems were proved to be Turing complete. Moreover, the computational efficiency of time-free P systems in solving intractable problems has been investigated widely, see, e.g., [8], followed by [5] and then by [6, 7]. Another variant of timed P systems, and a stronger freeness property called clock-free P systems was proposed in [4]. In clock-free P systems, rule applications may last for arbitrary real periods of time, and even applications of the same rule may have different durations. Similarly, freeness in this case means that such a P system generates (or accepts) the same family of vectors of natural numbers, independently of the value assigned to the execution time of each rule application. 1 About Instances We need to discuss what a rule application, or rule instance, actually means. Informally, a rule applied to different instances of objects means different instances of the rule, even though different copies of the same objects (in the same membrane or in indistinguishable membranes) are normally thought of as indistinguishable in a configuration of a P system. Here, membranes are indistinguishable if they have the same label and contents (including submembranes), and their parents either coincide, or are also indistinguishable. Hence, for a formal approach to clocked P systems (with per-instance timing function defined, but freeness not yet required), one would assume some sequentialization of a configuration. This is needed because we do want to allow even indistinguishable objects to have different behavior, e.g., time of rule execution. On the other hand, instances of rules applied to different configurations, or even to the same configuration but different computation histories, must also be considered different, to allow independent duration assignments.

127 Clock-freeness and Related Concepts in P Systems 127 Naturally, a question arises: aren t all rule instances different? If all of them were different, it could make the definitions easier, since instead of predefining some complicated per-instance rule timing function, timing could be decided during the computation. Unfortunately, we cannot permit this, because at least for generating sets (e.g., of vectors), for two computations C 1 and C 2 of the same P system Π, with seemingly different rule timings could be united by the same per-instance rule timing function τ, whose restriction to instances in C 1 provides the timings needed for C 1 and whose restriction to C 2 provides the timings needed for C 2. Such an approach later will lead us to the concept of un-clocked P systems [2], but clockfreeness concept becomes difficult to conceive: which computations correspond to the same timing function and their results need to be collected in a set, and which computations correspond to different timing functions, and their corresponding sets need to be compared? Therefore, two rule instances are the same if the computation histories are the same, including the current configurations, and the rule is applied to the same instances of objects. What is P s(π, τ) (the result of computations of P system Π under per-instance timing function τ) in this case? It is the set of results of all computations sharing one feature: for any two computations C 1 and C 2 (if their starting configuration is the same), C 1 and C 2 must behave identically until they diverge by different choices of rules due to non-determinism. Of course, after that point all rule applications are considered different and τ can assign their durations independently. The situation is much easier to imagine in the deterministic case. Here, by determinism we mean that the applicable rule multiset is always unique (but rule instances are allowed to have different durations). Then P s(π, τ) is the result of a single computation (if the starting configuration is fixed), and clock-freeness in this case is just a) let duration of all rule instances be arbitrary, and b) require the result of all computations to be the same. This concludes our discussion of what rule instances are, and how to think about concepts of clocked and clock-free systems without going through complicated formal definitions of what instances are. 2 Replacing a Per-instance Rule Timing Function by a Finishing Strategy In timed P systems, a configuration (for every region) consists of (a multiset of) available objects and also of (a multiset of) rules in execution, with timestamps. Upon the application of any rule r, its reactants are consumed, and this rule is now in application, with the timestamp initialized to the value e(r) given by the timing function e. Each step the timestamp is decremented by one, and when it reaches zero, the rule execution is completed and its products are released. For clocked P systems, the situation is the same, but the per-instance timing function allows a richer variety of computations. Ironically, in the clock-free case

128 128 A. Alhazov et al. it turns out that the actual timings no longer matter, what is important is just the order in which rules finish. Hence, the per-instance timing function can be replaced by a finishing strategy, and the concept stays equivalent. A finishing strategy is a function that, given a computation history (which includes the current configuration and thus also the multiset of rules currently in execution), returns a non-empty submultiset of rules that will finish first, i.e., simultaneously and before all other rules currently in execution. Each per-instance timing function induces a finishing strategy: just return the multiset of rules with minimal timestamps. It is interesting to note that timestamps can be removed, since the finishing function could compute them, recursively calling itself and following the computation history which it is given as input, from the beginning until the current configuration. Of course, we do not really need to focus on such low-level details in order to just use the concept of clock-freeness. 3 Event-driven Model The discussion above leads us to the following conclusion: for clock-freeness, it does not really matter whether rule durations are integers or reals, or whether per-instance rule function is required to be computable or not. There is only one possible problem, not envisioned in the original definitions of clock-freeness, and the root of this problem is that there is no minimal positive real number. In the event-based model, we no longer have computation steps in the original meaning, hence, in a way, no global clock. This was the original idea behind the meaning of the clock-freeness concept. Instead of steps, the computation consists of events, which are ordered in time, but the actual values of times are abstracted out of the model. (In principle, one could call a computation step a distance between the events, but of course this is not equivalent to the original meaning.) What is an event? It is a moment of change of the configuration (only decrementing the timestamps, without finishing or starting any rules, does not count as an event), and it would be nice to consider the start as an event, too. Hence, an event is when at least one rule starts and/or at least one rule finishes. In [2] followed by [1] we defined two properties: non-extendable for a derivation mode, and monotone for a class of P systems. It turns out that for a monotone P system working in a non-extendable derivation mode, a rule can only start when some rule finishes (thus producing the needed reactants), with the exception of the initial configuration. Note that in the event-based model we do not need to worry whether time is discrete or continuous, we just follow sequences of events. For each event, we select applicable rules according to the working mode, and wait until the next event. A finishing strategy describes precisely which of the rules have finished. For a computation to finish, two conditions are required: no more applicable rules and no rules in execution. In a non-monotone P system, a candidate for an event would be a moment when some objects are consumed a rule may start without waiting for any rule

129 Clock-freeness and Related Concepts in P Systems 129 to finish. Consider, for example, the situation in which an object consumed by a rule may enable another rule which was inhibited before. When should that rule start? As soon as possible, of course. It would mean after one step in classical P systems, but a problem appears if timings can be arbitrarily small reals, since there is no smallest positive real number. So, one should either define a minimal permitted time ε (which would deviate from the pure event-based model, and since the actual time does not matter, ε could probably be even scaled up to 1), or allow such a new rule to be applied after 0 steps. Does the latter mean simultaneously? Well, it is still the next event, but it would be same time in the time-based model. This sounds like constructing a new derivation mode based on the given one. For instance, if the original derivation mode was sequential, it would still be possible to start multiple rules in the same time, but at least in the event-based model such a possibility is sound. Luckily, there is a third interpretation: even if applicable, only allow to start a rule when some rule finishes, also in this case. Ironically, all these three variants are different generalizations, in some sense, of the classical model where all rule executions take one step. Now consider the case when the working derivation mode of the P system is not non-extendable. Then, again, after some rule multiset has started execution, another rule multiset may become applicable. Not necessarily because some rule was not applicable before, but because the first rule multiset might have not exhausted (the reactants of) all applicable rules. 4 Clock-freeness and Efficiency Many papers published recently have shown efficiency of time-free P systems in the sense of solving NP-hard problems in a polynomial number of rule-starting steps. Counting events would seem more fair, but at least for timed (and hence also timefree) P systems this would make no difference, since all simultaneously started instances of the same rule also finish simultaneously, and thus the number of rule starting moments multiplied by the constant number of rules stays polynomial. For clock-freeness, on the other hand, we argued that solving an NP-hard problem in a polynomial number of moments of configuration change cannot be possible, unless of course P = NP [2, 1]. 5 Un-timed and Un-clocked P Systems In time-freeness and clock-freeness, the results of computations obtained with different (rule or instance) timing functions were required to be the same. Consider the case when rule durations are considered to be non-deterministic, in the same way as is the choice of the applied multiset out of more than one applicable alternatives? Then instead of comparing P s(π, e) for different functions e, respectively P s(π, τ) for different functions τ, we would take their union. Such systems were called un-timed and un-clocked, respectively.

130 130 A. Alhazov et al. Hence, for un-clocked systems we obtain a more elegant, simpler definition equivalent to the ones we saw in the previous sections: we do not need to define any timing functions, per-instance timing functions or finishing strategies rules just start in a given derivation mode (e.g., maximal parallelism), and finish asynchronously, or are performed in a lazy way. Since the variants described in this section have been introduced just recently and are not yet well known, we would like to illustrate them by an example which yields different results when considered as an un-clocked or as an un-timed system, and which is neither time- nor clock-free: Example 1. We consider the one-membrane multiset rewriting P system Π with the following non-cooperative rules with inhibitors: r 1 : a aa c and r 2 : b c. Starting from the axiom ab, we have to apply both r 1 : a aa c and b c in parallel. Considering Π as an un-timed system, r 1 can be applied again and again in parallel to all symbols a being generated, until the application of the rule r 2 has finished which prevents r 1 from restarting by the appearance of the inhibitor c. In sum, we obtain L un timed (Π) = t timing function L(Π, t) = n N {a 2n c}. Of course, this system is neither time- nor clock-free. The infinite set is generated not due to the choice between applicable rule multisets, but due to the non-deterministic choice of the timing function. Considering Π as an un-clocked system, again r 1 : a aa c and b c have to be applied in parallel in the first step, and r 1 can be applied until the application of the rule r 2 has finished which prevents r 1 from restarting by the appearance of the inhibitor c. Yet in contrast to the un-timed version, the applications of the rule r 1 to the symbols a appearing in the meantime may end at arbitrary moments of time. After the first application of rule r 1 has finished, unless c has been already produced, r 1 has to be reapplied simultaneously to both newly produced symbols a, and from that moment on the different instances of rule r 1 may finish in an unsynchronized way. Hence, as we sum up all possible results, we may restrict ourselves to consider only the events when just one rule application ends. The only symbols to which a rule (i.e., r 1 ) can now be applied are the two symbols a having appeared as a result of the finished rule application, and to these two symbols two copies of r 1 have to be applied simultaneously. In fact, this means that the number of symbols a increases by one with each finishing of a rule r 1. Therefore, in sum we obtain L un clocked (Π) = L(Π, τ) = {a n c n N + \ {1, 3}}. τ per instance timing function

131 Clock-freeness and Related Concepts in P Systems 131 A similar result can be obtained by the one-membrane multiset rewriting P system Π with the following non-cooperative rules with promoters: r 1 : a aa b, r 2 : b b, and r 3 : b c Starting from the axiom ab, we now may assume that the execution time of rule r 1 is a multiple of the execution time of rule r 2. Otherwise r 1 would anyway wait until r 2 is finished, because the promoter b is needed to allow r 1 to be applied. Again, no further rules may start after promoter b is eliminated by applying r 3. For the un-timed mode, the application of rules r 1 is still synchronized, and we therefore obtain L un timed (Π ) = t timing function L(Π, t) = n N {a 2n c}. In the un-clocked mode, the finishing of rules execution may be arbitrary, yet the promoter b is still needed to continue applying rule r 1, i.e., rule r 1 can be applied only when the application of the rule r 2 : b b has finished. In sum, we again obtain L un clocked (Π ) = L(Π, τ) = {a n c n N + \ {1, 3}}. τ per instance timing function 6 Beyond Tuples of Multisets In the model with active membranes the definitions need further specification. Numerous efficiency proofs for P systems with active membranes already exist. Since in this model the resources are not only objects, but also membranes, we would like to point the attention of the reader to one question: how is the state of a membrane monopolized by a rule, while such a rule is in execution? We recall that monopolizing rules are those that may modify the structure (create, destroy or divide membranes), change the polarization of a membrane, and, generally, the rules required to be applicable at most once to a given instance of a membrane at the same time. Usually these are all rules except the rules of type (a) (evolution rules). In existing publications so far, e.g., [8], followed by [5], the choices of the state of the membrane during the execution of a rule monopolizing it are those that make the desired result possible. Yet, in general, one can imagine more possibilities. We discuss them in an example of an arbitrary rule changing the polarization of a membrane, say, from + to. During an execution of such a rule, any of the following may be consistently considered: 1. the polarization stays + (i.e., the old one, and is changed at the end of the execution of the rule); 2. the polarization instantly changes to (i.e., the new one; the change is therefore done at the beginning of the execution);

132 132 A. Alhazov et al. 3. the polarization is busy (forbidding to start any rules associated with the membrane in the meantime; the polarisation is thus changed both at the beginning and at the end of the execution of the rule). The actual space of possibilities is even larger. For instance, while in the first two cases we would normally assume non-monopolizing rules, i.e., of type (a), to be applicable, other rules may or may not be applicable. The applicability of other rules in the second case is easy, while in the first case multiple variants or even conflicts may arise. 7 Open Questions, Research Directions and Highlights We would like to mention and repeat the following thoughts and questions. While formal definitions may be very tedious, the defined concepts are interesting and popular. While the authors need to be careful with interpretations, one does not always need to go into the details of functions defined on instances to get results. Clock-freeness captures independence on durations of rule timings better, while time-free systems still possesses residual synchronization. Typically one would expect the universality proofs for time-freeness to automatically hold also for clock-free systems, but for efficiency it was shown not to be the case, see, e.g., [2] followed by [1]. One should not confuse clocked and clock-free: the first is a variant meaning durations are fixed, but not necessarily to 1, while the latter is a property. The first variant makes the class of systems larger, while the latter property makes the class of systems smaller! While clocked might be not the most successful name, it fits best in the existing family of concepts: timed and clocked means timings are defined, both freeness properties require associated timed or clocked systems to be independent on this timing, and un-timed/un-clocked systems are systems with two kinds of non-determinism in the choice of rule multisets and in the choice of rule durations. Event-based models allow different interpretations for classes of P systems that are not monotone or for derivation modes that are not non-extendable. What is the effect of this difference, e.g., how it affects the computational power and descriptional complexity? What about intermediate variants between per-rule and per-instance timing? Consider, for example, Markovian systems, in which the evolution history does not influence the assignment of durations by the timing functions. What would be the properties of such a variant? How does the power or efficiency of other classical models change under the un-timed/un-clocked semantics? Mode-freeness is another interesting concept, see [2, 1], and it has an interesting relationship with time/clock freeness, some of the aspects of which are still poorly understood.

133 Clock-freeness and Related Concepts in P Systems 133 This does not exhaust all open problems from [2]. Acknowledgements The work is supported by the National Natural Science Foundation of China ( , , , and ), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province ( ), and the China Postdoctoral Science Foundation (2016M600592, 2017T100554). References 1. Artiom Alhazov, Rudolf Freund, Sergiu Ivanov, Bosheng Song, and Linqiang Pan. Clock-freeness and mode-freeness in P systems. In G. Zhang, J. Wang, L. Pan, Q. Yang, Y. Zeng, and X. Zeng, editors, Pre-Proceedings of the Sixth Asian Conference on Membrane Computing, ACMC2017, Chengdu, 2017, pages , Artiom Alhazov, Rudolf Freund, Sergiu Ivanov, Bosheng Song, and Linqiang Pan. Time-freeness and clock-freeness and related concepts in P systems. In C. Graciani, Gh. Păun, A. Riscos-Núñez, and L. Valencia-Cabrera, editors, RGNC Report 1/2017, University of Seville, Fifteenth Brainstorming Week on Membrane Computing, Sevilla, pages Fénix Editora, Matteo Cavaliere and Dragoş Sburlan. Time independent P systems. In G. Mauri, Gh. Păun, M.J. Pérez-Jiménez, G. Rozenberg, and A. Salomaa, editors, Membrane Computing: 5th International Workshop, WMC 2004, Milan, Italy, June 14-16, 2004, Revised Selected and Invited Papers, pages Springer, Dragoş Sburlan. Clock-free P systems. In Pre-proceedings of the Fifth Workshop on Membrane Computing (WMC5), Milano, Italy, June 2004, pages , Bosheng Song and Linqiang Pan. Computational efficiency and universality of timed P systems with active membranes. Theoretical Computer Science, 567:74 86, Bosheng Song, Mario J. Pérez-Jiménez, and Linqiang Pan. An efficient time-free solution to QSAT problem using P systems with proteins on membranes. Information and Computation, 256: , Bosheng Song, Tao Song, and Linqiang Pan. A time-free uniform solution to subset sum problem by tissue P systems with cell division. Mathematical Structures in Computer Science, 27(1):17 32, Tao Song, Luis F. Macías-Ramos, Linqiang Pan, and Mario J. Pérez-Jiménez. Timefree solution to SAT problem using P systems with active membranes. Theoretical Computer Science, 529:61 68, 2014.

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135 Abstracts of PhD Theses Author: Richelle Ann B. Juayong: Title: Communication Complexity in P Systems with Energy Supervisor: Henry N. Adorna Algorithms and Complexity Laboratory ( Department of Computer Science, College of Engineering University of Philippines, Diliman, Quezon City, Philippines Thesis committee: Jaime D.L. Caro, University of the Philippines, Diliman Cedric Angelo M. Festin, University of the Philippines, Diliman John Paul C. Vergara, Ateneo de Manila University, Diliman Jan Michael C. Yap, University of the Philippines, Diliman Date of presentation: April 24, 2017 Thesis Abstract One crucial aspect in the understanding of P systems is investigating the role of communication since P systems are mainly parallel and distributed devices. Before 2009, measures considered for communication analysis in membrane computing are static measures. It is in a paper entitled Communication Complexity of Evolution- Communication P systems that dynamic communication measures were proposed. It is hoped that these dynamic measures can be utilized to develop results that are similar to results in classical communication complexity. The introduction of such measures were proposed for a specific model called Evolution-Communication P Systems with Energy (ECPe systems). An ECPe system has two main and

136 136 Abstracts of PhD Theses distinct types of rules: evolution and communication. A form of payment called energy is earned through the use of evolution rules and given off upon the use of a communication rule. Dynamical measures, such as considering the number of communication steps (ComN), communication rules (ComR) and total energy used for communication (ComW ), were proposed for ECPe systems. This dissertation contributes to exploring two of the proposed ways by which communication analysis can be applied in ECPe systems, i.e. as properties of a P system and as a standard complexity measure. Our work mainly focuses on further understanding ECPe systems by contributing to mainly three topics: (a) understanding the computing power of ECPe systems with bounded and unbounded communication; (b) analyzing communication of problems solved in ECPe systems, and eventually ECP systems, using the proposed dynamical communication measures and (c) relating computations in ECPe system and other P systems. The main contributions derived from the work object of this dissertation are the following: Understanding the computing power of ECPe systems with bounded and unbounded communication. One-membrane ECPe systems having the environment as the output region only generate a finite set of numbers when ComW is bounded and nonzero. On the other hand, such a system, when using an unbounded ComW, already generates semilinear sets of numbers. Any k-membrane ECPe system with bounded ComW can only generate semilinear sets of numbers when the output region only acts as a receiving end of communication. A 2-membrane ECPe system with bounded ComN and ComR can already generate non-semilinear sets of numbers. The set {k n + 1 n 0} for k > 1 can be computed using 2-membrane ECPe systems having only two communication steps, however the maximum energy needed in antiport rule is at least k. The set {k n n 0} for k > 1 can be computed using 2-membrane ECPe systems having only four communication steps and only symport rules are used, however the maximum energy needed in symport rule is at least k 1. A two-membrane ECPe system with two-way communication and having unbounded ComW already generate recursively enumerable sets of numbers. Communication analysis using dynamical communication measures for some hard problems solved in ECPe and ECP systems. Decision problems are solved by a family of so-called recognizer P systems. We propose a way to classify problems solved by families of recognizer ECPe systems based on dynamical communication measures. We then show example problems belonging to some such classes by analyzing the communication resource on two NP-hard problems, i.e. the Vertex Cover Problem (VCP) and 3-Satisfiability Problem (3SP).

137 Abstracts of PhD Theses 137 We also proposed communication classes based on dynamic measures in ECP systems; this is a slight modification of a similar analysis we did for ECPe systems (outlined above). We also analyze VCP and 3SP, this time using ECP systems solution. Investigating the relation of computations in a basic membrane computing model called Transition P systems (TP systems) and ECPe systems that function as string generators. When using only non-cooperative rules, i.e. rules that require only one copy of one object, one transition in the TP system is simulated by a 3-step computation in the simulator. The ECPe system simulator requires a minimal cost for each communication rule used. The number of communications is also not dependent on the number of communicated objects used in a rule, but only on the number of receiving regions. To investigate simulation for TP systems with cooperative rules, i.e. rules that require multiple copies or multiple objects, we first investigate TP systems having cooperative rules where the required objects can also evolve independently. We name the model a TP system with independent triggers (TP-ind systems). One transition in a TP-ind system corresponds to a time t in the ECPe system simulator where t is linear to the cardinality of the alphabet used in the TP-ind system. Additional membranes are also allocated for every cooperative rule in the original system. In such simulations, the copies of each object required for a rule is manifested as energy (or cost) imposed on communication rules. We are also able to show that the language of some other TP systems can be computed using TP-ind systems. These systems include cooperative rules where there is at least one independent trigger required in their left-hand side and cooperative rules where there are several copies of a single trigger required in their left-hand side. Contents Acknowledgements iv Abstract v List of Figures xi List of Tables xiii 1 Introduction 1 I Preliminaries 8 2 Technical prerequisites Set Notations Strings and Length Sets Multisets Grammars 10 3 Membrane Computing 12

138 138 Abstracts of PhD Theses 3.1 Basic Features of a P System Membranes Objects Rules, Non-determinism and Maximal Parallelism Configuration, Transition and Computation Computability and Complexity Practical Applications and Implementations 17 4 Evolution-Communication P systems with Energy Introduction Formal Definition ECPe Systems as Generators ECPe Systems as Number Generators ECPe Systems as String Generators Dynamical Measures for Communication Complexity Preliminary Results for ECPe Systems 24 II Body of Work 25 5 On ECPe Systems Having U nbounded Communication Matrix Grammars with Appearance Checking Main Results On ECPe Systems with Unbounded Communication On ECPe Systems with Bounded ComW On ECPe Systems with Bounded ComX, X {N, R} On ECPe Systems Having Symport Rules Only Summary 38 6 On Comm n Complexity of Some Hard Problems in ECPe Systems Definitions of Some NP-complete Problems Solving Problems in ECPe systems ECPe Systems Solutions to NP-hard Problems Summary 53 7 On Comm n Complexity of Some Hard Problems in ECP systems Evolution-Communication P Systems Solving Hard Problems in ECP Systems On ECP Solutions to VCP On ECP Solutions to 3SP Summary 65 8 Relating Computations in ntp Systems and ECPe Systems Non-cooperative Transition P Systems (ntp Systems) Main Result Summary 72 9 On Simulating Cooperative TP Systems in ECPe Systems Transition P systems and Independent Triggers Main Results 75

139 Abstracts of PhD Theses Informal Description of the Simulation Categories for coop-ind Rules and Additional Notations for TP Systems Formal Construction of the Simulator Summary Notes on Language Relations among TP Systems TP-ind Systems and TP-dep Systems TP-ind Systems vs. TP-dep Systems From TP-dep to TP-ind System: An Initial Approach TP-dep Systems with Cat 1 coop-dep Rules TP-dep Systems with Cat 2 coop-dep Rules TP-dep Systems with Cat 1 and Cat 2 coop-dep Rules Summary 102 III Final Remarks Conclusions and Open Problems 104 List of References 113

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141 Calls for Participation to MC and Related Conferences/Meetings Call for Participation Sixteenth Brainstorming Week on Me4mbrane Computing (BWMC 2018) Research Group on Natural Computing University of Sevilla, Spain January 30 February 2, 2018, Sevilla, Spain Goal: Similarly to the previous editions, the goal is to gather together researchers interested in Membrane Computing (theory and applications), for exchanging ideas, problems, solutions, for working together, in a friendly framework. To this aim, the meeting combines joint work sessions (usually on the afternoons), together with short and provocative talks, possibly presenting ongoing work and/or open problems. Those interested in having such presentations should inform the organizers ({cgdiaz,mdelamor}@us.es) some days before the meeting. Researchers from MC community, and from other areas as well, are strongly encouraged to circulate open problems and research ideas before the event, even if they cannot attend the meeting (please send to {cgdiaz,mdelamor}@us.es your proposals, and we will post them on the BWMC web page). The Brainstorming Week is an excellent opportunity to discuss about potential collaborations and interdisciplinary approaches. Dates: January 30, 2018 (Tuesday) February 2, 2018 (Friday) Organizing institution: Research Group on Natural Computing (RGNC) Department of Computer Science and Artificial Intelligence University of Sevilla, Spain Venue:

142 142 Call for Participation 16th BWMC E.T.S. Ingeniería Informática, module H, first floor Avda. Reina Mercedes s/n. (41012) Sevilla Web page: News will be communicated through the BWMC 2018 web page at the Research Group on Natural Computing: and through the P systems web page: Proceedings: As usual, after the meeting a proceedings volume will be published as a research report of RGNC. Then, a selection of final papers will be published in a special issue of a journal as it was the case with the previous editions. Here is the complete list (all journals, except two, are currently indexed in the ISI JCR): BWMC 2003: Natural Computing 2 (3), 2003, and New Generation Computing 22 (4), 2004; BWMC 2004: Journal of Universal Computer Science 10 (5), 2004, and Soft Computing 9 (5), 2005; BWMC 2005: Int. Journal of Foundations of Computer Science 17 (1), 2006); BWMC 2006: Theoretical Computer Science 372 (2-3), 2007; BWMC 2007: Int. Journal of Unconventional Computing 5 (5), 2009; BWMC 2008: Fundamenta Informaticae 87 (1), 2008; BWMC 2009: Int. Journal of Computers, Control and Communication 4 (3), 2009; BWMC 2010: Romanian Journal of Information Science and Technology 13 (2), 2010; BWMC 2011: Int. Journal of Natural Computing Research 2 (2-3), 2011; BWMC 2012: Int. Journal of Computer Mathematics 99 (4), 2013; BWMC 2013: Int. Journal of Unconventional Computing 9(5-6) 2013; BWMC 2014: Fundamenta Informaticae 134 (1-2), 2014; BWMC 2015: Natural Computing 15 (4), 2016; BWMC 2016: Theoretical Computer Science (to appear) and BWMC 2017: Theoretical Computer Science (to appear) Registration: The participants should Carmen Graciani (cgdiaz@us.es) in order to register. A registration fee of 100 euros will be requested on arrival at the registration desk. This will cover workshop materials, coffee breaks, lunch, social dinner and proceedings. Several accommodation options are listed on the webpage of the BWMC 18 that need to be booked by the participants. Important advice: Bookings should be arranged the sooner the better, in order to avoid availability restrictions. In particular, reservations for university residences should be made not later than January 20th 2018 (by to cgdiaz@us.es). Organizing committee:

143 Gh. Păun, Co Chair M.J. Pérez Jiménez, Co Chair C. Graciani M.Á. Martínez del Amor David Orellana Martín Supports: Call for Participation 16th BWMC 143 I. Pérez Hurtado de Mendoza A. Riscos Núñez Á. Romero Jiménez L. Valencia Cabrera Dpto. Ciencias de la Computación e Inteligencia Artificial, Univ. Sevilla VI Plan Propio, Vicerrectorado de Investigación de la Universidad de Sevilla Plan Andaluz de Investigación, Desarrollo e Innovación (PAIDI) 2020

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145 Call for Papers 19 th International Conference on Membrane Computing (CMC19) 4 7 September, 2018, Dresden, Germany organised by the Friedrich-Schiller University Jena (Germany) under the auspices of the International Membrane Computing Society (IMCS) Aim and Scope Membrane Computing is an area of computer science aiming to abstract computing ideas and models from the structure and the functioning of living cells, as well as from the way the cells are organized in tissues or higher order structures. It deals with membrane systems, also called P systems, which are distributed and parallel algebraic models processing multisets of objects in a localised manner (evolution rules and evolving objects are encapsulated into compartments delimited by membranes), with an essential role played by the communication among compartments and with the environment. CMC19 aims at continuing the fruitful tradition of 18 previous editions enriched by some new ideas and inspirations emphasising multidisciplinarity and innovative capacity. The conference is intended to bring together researchers working in Membrane Computing and related areas in a friendly atmosphere enhancing communication and cooperation. We are pleased to hold CMC19 at the hotel venue NH Dresden-Neustadt, Hansastr. 43, D Dresden, Germany. Dresden is said to be one of the most beautiful and picturesque cities in Germany. The former Saxonian Residence and now the capital of the German free state Saxony with approximately 560,000 inhabitants belongs to the most visited in Germany, with 4.3 million overnight stays per year. Along the Elbe river, the royal buildings are among the highly impressive buildings in Europe. Further highlights like the Frauenkirche (Church of our Lady), famous museums, fascinating exhibitions of art, exquisite vineyards, and many further cultural attractions complete the long list of outstanding sights worth to be visited in Dresden and its environment.

146 146 Call for Papers 19th CMC Submission and Proceedings Authors are invited to submit papers presenting original, unpublished research as a PDF document. There are two tracks for submission: (1) full paper (of a reasonable length) (2) extended abstract for poster presentation (maximum four pages). Typical extended abstracts present significant work-in-progress, late-breaking results, or contributions from students new in the field or at the start of their research career. Only electronic submissions are accepted. Papers should be formatted according to the usual LNCS article style which can be downloaded at Springer s LNCS website ( Simultaneous submission to other conferences or workshops with published proceedings is not allowed. Submissions will be handled via EasyChair. To submit your contribution, please follow this link: All submissions will be reviewed by at least three referees. To ensure inclusion of an accepted contribution into the (pre)proceedings volume, at least one of its authors must be registered for CMC19. The pre-proceedings volume will be available during the conference. The final proceedings, a volume in Springer Lecture Notes in Computer Science series (LNCS), for selected and additionally refereed papers, will be published after CMC19. Dates and Deadlines Submission deadline: April 30, 2018 Notification of acceptance: June 04, 2018 Early bird registration: june 11, 2018 Camera-ready version: july 02, 2018 Registration deadline: July 02, 2018 Conference: September 04-07, 2018 Program Committee Henry Adorna (Quezon City, Philippines) Bogdan Aman (Iasi, Romania) Jörn Behre (Jena, Germany) Matteo Cavaliere (Edinburgh, UK) Erzsebet Csuhaj-Varju (Budapest, Hungary) Lucas Eberhardt (Leipzig, Germany) Giuditta Franco (Verona, Italy) Rudolf Freund (Vienna, Austria)

147 Call for Papers 19th CMC 147 Marian Gheorghe (Bradford, UK) Thomas Hinze (Jena, Germany) - co-chair Florentin Ipate (Bucharest, Romania) Savas Konur (Bradford, UK) Shankara N. Krishna (Bombay, India) Raluca-Elena Lefticaru (Bradford, UK) Alberto Leporati (Milan, Italy) Giancarlo Mauri (Milan, Italy) Radu Nicolescu (Auckland, New Zealand) Taishin Y. Nishida (Toyama, Japan) Linqiang Pan (Wuhan, China) Gheorghe Păun (Bucharest, Romania) Mario J. Perez-Jimenez (Sevilla, Spain) Dario Pescini (Milan, Italy) Antonio E. Porreca (Milan, Italy) Agustin Riscos-Nunez (Sevilla, Spain) Jose M. Sempere (Valencia, Spain) Petr Sosik (Opava, Czech Republic) Heiko Tiedtke (Halle, Germany) György Vaszil (Debrecen, Hungary) Sergey Verlan (Paris, France) Claudio Zandron (Milan, Italy) - co-chair Gexiang Zhang (Chengdu, China) Steering Committee Henry Adorna (Quezon City, Philippines) Artiom Alhazov (Chisinau, Moldova) Bogdan Aman (Iasi, Romania) Matteo Cavaliere (Edinburgh, UK) Erzsebet Csuhaj-Varju (Budapest, Hungary) Giuditta Franco (Verona, Italy) Rudolf Freund (Vienna, Austria) Marian Gheorghe (Bradford, UK) Honorary member Thomas Hinze (Jena, Germany) Florentin Ipate (Bucharest, Romania) Shankara N. Krishna (Bombay, India) Alberto Leporati (Milan, Italy) Taishin Y. Nishida (Toyama, Japan) Linqiang Pan (Wuhan, China) Gheorghe Păun (Bucharest, Romania) Honorary member Mario J. Perez-Jimenez (Sevilla, Spain) Agustin Riscos-Nunez (Sevilla, Spain)

148 148 Call for Papers 19th CMC Jose M. Sempere (Valencia, Spain) Petr Sosik (Opava, Czech Republic) Kumbakonam G. Subramanian (Chennai, India) György Vaszil (Debrecen, Hungary) Sergey Verlan (Paris, France) Claudio Zandron (Milan, Italy) Gexiang Zhang (Chengdu, China) Organizing Committee Jörn Behre (Jena, Germany) Thomas Hinze (Jena and Dresden, Germany) Contact Information Thomas Hinze Friedrich-Schiller University Jena Department of Bioinformatics Ernst-Abbe-Platz 2 D Jena, Germany Please do not hesitate to contact us if you have any question, preferably via Thomas Hinze thomas.hinze@uni-jena.de Jörn Behre mail@joern-behre.de

149 Reports on MC Conferences/Meetings The 18 th International Conference on Membrane Computing Bradford, UK, July 25 28, 2017 Marian Gheorghe School of Electrical Engineering and Computer Science, The University of Bradford, UK The 18 th edition of the International Conference on Membrane Computing (CMC18) returned back to the UK after 9 years. An workshop by that time, the 9 th edition has been held in Edinburgh, Scotland, in This time the conference was held in Bradford, England. Bradford is a multicultural, vibrant, and well-connected city in the heart of Yorkshire. Full of history, it was once the wool capital of the world, also the first UNESCO City of Film. Home to the National Media Museum, Bradford is famous for some of the finest Asian food in the UK, being crowned Curry Capital of Britain for six consecutive years; more about Bradford and CMC18 can be found at We all proudly remember the CMC series started with three workshops organized in Curtea de Argeş, Romania, in the three consecutive years 2000, 2001 and 2002, by the initiator of the research field, Gheorghe Păun. The European itinerary of the CMC series included Tarragona, Spain (2003), Milan, Italy (2004), Wien, Austria (2005), Leiden, The Netherlands (2006), Thessaloniki, Greece (2007), and Edinburgh, UK (2008). The 10 th edition returned back to its origins, Curtea de Argeş, in August 2009, where it was decided to continue the series as the Conference on Membrane Computing. The following editions were held in Jena, Germany (2010), Fontainebleau, France (2011), Budapest, Hungary (2012), Chişinău, Moldova (2013), Prague, Czech Republic (2014), Valencia, Spain (2015), and Milan, Italy (2016). A regional version of CMC, the Asian Conference on Membrane Computing (ACMC) started in 2012 in Wuhan, China, and continued in Chengdu, China

150 150 M. Gheorghe (2013), Coimbatore, India (2014), Hefei, Anhui, China (2015), and Bangi, Selangor, Malaysia (2016). In 2016, at CMC17, in Milan, Italy, it was announced the official establishment of the International Membrane Computing Society (IMCS), an organization bringing together the researchers from all over the world and facilitating collaboration and cooperation on all the topics related to membrane computing. CMC18 consisted of two different parts: standard sessions, held from Tuesday to Thursday, and an interaction day between participants, held on Friday. Monday was the arrival day for most of the participants. The standard sessions included invited lectures given by Erzsébet Csuhaj-Varjú (Budapest, Hungary), Harold Fellermann (Newcastle, UK), Michael Fessing (Bradford, UK) and Maciej Koutny (Newcastle, UK), 21 regular and 2 short papers, and 3 extended abstracts. The Best Student Paper Award, sponsored by Springer-Verlag, was given to the paper Generalized P Colony Automata and Their Relation to P automata, by Kristóf Kántor and György Vaszil. The CMC18 program included also a trip to Soltaire, a Victorian model village located in Shipley, part of the City of Bradford Metropolitan District, a designated UNESCO World Heritage Site. Saltaire was built in 1851 by Sir Titus Salt, a leading industrialist, as a site for his large textile mill. Salt built neat stone houses for his workers, wash-houses with tap water, bath-houses, a hospital and an institute for recreation and education, with a library, a reading room, a concert hall, billiard room, science laboratory and a gymnasium (for more details see A wonderfully entertaining and educational guided tour of the village provided by Maria Glot and Anne Heald was full of historical facts, fun and fiction. They are very knowledgeable and provided witty descriptions, impersonations and stories about the life in Saltaire. We all keep fond memories of that place. Finally, we would like to acknowledge the support provided by the School of Electrical Engineering and Computer Science of the University of Bradford and the Prize for the Best Student Paper award granted by Springer-Verlag. We would also like to thank members of the the Steering and Programme Committees Henry Adorna (Quezon City, Philippines), Artiom Alhazov (Chişinău, Moldova), Bogdan Aman (Iaşi, Romania), Matteo Cavaliere (Edinburgh, UK), Lucie Ciencialová (Opava, Czech Republic), Erzsébet Csuhaj-Varjú (Budapest, Hungary), Giuditta Franco (Verona, Italy), Rudolf Freund (Wien, Austria), Marian Gheorghe (Bradford, UK), Thomas Hinze (Jena, Germany), Florentin Ipate (Bucharest, Romania), Shankara N. Krishna (Bombay, India), Alberto Leporati (Milan, Italy), Vincenzo Manca (Verona, Italy), Giancarlo Mauri (Milan, Italy), Radu Nicolescu (Auckland, New Zealand), Taishin Y. Nishida (Toyama, Japan), Linqiang Pan (Wuhan, China) SC Co-Chair, Gheorghe Păun (Bucharest, Romania), Mario J. Pérez-Jiménez (Sevilla, Spain), Antonio E. Porreca (Milan, Italy), Agustín Riscos-Núñez (Sevilla, Spain, José M. Sempere (Valencia, Spain), Petr Sosík (Opava, Czech Republic), Kumbakonam Govindarajan Subramanian (Penang, Malaysia), György Vaszil (Debrecen, Hungary), Sergey Verlan (Paris,

151 The 18th CMC, Bradford, UK, France), Claudio Zandron (Milan, Italy) SC Co-Chair, and Gexiang Zhang (Chengdu, China), that also acted as conference reviewers, together with Luca Manzoni (Milan, Italy). Special thanks to the Organizing Committee of CMC18 Marian Gheorghe (Bradford, UK) Co-chair, Savas Konur (Bradford, UK) Co-chair, Raluca Lefticaru (Bradford, UK) Communication Chair, and Daniel Neagu (Bradford, UK) Publicity Chair. We hope to see many of the membrane computing community members next year in Dresden, Germany, at CMC19, 4 7 September.

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153 A Summary of The 6th Asian Conference on Membrane Computing (ACMC 2017) Gexiang Zhang 1,2, Qiang Yang 1, Linqiang Pan 3 1 Robotics Research Center, Key Laboratory of Fluid and Power Machinery of Ministry of Education, Xihua University, Chengdu, , China 2 School of Electrical Engineering, Southwest Jiaotong University, Chengdu , China 3 School of Automation, Huazhong University of Science and Technology, Wuhan , China The 6th Asian Conference on Membrane computing (ACMC 2017) was held in Xihua Unversity (XHU), Chengdu, China, from September This conference was organized by a joint effort of the International Membrane Computing Society(IMCS) and Xihua University. ACMC 2017 is one of the flagship conference on Membrane Computing aiming to provide a high-level international forum for researchers working in membrane computing and related areas, especially for the ones from the Asian region. ACMC 2017 has received 31 submissions. Each submission was reviewed by at least three programme committee members. Thirty papers were accepted for oral presentations during the conference. All the 30 papers have been published in the pre-proceeding of ACMC 2017 and recommended to be published in several international journals. Ten papers are selected to be published in the SCI-indexed journal International Journal of Computers, Communications and Control (IJCCC). A selection with about 5 papers are considered to be published in the SCI-indexed journal Fundamenta Informaticae. The rest accepted papers are considered to be published in international EI-indexed journals. This conference has been organized as a friendly interactive platform with 7 keynote lectures, 27 oral presentations, covering a wide range of topics on membrane computing, including theory, applications, implementation and various related aspects. ACMC 2017 was well organized and was a very successful conference. A special thanks is given to the Organising Committee and Supporting Committee at the Xihua University, all the authors and participants, the Steering Committee and Programme Committee. In the opening ceremony of ACMC 2017, the President of IMCS, Prof. Gexiang Zhang, delivered an introductive report on IMCS and ACMC. The invited talks are summarized as follows:

154 154 G. Zhang, Q. Yang and L. Pan Invited Talk 1: A Parallel Big Data Processing & Intelligent Analysis System and its Industrial Applications Kenli Li, Professor Hunan University, Changsha, China lkl@hnu.edu.cn The aim of this talk is to present the current development of the big data and artificial intelligence (AI 2.0), and point out the technical threshold of the traditional enterprises when they hope to implement the data mining and intelligent analysis based on their historical big data. This talk also discusses how to propose and implement a platform of parallel data processing and analysis, which can help the traditional enterprises to across these thresholds. Finally, several typical practice industrial applications based on this platform are introduced, such as the data analysis system for the medical behavior, the auto failure detection and identification system for the train, and the data collection and analysis system for the CNC machines. Invited Talk 2: Modelling the dynamics of complex systems: A membrane computing based framework Luis Valencia Cabrera (on behalf of Prof. Mario J. Pérez-Jiménez) University of Seville, Sevilla, Spain lvalencia@us.es This talk reviews the research context on why we use the Nature way to perform certain processes in our real-life. Membrane computing is a computational paradigm aiming to abstract computing models, called P systems. In this talk, a uniform Membrane Computing framework providing predictive models of real-life phenomena is presented. Practical applications both at a micro (i.e. molecular mechanism in cellular processes) and a macro (i.e. population dynamics of real ecosystems) level are studied, being presented during the talk to illustrate some relevant features of the models and the underlying framework. Invited Talk 3: Grammar Systems and Rewriting P Systems : Connections K.G. Subramanian, Professor Madras Christian College, Chennai, India kgsmani1948@gmail.com This talk introduces the theory and development of grammar systems, which can be used for modelling distributed computation at symbolic level in terms of syntactic methods, especially grammar-based methods. With the development of P systems in theory and application, a variety of grammar systems and P systems with different motivations, have been introduced and studied in this talk. Actually, this talk is intended to highlight the connections between rewriting P systems and grammar systems, and point out unexplored areas as well.

155 A Summary of ACMC Invited Talk 4: Kernel P systems - A powerful modelling and verification framework Marian Gheorghe, Professor University of Bradford, Bradford, UK m.gheorghe@bradford.ac.uk This talk first presents the main computational and complexity theoretical results as well as verification aspects of the kernel P systems. Then, in this talk several applications modelled with kernel P systems are discussed, and their behaviour are analysed with the kp-workbench tool. Finally, future research directions of this computational model are summarised. Invited Talk 5: A formal framework for definition and analysis of P systems Sergey Verlan, Associated Professor Laboratoire d Algorithmique, Complexité et Logique, Département Informatique, Université Paris Est, France verlan@u-pec.fr In this talk, the model called the formal framework for P systems is discussed. It provides a descriptional language to represent most of the variants of P systems in a simple way. This allows an unambiguous definition of a P system as well as a uniform way for extensions and also a powerful tool for the comparison of different models. In order to understand, extend, compare and explain different models of P systems, a series of concrete examples of the application of the formal framework will be presented in future work. Invited Talk 6: Spiking neural P systems Linqiang Pan, Professor School of Automation, Huazhong University of Science and Technology, Wuhan, China lqpan@mail.hust.edu.cn This talk focuses on spiking neural P systems, which are a class of parallel computing models inspired by neurons. At the beginning, this talk gives an overview of spiking neural P systems, including motivation, biological background, the basic ingredients and functioning of a spiking neural P system, and the relationship and difference with the traditional spiking neural networks. Then, some classical or recent results on spiking neural P systems are reviewed. Finally, an outlook is given with a discussion of applications. Invited Talk 7: Fuzzy membrane systems and unsupervised learning algorithm of membrane computing: Recent Advances Hong Peng, Professor (on behalf of Prof. Jun Wang) School of Computer and Software Engineering, Xihua University, Chengdu, China

156 156 G. Zhang, Q. Yang and L. Pan This talk first reviews the theory of fuzzy membrane systems and fuzzy spiking neural P systems. Then, the recent works of their research group are introduced, including fuzzy reasoning spiking neural P systems, weighted fuzzy spiking neural P systems, adaptive fuzzy spiking neural P systems, and its applications for fault diagnosis of power system. This talk also discusses the learning mechanism of membrane computing models and several efficient distributed parallel clustering algorithms. Finally, some applications are given. In the closing ceremony of ACMC 2017, the President of IMCS, Prof. Gexiang Zhang, extended thanks to the invited speakers, authors, conference organizers, conference volunteers and so on. Then he awarded two The best paper and two The best student paper prizes selected from all accepted papers, respectively. This is the first time to have the two kinds of awards on ACMC, aiming to encourage participants to bring novel ideas and applications to ACMC. Below is the list of papers with authors during the ACMC James Cooper, Radu Nicolescu, The Travelling Salesman Problem in cp Systems. 2. S. Raghavan, K. Chandrasekaran, Enzymatic Numeriacl P System for Improved Analytical Hierarchy Process. 3. Yuzhen Zhao, Xiyu Liu, Wenxing Sun, The Chained P System with Application in Graph Clustering. 4. Shuo Liu, Kang Zhou, Shan Zeng, Xing Chen, A Small Universal SN P System with Rules on Synapses. 5. Fangxiu Wang, Kang Zhou, Huaqing Qi, Improvement of Serial Input Adder Based on Spiking Rules. 6. Juan Hu, Guangchun Chen, Hong Peng, Jun Wang, The Edited K-nearest Neighbor Classifiers Evolved by P Systems. 7. Xuan Hou, Jun Wang, Tao Wang, Hongjun Li, Multi-objective Particle Swarm Membrane Optimization Algorithm and Its Application in Energy Management of Microgrids. 8. Jun Ming, Tao Wang, Jun Wang, Zhang Sun, Chuanxiang Wei, Application of Neural-like P Systems with State Values for Power Coordination of Microgrids under Connected Operation. 9. Ping Guo, Xixi Peng, The P System Design Based on P Module. 10. Guangchun Chen, Jinyu Yang, Hong Peng, Jun Wang, A Spectral Clustering Algorithm Improved P Systems. 11. Huifang Wang, Kang Zhou, Gexiang Zhang, Arithmetic Operations with Spiking Neural P Systems with Rules and Weights on Synapses. 12. Yingying Duan, Kang Zhou, A P-based Hybrid Evolutionary Algorithm with Exchange-tree Mechanism for Traffic Network Transportation Optimization Problem.

157 A Summary of ACMC Jherico Gabriel Torres, Kelvin Buño, Francis George Cabarle, Some Notes on Spiking Neural dp Systems. 14. Yun Jiang, Yuan Kong, Chaoping Zhu, Implementation of Arithmeti Operations by Spiking Neural P Systems with Communication on Request. 15. Jym Paul Carandang, Francis George Cabarle, Henry Adorna, Nestine Hope Hernandez, Miguel Ángel Martínez-Del-Amor, Nondeterminism in Spiking Neural P Systems: Algorithms and Simulations. 16. Yunyun Niu, Yongpeng Zhang, Jianhua Xiao, Walking P System for Evacuation Simulation. 17. Yuan Kong, Yun Jiang, Yunyun Niu, Yong Fang, On Languages Generated by Asynchronous Spiking Neural P Systems with Astrocytes. 18. Erzsébet Csuhaj-Varjú, Sergey Verlan, Computationally Complete Generalized Communicating P Systems with Three Cells. 19. Xiangrong Liu, Xiaoshan Yan, Shiyue Xie, Alfonso Rodríguez-Patón, Xiangxiang Zeng, On String languages Generated by Spiking Neural P Systems with a Generalized Use of Rules. 20. Ren Tristan de La Cruz, Francis George Cabarle, Xiangxiang Zeng, Arithmetic and Memory Module using Spiking Neural P Systems with Structural Plasticity. 21. D.G. Thomas, Robinson Thamburaj, Atulya Nagar, Parallel Contextual Hexagonal Array Insertion Deletion P System. 22. Artiom Alhazov, Rudolf Freund, Sergiu Ivanov, Linqiang Pan, Tingfang Wu, Spiking Neural P Systems over Tree Structures. 23. Artiom Alhazov, Rudolf Freund, Sergiu Ivanov, Linqiang Pan, Boseng Song, Clock-freeness and Mode-freeness in P Systems. 24. K.G. Subramanian, Somnath Bera, Boseng Song, Linqiang Pan, Gexiang Zhang, Array P Systems Based on Parallel Rewriting with Array Contextual Rules. 25. Gexiang Zhang, Ming Zhu, Qiang Yang, Haina Rong, Weitao Yuan, Mario J. Pérez-Jiménez, P Systems based Computing Polynomials with Integer Coefficients: Design and Formal Verification. 26. Naeimeh Elkhani, Ravie Chandren Muniyandi, Kernel P System Multi Objective Binary Particle Swarm Optimization on GPU. 27. Henry N. Adorna, Linqiang Pan, Boseng Song, On Distributed Solution to k-sat on Membrane Computing. 28. Zeyi Shang, Sergey Verlan, Gexiang Zhang, Miguel A. Martínez-Del-Amor, Luis Valencia-Cabrera, An Overview of Hardware Implementations of P Systems. 29. Mianjun Ge, Haina Rong, Ming Zhu, Gexiang Zhang, A Novel Approach for Detecting Fault Lines in a Small Current Grounding System Using Fuzzy Reasoning Spiking Neural P Systems. 30. Ignacio Pérez-Hurtado-De-Mendoza, Mario J. Pérez-Jiménez, Generation of Rapidly-exploring Random Trees by Using a New Class of Membrane Systems.

158 158 G. Zhang, Q. Yang and L. Pan The list of authors as well as their addresses are given below, with the aim of facilitating the further communication and interaction: 1. Henry N. Adorna, Algorithms & Complexity Lab Department of Computer Science, University of the Philippines Diliman, Diliman 1101 Quezon City, Philippines 2. Artiom Alhazov, Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan , China 3. Somnath Bera, Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan , China 4. Kelvin C. Buño, Department of Computer Science (Algortihms & Complexity), University of the Philippines Diliman, Diliman 1101 Quezon City, Philippines 5. Francis George C. Cabarle, Algorithms and Complexity Laboratory, Department of Computer Science, University of the Philippines Diliman, Quezon City, Philippines 6. Jym Paul Carandang, Algorithms and Complexity, Department of Computer Science, University of the Philippines Diliman, Diliman 1101 Quezon City, Philippines 7. K Chandrasekaran, National Institute of Technology Karnataka Surathkal, Mangalore, India kch@nitk.ac.in 8. Guangchun Chen, School of Computer and Software Engineering, Xihua University, Chengdu, , Sichuan, China 9. Xing Chen, Department of Math and Computer, Wuhan Polytechnic University, Wuhan , Hubei, China chenxing@163.com 10. James Cooper, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand jcoo092@aucklanduni.ac.nz 11. Ren Tristan A. de la Cruz, Algorithms and Complexity Laboratory, Department of Computer Science, University of the Philippines Diliman, Quezon City, Philippines rentristandelacruz@gmail.com 12. Erzsébet Csuhaj-Varjú, Department of Algorithms and Their Applications, Faculty of Informatics, ELTE Eötvös Loránd University, Budapest, Pázmány Péter Sétány, 1117, Hungary

159 A Summary of ACMC Yingying Duan, School of Math and Computer, Wuhan Polytechnic University, Wuhan , China Naeimeh Elkhani, Centre for Software Technology and Management, Faculty of Information Sciene and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia Rudolf Freund, Faculty of Informatics, TU Wien, Vienna, Austria Mianjun Ge, School of Electrical Engeneering, Southwest Jiaotong University, Chengdu, , China Marian Gheorghe, School of Electrical Engineering and Computer Science, University of Bradford, Bradford BD7 1DP, United Kingdom Ping Guo, College of Computer Science, Chongqing University, Chongqing, , China Nestine Hope S Hernandez, Algorithms and Complexity, Department of Computer Science, University of the Philippines Diliman, Diliman 1101 Quezon City, Philippines nshernandez@dcs.upd.edu.ph 20. Xuan Hou, School of Electrical Engineering and Electronic Information, Xihua University, Chengdu , China @qq.com 21. Juan Hu, School of Computer and Software Engineering, Xihua University, Chengdu, , Sichuan, China 22. S James Immanuel, Department of Mathematics, Madras Christian College, Tambaram, Chennai , India James imch@yahoo.co.in 23. Sergiu Ivanov, LACL, Université Paris Est C Créteil Val de Marne 61, av. Général de Gaulle, 94010, Créteil, France sergiu.ivanov@u-pec.fr 24. S Jayasankar, Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai , India fksjayjay@gmail.com 25. Yun Jiang, Detection and Control of Integrated Systems Engineering Laboratory, Chongqing Technology and Business University, China jiangyun@ .ctbu.edu.cn 26. Yuan Kong, College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao , China 27. Shuo Liu, Department of Math and Computer, Wuhan Polytechnic University, Wuhan , Hubei, China

160 160 G. Zhang, Q. Yang and L. Pan Xiangrong Liu, Department of Computer Science, Xiamen University, Xiamen , Fujian, China 29. Xiyu Liu, School of Management Science and Engineering, Shandong Normal University, Jinan, , China Miguel Ángel Martínez-del-Amor, Research Group on Natural Computing, Department of Computer Science and Articial Intelligence, University of Seville Jun Ming, School of Electrical Engineering and Electronic Information, Xihua University, Chengdu , China Ravie Chandren Muniyandi, Centre for Software Technology and Management, Faculty of Information Sciene and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia Atulya K Nagar, Department of Mathematics and Computer Science, Liverpool Hope University, Liverpool, United Kingdom nagara@hope.ac.ukg 34. Radu Nicolescu, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand r.nicolescu@auckland.ac.nz 35. Yunyun Niu, School of Information Engineering, China University of Geosciences in Beijing, Beijing , China niuyunyun1003@163.com 36. Linqiang Pan, Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan , China lqpan@mail.hust.edu.cn 37. Hong Peng, School of Computer and Software Engineering, Xihua University, Chengdu, , Sichuan, China ph.xhu@foxmail.com 38. Xixi Peng, College of Computer Science, Chongqing University, Chongqing, , China 39. Ignacio Pérez-Hurtado, Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, Universidad de Sevilla, Spain perezh@ue.es 40. Mario J. Pérez-Jiménez, Research Group on Natural Computing, University of Seville, Sevilla, Spain marper@us.es 41. Huaqing Qi, Department of Economics and Management, Wuhan Polytechnic University, Wuhan , Hubei, China

161 A Summary of ACMC S Raghavan, National Institute of Technology Karnataka Surathkal, Mangalore, India raghavan.sm2005@gmail.com 43. Alfonso Rodríguez-Patón, Department of Artificial Intelligence, Faculty of Computer Science, Polytechnical University of Madrid Campus de Montegancedo Boadilla del Monte 28660, Madrid, Spain 44. Haina Rong, School of Electrical Engeneering, Southwest Jiaotong University, Chengdu, , China ronghaina@126.com 45. Zeyi Shang, School of Electrical Engeneering, Southwest Jiaotong University, Chengdu, Sichuan, China sibanordol@126.com 46. Bosheng Song, Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan , China boshengsong@hust.edu.cn 47. K.G. Subramanian, Department of Mathematics, Madras Christian College, Tambaram, Chennai , India kgsmani1948@gmail.com 48. Wenxing Sun, School of Management Science and Engineering, Shandong Normal University, Jinan , China @qq.com 49. Zhang Sun, School of Electrical Engineering and Electronic Information, Xihua University, Chengdu , China 50. Robinson Thamburaj, Department of Mathematics, Madras Christian College, Tambaram, Chennai , India robin.mcc@gmail.com 51. D.G. Thomas, Department of Mathematics, Madras Christian College, Tambaram, Chennai , India dgthomasmcc@yahoo.com 52. Jherico Gabriel Q. Torres, Department of Computer Science (Algortihms & Complexity), University of the Philippines Diliman, Diliman 1101 Quezon City, Philippines jqtorres@up.edu.ph 53. Luis Valencia-Cabrera, Research Group on Natural Computing, University of Seville, Sevilla, Spain lvalencia@us.es 54. Sergey Verlan, Laboratoire d Algorithmique, Complexité et Logique, Département Informatique, Université Paris Est, France verlan@u-pec.fr 55. Fangxiu Wang, School of Math and Computer, Wuhan Polytechnic University, Wuhan , China wfx323@126.com

162 162 G. Zhang, Q. Yang and L. Pan 56. Jun Wang, School of Electrical Engineering and Electronic Information, Xihua University, Chengdu , China Tao Wang, School of Electrical Engineering and Electronic Information, Xihua University, Chengdu , China Chuanxiang Wei, School of Electrical Engineering and Electronic Information, Xihua University, Chengdu , China 59. Tingfang Wu, Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan , China Jianhua Xiao, School of Information Engineering, China University of Geosciences in Beijing, Beijing , China Shiyue Xie, Department of Computer Science, Xiamen University, Xiamen , Fujian, China 62. Xiaoshan Yan, Department of Computer Science, Xiamen University, Xiamen , Fujian, China 63. Jinyu Yang, School of Computer and Software Engineering, Xihua University, Chengdu, , Sichuan, China 64. Qiang Yang, Control Engineering College, Chengdu University of information Technology, Chengdu , China Weitao Yuan, School of Electrical Engeneering, Southwest Jiaotong University, Chengdu, , China 66. Shan Zeng, Department of Math and Computer, Wuhan Polytechnic University, Wuhan , Hubei, China Xiangxiang Zeng, Department of Computer Science, Xiamen University, Xiamen , Fujian, China 68. Gexiang Zhang, Robotics Research Center, Xihua University, Chengdu, , China Jieqiong Zhang, School of Information Engineering, China University of Geosciences in Beijing, Beijing , China 70. Yongpeng Zhang, School of Information Engineering, China University of Geosciences in Beijing, Beijing , China 71. Yuzhen Zhao, School of Management Science and Engineering, Shandong Normal University, Jinan , China zhaoyuzhen 72. Kang Zhou, Department of Math and Computer, Wuhan Polytechnic University, Wuhan , Hubei, China zhoukang

163 A Summary of ACMC Chaoping Zhu, Detection and Control of Integrated Systems Engineering Laboratory, Chongqing Technology and Business University, Chongqing , China 74. Ming Zhu, Control Engineering College, Chengdu University of information Technology, Chengdu , China Fig. 1. Group photo from the 6th Asian Conference on Membrane Computing

164

165 Miscellanea Call for a Position Prithwineel Paul Phone: prithwineelpaul@gmail.com I wish to know if there is a postdoctoral position available in a research group. I just submitted my doctoral research thesis on Finite Splicing Systems Computation and Control, under the supervision of Professor Kalpana Mahalingam, at the Indian Institute of Technology Madras, Chennai, India. For my postdoctoral research, I am particularly interested in (1) computational capabilities, complexity aspects, exploring language theoretic properties of different variants of DNA and Membrane computing models, (2) construction of algorithms using biologically inspired models to solve computationally hard problems, (3) exploring the properties of different variants of DNA and Membrane computing models when the ideas from regulated rewriting are incorporated into these models, (4) Investigate the properties, descriptional complexity measures etc. of Network of evolutionary processors, Network of splicing processors, accepting hybrid network of evolutionary processors and their variants. Also, developing techniques using these processors to solve NP-hard, NP-complete problems and so forth. I have attached herewith my curriculum vitae. Citizenship: Indian Date of Birth: 17 th September 1987 Education: (a) 2012 (dec)- PhD thesis submitted, under the supervision of Dr. Kalpana Mahalingam, Department of Mathematics, Indian Institute of Technology Madras, Chennai, India. (b) (dec): Guest faculty (R.C.C Institute of Information Technology, Government College of Engineering and Ceramic Technology).

166 166 Call for Position (c) : M.Sc. Pure Mathematics (Percentage: 79.1, Rank-3rd in the University), Department of Pure Mathematics, University of Calcutta, Kolkata, India. (d) : B.Sc. Mathematics (Honours), Ramakrishna Mission Residential College, Narendrapur affiliated to University of Calcutta, West Bengal, India. Ph.D. Thesis: Finite Splicing Systems: Computation and Control Awards and Fellowships: (a) CSIR-NET/JRF from India, June-2012, (b) GATE in Mathematics, Research Experiences: (a) Theory: (i) Descriptional complexity of computing models from DNA computing and Membrane computing, (ii) Computing capabilities of DNA and Membrane computing models, (iii) Time and space complexity of splicing systems, (iv) Computing by observing paradigm (membrane systems, G/O systems, splicing systems, insertion/deletion systems, sticker systems). (v) Membrane Computing (Splicing P systems, Tissue P systems, Spiking Neural P systems), (vi) Splicing grammar systems, Communicating distributed H systems, Time varying distributed H systems, Replication systems, flat splicing systems, Circular splicing systems, Alphabetic systems, Semi-simple splicing systems, Marked systems, Hybrid systems, monotone complete systems, reflexive splicing systems, (vii) Accepting splicing systems, accepting splicing systems with permitting and forbidding words, decidability problems in accepting splicing systems, (viii) Schützenberger constants, structure of reflexive splicing languages via Schützenberger constants, existence of Schützenberger constants in regular splicing languages, (ix) Decidability problems in Splicing systems, (x) Insertion/deletion systems and their variants, Matrix insertion/deletion systems. (xi) Sticker systems and their variants. (xii) Derivation languages (Chomsky grammars, Matrix grammars, I/ O grammars, Parallel communicating grammar systems, Communicating distributive grammar systems), Left Szilard languages, homomorphic images of Szilard languages of context-free grammars, ranking and unranking algorithm for left Szilard languages, relation of classes of Szilard languages and parallel complexity classes, (xiii) Regulated rewriting in formal language theory. (xiv) Network sof evolutionary processors, Networks of splicing processors, Accepting hybrid networks of evolutionary processors and their variants. (b) Programming & Computations: (iv) Unix/Windows, (iv) C, (v) Latex. Research Articles (Published and Communicated): (a) Kalpana Mahalingam, Prithwineel Paul (2016). Locally evolving splicing systems. Romanian Journal of Information Science and Technology, 19(4), (Impact factor 0.422).

167 Call for Position 167 (b) Kalpana Mahalingam, Prithwineel Paul and Erkki Mäkinen (2017). On derivation languages of a class of splicing systems. Acta Cybernetica (accepted). (c) Kalpana Mahalingam, Prithwineel Paul, Boseng Song, Linqiang Pan and K.G. Subramanian (2017). Derivation languages of splicing P systems. 12 th International Conference on Bio-inspired Computing: Theories and Applications (BIC-TA 2017) (accepted). (d) Kalpana Mahalingam and Prithwineel Paul (2017). Splicing recognizers of radius two can accept recursively enumerable languages. (Submitted for Publication). Office: Research Scholar, Residence: Prithwineel Paul, Room No.- 240A, Horizon Plaza, Block-C/1, Flat No- 113, Department of Mathematics, Deshbandhu Nagar, Indian Institute of Technology Madras, P.S.- Baguiati, Chennai , India. Kolkata , India. Teaching Experiences: Teaching assistant in Indian Institute of Technology Madras for (i) Series and Matrices (B. Tech), (ii) Basic Graph Theory (B. Tech.), (iii) Algebraic Theory of Codes and Automata (M. Sc.), Teaching Interests: I would like to teach the following area in the Department of Mathematics and Computer Science. (a) Automata theory Finite state automata, Pushdown automata, Turing machines, The Church-Turing thesis, Decidability, Reducibility. (b) Complexity theory Time complexity, Space complexity, P, NP, NPcompleteness, NP-hard, PSPACE, L, NL, co-nl, Intractability. (c) Unconventional models of computation DNA computing, Membrane computing. (d) Discrete mathematics, (e) Graph theory. Research Accomplishments: My research works are as follows: (a) The recombination behaviour of the DNA molecules was mathematically formalized in the form of splicing systems. Finite splicing systems can generate only regular languages. But, finite splicing systems can be computationally complete with different restrictions on the strings present in the system and splicing rules. At first, we observed computational power and descriptional complexity of a variant of splicing systems called locally evolving splicing systems. The descriptional complexity of the parameters (i.e., splicing rules and insertion/deletion rules) of this variant of splicing systems called locally evolving splicing systems has been examined. Also, I have introduced Matrix restricted locally evolving splicing systems and investigated descriptional complexity of the splicing as well as insertion/deletion rules.

168 168 Call for Position (b) In an another independent work, we have studied derivation languages of finite splicing systems and compared the family of languages generated as Szilard or Control languages by finite splicing systems with the families of languages in Chomsky hierarchy. Although the splicing operation is a string rewriting operation, we observed that the properties of the Szilard languages of context-free grammars are different from the Szilard languages of splicing systems. (c) We also investigated the derivation languages for splicing P systems, which is a computational model in membrane computing and compared the family of derivation languages generated by these models with the families of languages in Chomsky hierarchy. In this work, we defined derivation languages of a splicing P system in two different manners. Using the first definition we proved that any recursively enumerable language can be written as a homomorphic image of a Szilard language generated by a splicing P system with two membranes. Using the second definition we proved that any recursively enumerable language can be generated as a control language of a splicing P system with one membrane. (d) Splicing recognizers work as a language accepting device instead of language generating device. We constructed a splicing recognizer of radius two which can accept recursively enumerable languages which eventually proves a conjecture present in the literature. Also, I have constructed a new restricted variant of splicing recognizers of radius one which can accept recursively enumerable languages. In this work, the splicing recognizer is constructed by simulating a Turing machine. Research Interests: (i) Automata and formal languages theory, (ii) Unconventional models of computing (DNA computing, Membrane computing), (iii) Algorithmic self-essembly (iv) Algorithms and complexity theory, (v) Combinatorics on Words, (v) Regulated rewriting in formal language theory. References: (a) Dr. Kalpana Mahalingam, Department of Mathemaics, Indian Institute of Technology Madras, Chennai ( (office), kmahalingam@iitm.ac.in).

169 IMCS Prize Arriving in Sevilla As already known, one of the IMCS Prizes for The Application of the Year 2016 was awarded to the paper Improving Simulations of Spiking Neural P Systems in NVIDIA CUDA GPUs: CuSNP, by J.P. Carandang, J. Matthew, B. Villaflores, F.G.C. Cabarle, H.N. Adorna, M.A. Martínez-del-Amor, published in Proceedings of the Fourteenth Brainstorming Week on Membrane Computing. The prizes were awarded during CMC 2017, Bradford, UK, but the diploma and the Hamangia thinker statue reached Sevilla authors around the middle of October, when José M. Sempere visited Sevilla group and brought them. The next photos show the moments when Miguel Angel Martínez-del-Amor receives the prize.

170 170 IMCS Prizes

171 Interview with Gheorghe Păun Thomas Hinze Friedrich Schiller University Jena, Germany Nowadays, the membrane computing research area is going to celebrate its 20 th anniversary. Two decades in which a worldwide community emerged reflect remarkable results and sustained successes, but also challenges and open questions. More than 20 textbooks, around 100 PhD theses, more than 2,000 scientific papers and last but not least two annual conferences and the recently established International Membrane Computing Society witness the prosperous development of the underlying ideas behind membrane computing: mainly algebraic modelling approaches based on multisets for description of aspects of biological information processing inspired by living cells and related issues. Twenty years for humans, this time span marks the period from birth up to becoming an adult person. This gives a good reason to discuss milestones, strategies, and challenges to cope with in the future. Thomas Hinze (Friedrich Schiller University Jena, Germany) assumed the role of an investigative journalist and conducted an interview with Gheorghe Păun (Romanian Academy, Bucharest), the father of membrane computing: Twenty years ago, mid and end of the 1990 s, DNA computing was quite popular. After the famous Adleman experiment, many theoretical models and abstract descriptions of this unconventional approach to process information have been developed. Soon it turned out that there is a gap between experimental results obtained in the laboratory and highly idealised models neglecting side effects and malfunctions in the operational scheme. As a consequence, DNA computing more and more focused on nanotechnologies and specialised experimental setups to increase its practicability while the impact of new theoretical models became lower. At the same time, the idea of membrane computing was born, again based on abstract models and providing a field of activity for theoreticians after the pioneering era of DNA computing. Is there indeed any historically induced correlation between DNA computing and membrane computing? In principle, the membrane computing idea could have been invented much earlier...

172 172 T. Hinze In some sense, membrane computing existed before... Multisets appeared in various places, the Gamma language computing with multisets also existed before, the Chemical Abstract Machine, introducing the idea of membranes, arranged in a cell like manner, also existed. However, all these were only remotely linked with (inspired by) biology and were investigated with somewhat narrow goals and without getting together, without having a common name. (I said many times: maybe membrane computing was not the best choice, cellular computing is probably more appropriate, but it looks too general.) At the personal level, there is indeed a direct correlation between DNA and membrane computing: at that time, in the autumn of 1998, I was in Turku, Finland, after almost four years of intense research in DNA computing (I started in April 1994 hence before Adleman s experiment during a conference in Graz, Austria, when I got a copy of Tom Head paper where the splicing operation was introduced; I was immediately and totally intoxicated...). Together with the great seniors Arto Salomaa and Grzegorz Rozenberg, I just had at that time the Springer book DNA Computing. New Computing Paradigms. A real success (translations in Japan, China, Russia), but, as you said, idealised models, mathematics, automata and language theory. At the same time, it was already clear in the DNA community that the enthusiasm which followed after Adleman s experiment should be recalibrated (J. Hartmanis stated this in a paper published in Bulletin of the EATCS, immediately after Adleman s paper in Nature). The feeling was that DNA behaves better in its natural framework, in the cell, than in a test tube. I remember that Tom Head gave me illustrations of this observation, obtained when trying to test his splicing based models in a laboratory. Then, the idea came: let s go to the cell! I was not aware at that time about Gamma language, CHAM, the theory of multisets (called bags in some papers). But I had the long experience of working in automata and language theory, in particular, of working with grammar systems, sequential or parallel, and the recent experience of DNA computing (mainly based on splicing). The result of some long days and nights in Turku, in the autumn of 1998, was a computing model I called super-cell system (this is the terminology used in the TUCS Report where I published, in November 1998, the initial paper; the journal paper appeared about two years later). A sort of distributed parallel grammar working with multisets. With a lot of possibilities to restrict, extend, modify the model. I felt a distinct thrill and I have mentioned the report to a series of colleagues around the world. And the avalanche started... In a great extent, unexpected. Already in 2000 it was possible to organize a dedicated workshop, in Curtea de Argeş, my home town in Romania. (By the way, Kamala Krithivasan was the first one to change super-cell into P I used to say that this comes from promising, but soon the initial became totally neutral to me...) Anyway, membrane computing appeared at the right time. Natural computing was growing, DNA computing was at the beginning, but already discovering its limits (from the computational point of view), formal language theory was mature, already considered an old area, similarly the computational complexity (this

173 Interview with Gheorghe Păun 173 does not mean that these research areas were/are exhausted, but that they were out of fashion for accounters who provided money for projects...), biology was looking for mathematical and computer science tools. (After the completion of the Genome Project, it was said that the main challenge which follows is the simulation of the biological cell as a whole.) Numerous processing schemes found in nature have been proved to be Turingcomplete in principle: chemical computing based on substrate concentrations along with reaction kinetics, DNA computing, molecular assembly, evolutionary computing, artificial immune systems, neural computing, amorphous computing, quantum computing, membrane computing, and many further approaches. In contrast, it seems that real information processing in vivo carried out by signalling pathways, gene expression, or biological control systems are aimed at optimisation on the fly to keep the organism alive at a low energy consumption level. These tasks seem to be far away from the notion of enumerating a search space from scratch in an algorithmically controlled manner typically done in computer science. Do you think that the classical complexity measures from computer science are useful when considering biological information processing in vivo? Let me address a more general question: whether or not the theoretical computer science results, both dealing with computing power and complexity, are relevant for biology. My answer is YES, with arguments, not only because I am a mathematician. First, natural computing is both looking into biology, in nature in general, for ideas useful to computing, but it also aims to provide new tools, ideas, points of view to biology, physics and other sciences. There is nothing new here. Already in sixties there were discussions about the use in and the usefulness for biology of systems theory. There is a famous saying of Mihajlo Mesarovic, one of the important names of systems theory, which tells us something like that: biology will really take a profit from systems biology not by reformulating old questions in new terms, but by formulating new questions, taking the advantage of the new paradigm, and then we will not have the application of engineering principles to biological problems, but rather a field of systems biology with its own identity and in its own right. System biology and synthetic biology avant la letter... It should be also mentioned the crucial difference between short term objectives (unfortunately, specific to capitalistic thinking) and long term objectives. If we focus on applied mathematics, soon we will have no mathematics to apply this is another classic saying. Furthermore, at the educational level: for the standard muscles we have a lot of ways, tools, sport camps to train them. What about the brain muscles?... How can we train them? Just thinking rigorously (algorithms included), through mathematics, computer science, logical games... You have mentioned the Turing completeness of many computing models inspired from biology this particular result, a purely theoretical one at the first

174 174 T. Hinze sight, weird and almost non-sense for the traditional biologist, can have a really practical interpretation for the collaboration between computer science and biology. The reasoning is the following. We, the theoreticians, have the Rice theorem: any non-trivial question (of a semantic nature) about a Turing complete computing model is non-decidable (algorithmically). This means that there is no algorithmic way to answer any non-trivial question about a model of a biological proces/system, e.g., about the life of a cell. Such questions are important for biologists, but the answer should be found by ad-hoc means or by simulations. Hence, the need for simulating tools and techniques. And for computer scientists who can perform these simulations. A generalization: it is highly plausible that the biology modeling is constrained by various impossibility theorems, similar, for instance, to the Arrow theorem from social choice (multicriterial decision making). The models should have a series of properties, imposed/desired either by the computer scientist or by the biologist: adequacy to the modeled reality, abstract enough in order to capture essential features, understandability, easy programmability, scalability and so on. I bet that there are combinations of such properties which are contradictory, no model exists which has all the properties from a carefully chosen combination. Such a theorem should be formulated and proved in a mathematical framework, far from the language used by the biologist and not understandable by him, but the relevance for the biologist is obvious. (In computer science there are examples of this kind, for instance, Conrad theorems, of the form: there is no computing model at the same time programmable, efficient and able to evolve.) Now, coming to your precise question, the answer depends on the concrete framework, model, situation. Restricting our concern to biology, to the life itself, we have to be rather modest, aware of the fact that the goal of life is life, not computing. Moreover, what we interpret as being computations, are, in most cases, purely chemical and physical processes, we can also say analogical computations and for such a framework we do not have well developed computational theories. Typically, a specialised research area comes with a vision at its beginning. Appropriate modelling of information processing in living cells and organisms could be the central vision of membrane computing. Having in mind that a biological cell in average consists of around one billion molecules able to interact, and an adult human being comprises around cells, do you think that the corresponding entire system can be captured in a comprehensive manner which includes side effects and allows a detailed prediction of the cell s or organism s behaviour? I recalled before the saying that the simulation of the cell is the main challenge for this century. From the cell to the organism, the distance is rather large, of course. On the other hand, we do not have to underestimate the progresses, both in science and technology. We have an example at hand, the case of the game of GO. For decades, it was said that GO is the ultimate challenge for artificial intelligence. A prize of one million and a half dollars was open in eighties for the first

175 Interview with Gheorghe Păun 175 GO program able to defeat a human player of 1 dan level. The prize was never claimed, but, last year, in March 2016, AlphaGO, a GO playing project (I am not calling it a program), organized by Google, defeated Lee Seedol, 9 dan, from South Korea. The background was a deep learning strategy (neural computing) combined with fast searching techniques, starting with millions of human played GO games placed on thousand of computers... In short, natural computing plus brute force plus connectivity plus... I stress the term connectivity, which is said to be the engine of the fourth technological revolution (which just started). Actually, the AlphaGO achievement is simultaneous with other unexpected progresses of artificial intelligence in the last very few years: IBM Watson provides better medical diagnostics than human doctors and better advices that human lawyers, the automated theorem provers started to be useful to mathematicians, automated driving is imminent, and so on. Then, big systems can be approached also by means of statistical techniques, they also have a structure, regularities, recurrences, it is not necessary to separately take into account all components, all details. It is also true that in modeling such complex and subtle issues, like life and intelligence, it is possible that our current mathematics (and computer science) is not sufficient, one needs a new mathematics such an assumption was made also by Rodney Brooks, John McCarthy and other really knowledgeable people in artificial intelligence. I have the impression that our membrane computing community becomes smaller and smaller during the last three or four years, at least in Europe. That s why it is important to attract young scientists and students for this field of research. When I discuss with potential strong candidates about a PhD in membrane computing, I often hear that the students expect after the three-year PhD scholarship a long-term professional perspective, or at least a significant number of job announcements inside or outside universities mentioning skills in membrane computing as a prerequisite. Unfortunately, despite the high number of textbooks and papers in our field, there is a lack of attractive job offers in membrane computing. Particularly, senior community members with influence to relevant European boards and panels should be in charge here. Do you know any relevant initiative in this direction? What should be done to bring young researches into our community providing them a long-term perspective? There is a sociological problem here, with particular aspects related to our domain, but also with more general features, dealing with computer science in general. The things are connected, not completely clear to me especially difficult to forecast. All these are related to the idea of strong guys in a discipline, of the maturity age of a research area, of the (st)age of computer science itself. Typically, a research group appears and grows around a prominent personality (ideally, prominent both from a scientific point of view and in what concerns the socio-political influence, the capacity to get funds, for instance). In the last few years, several MC seniors from Europe, US and Japan retired, partially or totally, just because they have

176 176 T. Hinze reached the retirement age. This cannot happen without consequences, but I am still optimistic, as there are sufficiently many active (both scientific and from the organizatoric point of view) persons, enough strong groups while membrane computing is vigorously growing in China. Where I am not so optimistic is in what concerns the age of computer science. A convincing symptom that there is something to change in this area is the fact that almost all conferences in computer science have problems with the submissions, they repeatedly and systematically extend the deadlines. On the one hand, there probably are too many conferences, on the other hand, they are too specialized, addressed to a small community. I think that there is the time to change the style, to imitate the way the conferences in mathematics are organized: large, without ambitious goals in what concerns the selection and the publications, mainly an opportunity to meet colleagues of interest. In exchange, what mathematics should borrow from us is the Brainstorming! Relatively small size meetings, mainly if not fully devoted to working together. This is necessary in al sciences, because, in spite (maybe, also because) of internet, people have difficulties and have lost the habit of getting together, of interacting directly. It comes here another unfortunate situation, the shortage of money, as a consequence of the economic recession of about one decade ago. This influences everything, including our research area. Following the economic recession, we have to face the deterioration of international relations, with all bad things we see around, which also consume money and make more difficult the circulation of persons. Concerning the jobs, I can again invoke some ideas mentioned above: any way to train the brain is good, membrane computing or any other research area. With something additional in favor of membrane computing: this area is simultaneously rigorous, mathematically grounded, and rather flexible, with endless possibilities to prove your creativity; this area is connected with all branches of theoretical computer science, and also with practical computer science, from hardware to software engineering and programming; this area contains both pure mathematics, even speculative (e.g., hypercomputing), and direct interaction with real life, from biology to engineering, robot control, computer graphics, economics. All these can really contribute to build a researcher personality of success. If an institution looks for hiring employees qualified for today jobs, they will have problems tomorrow and after tomorrow; the jobs, especially in computer science, are rapidly evolving, changing. Training, flexibility, having an open mind are more important that specialization. Maybe I look biased, as advocating pro domo sua, but similar reasonings can be found in many places let us hope that also people involved in selection commissions for jobs and projects, if not aware of this, will learn it in time... In particular, the students should be aware of this. It is highly possible that the job they had in mind when starting the university studies will no longer exist after four or more years, when they graduate...

177 Interview with Gheorghe Păun 177 Who outside our membrane computing community is really interested in the results produced by our community? Bioinformaticians, systems biologists, medical scientists, and pharmacologists commonly employ different modelling approaches and do not currently use membrane systems. How can we convince them and attract them to join our community? It seems that their problems and open questions differ from proving Turing-completeness, from artificial termrewriting algorithms in maximally parallel manner and from other topics under study in our community. Should our community be open for interdisciplinarity in general? Inter-, multi-, trans-disciplinarity. It is an important point here: computer science is an universe, biology is another one, with different languages, goals, tools, interests at least in this moment, at least at the first sight. See also the previous discussion. It is rather difficult for an old mathematician/computer scientist to collaborate, in the comprehensive sense of the term, with an old biologist. According to my (limited, of course) experience, you need about one year of discussions until understanding each other. One solution would be to start simultaneously courses of biology and of mathematics/computer science, from the first year of the university studies. There are schools of this type (e.g., in Verona, Italy) I use to say that they are schools for Nobel Prize candidates... On the other hand, young researchers, with theoretical interests or with more practical interests, from both mathematics/computer science and biology (biomedicine, ecology, etc.) should be aware of the fact that the collaboration with researchers in other areas is highly rewarding, for both sides. Biology is making efforts to follow the old examples of physics, astronomy, chemistry, as well as the more recent examples of economics, linguistics, some social sciences, to get mathematized with the particular detail that the computer simulations are crucial in biology and related areas. The only way to get predictions concerning the evolution of processes, systems, phenomena is to use mathematical models, treated analytically when this is possible and through computer simulations when analytical solutions are not available. Then, such simulations, even using complex hardware and/or software supports, are cheaper, faster, safer than laboratory or economic simulations and experiments. This is common sense, but this is not fully understood by all interested people. Our community is open, by definition, I would say, to inter-disciplinarity but again there is a sociology issue here: personal contacts matter, the personal initiatives, the pro-active efforts in contacting people from the environment of our discipline. It is also important to (1) get convincing applications and (2) to present them in a marketable manner. Especially in the latter direction we still have a lot to do, until passing a certain threshold beyond which out tools will become standard tools. An important detail/argument in this context: the attractive features of membrane computing, especially from the point of view of the biologist. Understandability is obvious, the same with programmability, scalability, adequacy and so on. Two further crucial features: the discrete nature of the models and the emergent

178 178 T. Hinze behavior of the models. Using differential equations in biology only/mainly because they were so useful for physics and other areas where the reality is clearly of a continuous nature is simply wrong (and proved to provide wrong results in various cases). This does not mean to say that the cell does not have continuous aspects, but just that the discrete aspect are prevalent. Multisets! Biochemistry! The obvious link to membrane computing is forced. Then, it is important the fact that the behavior of a multiset of chemicals which evolve by means of a set of multiset rewriting rules cannot be predicted, neither by an intuitive analysis, nor by any analytic tool. Just have a look to a map describing the control pathways in a cell, involving tents if not hundred of chemical compounds and hundred if not thousand of reactions, most of them conditional, taking place according to given promoters or inhibitors... The socalled system effect, or emergent behavior (it can be compared with the non-linear evolution of solutions of differential equations or even with their singularities, catastrophic points in terms of René Thom) cannot be discovered and described without simulations on computer. Convincing enough reasons to have the biologist interested in membrane computing. Talks and forums with representatives from industry, especially from research and development departments appear to mark a new trend in suitable conferences to bring together scientists from universities with potential investors and their demanding open problems. Do you think that this could be a promising platform also for CMC? I am not familiar with this kind of initiatives, but any contact with people outside our community, in particular with possible users of our models, results, programs and so on could be useful. And, before trying it we cannot evaluate it... High-quality simulation software is unequivocally a helpful tool in many facets of membrane computing. Indeed, there are many separate solutions like P-Lingua and others. Maintenance, administration, and curation of those software products are essential to achieve and to ensure its high quality. I have the impression that papers addressing improvements and error-corrections for existing simulation software get a lower impact when reviewed for publication in comparison to papers presenting a new tool (often with numerous malfunctions). How can we cope with this situation? Again, I am not the right person to answer this very precise question, but it should be a problem of audience, market, topicality. Submitted to the right place, any professional paper will get a professional evaluation. In particular, the internal conferences and publications of membrane computing community are for sure open to such papers. As publications, I have in mind the already two years old Bulletin of IMCS and the forthcoming international Journal of Membrane Computing, to be produced and distributed by Springer-Verlag.

179 Interview with Gheorghe Păun 179 What are from your point of view the three most important open scientific questions and problems to be answered by research activities within our community during the next years? What is the benefit from the corresponding answers and solutions for the mankind, especially for people outside our community? After (almost) twenty years of research, membrane computing is a rather mature area, well developed at the theoretical level. We know already that (1) our models are rather powerful, in most cases equivalent in power with Turing machines, (2) that they are also efficient, able to solve computationally hard problems in a feasible time, and (3) that they are very versatile in modeling processes and systems from many real life areas, such as biology, biomedicine, ecology, economics, technology etc. Still, there are plenty of open problems and research topics. It is impossible to list three precise problems, but, in my opinion, there are certain research directions which deserve to be explored more systematically and which promise to have a relevance also outside our community. First, it would be good to have more elaborate and convincing applications in biology and related areas. This supposes to have a team comprising both computer scientists and biologists/ecologists/medical doctors, to build models and to check research hypotheses coming from the application areas. That is, looking for predictions, not for postdictions, for confirming the reliability of models. Of course, suitable software and, possibly, implementations on special hardware are also necessary, in order to perform experiments and simulations with data of a real life size. Then, I think that the whole philosophy of membrane computing (multisets, distribution, parallelism, unstructured programs given as opportunistic sets of instructions, communication, and so on) could be useful for conceiving new types of computer architectures, of single computers or of networks, clouds, whatever, as well as for imagining new types of software, closer to the way nature has build cells, communities of cells, processes inside them. My formulation is vague, but I feel that there is a lot to learn from nature at this level. Finally, I am confident that the more systematic study of spiking neural P systems and of numerical P systems can lead to interesting results, both at the theoretical level and from the application point of view. Connections with the brain functioning in the former case and the possibility to compute faster mappings with a practical significance, in the latter case, are to be expected. In particular, the result of the Milano team showing that numerical P systems using in their programs all the four operations, +,,,, working a polynomial time, characterize PSPACE (without using the usual space-time trade-off) is intriguing for me. No membrane division, no exponential space just the intrinsic power of programs (which, in each time unit perform complex computations). Can this intrinsic power be embedded in a computer hardware, can it be simulated in an efficient way? This would entail beautiful applications. How do you summarise membrane computing in ten years from now which trends, developments, and progress could be made in the meantime?

180 180 T. Hinze Well, the past is difficult to remember, the present is difficult to understand, the future is difficult to forecast... In a number of years, maybe not only ten, I think that membrane computing, together with other related areas, will... disappear, dissolved in a new age of biology, which will intimately incorporate mathematical and computer science techniques, to the benefit of the study of living systems and medicine. I have predicted this also in the end of my Reception Discourse in the Romanian Academy, a couple of years ago, also proposing the name infobiology for the new science. In his Springer book, Vincenzo Manca proposes the name infobiotics. Whatever name, the idea is to essentially consider also information as a basic ingredient of living systems, thus making necessary to use also information theory and informatics (computer science in a more European terminology) as supporting sciences, besides chemistry and physics as it is the case now. On shorter terms, I am pretty optimistic with respect to the popularity of membrane computing in China and other Asian countries, about the International Membrane Computing Society and its activity, the forthcoming journal especially. In two years, I will organize the twentieth edition of CMC back in Curtea de Argeş, where it started, initially as a workshop, two decades ago. I am convinced that this will be one of the largest meetings in this series, with many friends brought back, by nostalgia, to membrane computing and to Curtea de Argeş... This is already an invitation to the reader. Gheorghe, I express my gratitude being aware of the demanding nature of many questions.

181 Transdisciplinarity, Creativity, Elegance (Obituary for Tom Head) Natasha Jonoska 1, Gheorghe Păun 2, Grzegorz Rozenberg 3 1 University of South Florida, Department of Mathematics 4202 e. Fowler Av. CMC345, Tampa, Florida, USA 2 Romanian Academy Calea Victoriei 125, Bucharest, Romania 3 Leiden University, LIACS Niels Bohrweg 1, 2333 CA Leiden, The Netherlands Thomas J. Head, known by friends and collaborators as Tom, passed away on November 10, 2017, at the age of 83 (he was born in Tonkawa Oklahoma, on Jan ). His undergraduate and graduate studies were in pure mathematics, at