Process Algebras for Petri Nets

Transcription

1 Roberto Gorrieri Process Algebrs for Petri Nets The Alphbetiztion of Distributed Systems

2 Roberto Gorrieri Diprtimento di Informtic - Scienz e Ingegneri Università di Bologn Bologn, Itly ISSN ISSN (electronic) Monogrphs in Theoreticl Computer Science. An EATCS Series ISBN ISBN (ebook) DOI / Librry of Congress Control Number: Springer Interntionl Publishing AG 2017 This Springer imprint is published by Springer Nture The registered compny is Springer Interntionl Publishing AG The registered compny ddress is: Gewerbestrsse 11, 6330 Chm, Switzerlnd

3 Foreword Distributed, open-ended systems re ubiquitous in tody s informtion nd communiction technology: in most cses they exhibit n independent, concurrent behvior. The process of specifiction, design nd development requires forml models ble to express interesting properties nd to llow for efficient vlidtion nd verifiction procedures. In ddition to their role s design tools in ICT engineering, distributed, concurrent models re very successful in formlizing key concepts of biology, economics, complex systems nd so on. In mny cses it is convenient to develop specilized models for the clsses of systems we re interested in, equipped with convenient fetures. For instnce, while in ll cses forml semntics is required, there is conflict between model expressiveness nd complexity of verifiction lgorithms. Another importnt choice is between grphicl model nd text-bsed model, both equipped with observtion criteri nd consequent semntics: the former is often more perspicuous, but less convenient for specifying lrge, modulr systems in compositionl wy. The min chievement of this book is to present hierrchy of six models of incresing expressiveness, ech of them described both in grphicl form nd in textul, term-like form. Plenty of results gurntee tht there is one-to-one correspondence between the two representtions, nd tht corresponding forms hve the sme interleving nd step semntics. In relity, grphicl nd textul forms re not equivlent: the former is more expressive since its semntics cn be enriched with observtionl criteri to mke certin relevnt properties evident, which re present, but hidden, in the ltter. For instnce, the presenttion in grphicl form of the solution of the dining philosophers problem cn be shown to be correct, while the textul version, once the choice is mde of observing the behvior sequentilly, hs too restricted observtion cpbilities. In other words, s soon s the textul form is trnslted into the grphicl form, new observtion nd nlysis methods become pplicble. Thus the trnsltion hs the flvor of ssociting with the lnguge more expressive concurrent semntics. In the book, cler choice is mde bout which models to hndle: process description lnguge CCS nd Plce/Trnsition (P/T) Petri nets, with vrious syntctic

4 restrictions to yield the expressiveness hierrchy. Only the top level, Nonpermissive nets, is n extension of P/T nets: the extension is needed to chieve Turingcompleteness. I fully gree. There re severl resons for the choice. CCS is equipped with rich theory, s testified by severl textbooks nd monogrphs, some of them very recent. Also, CCS is the strting point of the most populr lnguge for mobile systems, i.e. π-clculus, nd of its extensions for security nd service-oriented rchitectures. P/T Petri nets were the first concurrency model (Crl Petri s thesis is dted 1962), nd they re mybe the model with the richest theory, the lrgest ppliction experience nd the most extensive collection of tools. Furthermore, (unsfe) finite P/T nets hve n esy-to-understnd cyclic behvior with possibly infinite set of sttes, but some importnt properties, e.g. rechbility, re still decidble. The technicl content of the book is of extremely high qulity. Actully, the uthor personlly contributed with number of seminl, well-known ppers to the historicl development of the min concepts nd results in the book. Also, the presenttion of complex results, which ppered over lrge spn of yers (the bibliogrphy contins bout 150 cittions), required remrkble effort to mke them consistent nd uniform. In ddition, some importnt results, needed to complete the full picture, re published in the book for the first time, s pproprite in reserch monogrph. Finlly, the presenttion nd justifiction of the conceptul developments nd of the forml chievements is well rticulted nd explined, nd supported by plenty of exmples s required in n dvnced textbook. The book will be precious for those interested in the truly concurrent semntics of distributed communicting systems. A system designer, given specifiction in CCS, might wnt to better understnd its behvior by looking t the corresponding net, by checking there the given concurrency requirements, nd possibly by tking dvntge of existing tools. More relevntly from methodologicl point of view, computer scientist, iming t studying prticulr clss of concurrent distributed systems, might define specifiction lnguge bsed on vrints of CCS nd Petri nets by pplying the pproch in the book. Ugo Montnri Pis, June 2016

5 Contents 1 Introduction The Alphbetiztion of Distributed Systems The Hierrchy Structure of the Book Interleving vs True Concurrency Beyond Turing-Completeness Lbeled Trnsition Systems Lbeled Trnsition Systems Behviorl Equivlences Strong Equivlences Wek Equivlences Step Trnsition Systems Petri Nets Introduction Plce/Trnsition Petri Nets Some Clsses of Petri Nets Dynmiclly Rechble nd Stticlly Rechble Subnets Decidble Properties Coverbility Tree Rechbility, Liveness nd Dedlock Behviorl Equivlences Net Isomorphism Interleving Semntics Step Semntics Nonpermissive Petri Nets Behviorl Equivlences Turing-Completeness... 74

6 4 The Bsic Clculus: SFM Syntx Opertionl LTS Semntics Expressiveness Congruence Opertionl Net Semntics Representing All Sequentil Finite-Stte Mchines Denottionl Net Semntics Adding Asynchronous Prllel Composition: CFM nd BPP CFM Interleving LTS Semntics Step Semntics Opertionl Net Semntics Representing All Concurrent Finite-Stte Mchines Soundness Denottionl Net Semntics BPP: Bsic Prllel Processes Expressiveness Opertionl Net Semntics Representing All BPP Nets Denottionl Net Semntics Adding Communiction nd Restriction: FNC Syntx Restricted Actions nd Extended Processes Syntctic Substitution Sequentil Subterms Opertionl LTS Semntics Expressiveness Step Semntics Opertionl Net Semntics Plces nd Mrkings Net Trnsitions The Rechble Subnet Net(p) Representing All Finite CCS Nets Soundness Denottionl Net Semntics RCS Adding Multi-prty Communiction: FNM Preliminries Syntx nd Informl Semntics Extended Processes nd Sequentil Subterms Well-Formed Processes

7 7.2 Opertionl LTS Semntics Expressiveness Congruence Problem Step Semntics Step Bisimilrity Implies Interleving Bisimilrity Step Bisimilrity Is Congruence Opertionl Net Semntics Plces nd Mrkings Net Trnsitions Properties of Net Trnsitions The Rechble Subnet Net(p) Representing All Finite P/T Nets Expressiveness Soundness Denottionl Net Semntics RMCS Adding Atomic Tests for Absence: NPL Syntx Opertionl LTS Semntics Expressiveness Congruence Problem Step Semntics Opertionl Net Semntics Plces nd Mrkings Net Trnsitions Properties of Net Trnsitions The Rechble Subnet Net(p) Representing All Finite NP/T Nets Soundness Denottionl Net Semntics RNPL Generliztions nd Vrint Semntics Communicting Petri Nets Vrint Net Semntics Generl Restriction Asynchronous Communiction Other Lnguges? Future Reserch Glossry References Index...299

8 Chpter 1 Introduction Abstrct This introductory chpter outlines the min problem delt with in this book: finding suitble lnguges for representing clsses of Petri nets, tking inspirtion from the process lgebrs developed in the lst four decdes. The structure of the book is outlined nd some hints on how to red it re presented. Finlly, it is lso rgued tht Turing-completeness is not sufficient criterion to compre the expressive power of different process lgebrs, becuse the sets of problems in distributed computing tht two lnguges cn solve my be different, even if both include ll the Turing-computble functions. 1.1 The Alphbetiztion of Distributed Systems A distributed system is computer system mde of severl components, implemented in hrdwre or softwre or s combintion of both, tht my be locted t different sites, even t geogrphicl distnce, nd tht cooperte to ccomplish tsk or coordinte to offer service by mens of suitble communiction protocols bsed on messge pssing. The most notble exmple of distributed system is the Internet, whose most importnt service is the World Wide Web. At very bstrct level of detil, the min feture of distributed system is distribution: the globl stte of the system is composed of collection of locl sttes, physiclly locted t different sites, nd ech ctivity tht the system performs my ctully involve only subset of these locl sttes. Another importnt feture is tht communiction tkes plce only by messge pssing, so tht ny informtion exchnge hppens by mens of explicit communiction primitives of send or receive; in other words, there is no globl memory, shred by the components. The communiction mechnism cn be synchronous or synchronous: the former when the send ction nd the receive one re performed by the intercting prtners t the sme time; the ltter when the send ction is decoupled from the receive one. Mny other fetures of distributed systems re relevnt, such s heterogeneity of the components, possible independent filure of components, the bsence of Springer Interntionl Publishing AG 2017 R. Gorrieri, Process Algebrs for Petri Nets, Monogrphs in Theoreticl Computer Science. An EATCS Series, DOI / _1 1

9 2 1 Introduction globl clock, sclbility etc., but for the ims of this book, only distribution nd communiction re considered. In prticulr, communiction is ssumed to be synchronous, becuse this mechnism cn lso esily implement the synchronous one (it is enough to put medium, such s buffer, between the two prtners of the communiction to get n synchronous communiction), while the reverse is more difficult to chieve. Mny semntic models of computtion hve been proposed to model distributed systems; here we give short, not exhustive, list: Petri nets [Petri62, Hck76b, Pet81, Rei85, MM90, JK95, MR95, Bus02, Rei13]; Trnsition systems [Kel76, Mil80, Plo04b, Gl01, Sn12, GV15]; Event structures [NPW81, Win87, Win88, BMM06]; Cusl trees [DD89, DD90, BMS15]; Concurrent histories [DM87]; Sttechrts [Hr87, HPSS87]; Messge sequence chrts [RGG, DH99]; Khn process networks [Kh74]. Ech of these hs its own pros nd cons. However, mong them, we chose Petri nets, for the following resons: 1. distribution is first-clss concept (which is not the cse for, e.g., lbeled trnsition systems); 2. Petri nets cn model recursive behvior with finite structure (which is not the cse for, e.g., event structures); 3. they re widely studied model (see, e.g., [RR98, DRR04] nd the references therein), equipped with simple, precise, forml semntics, both for the so-clled liner-time nd brnching-time semntics (which is not the cse for, e.g., messge sequence chrts); 4. they re equipped with nlysis techniques tht re decidble in mny cses, s described in Section 3.3 (which is not the cse for, e.g., concurrent histories), nd tht re sometimes supported by utomtic or semi-utomtic softwre tools (see, e.g., [TA15, Tool] for surveys on Petri net tools); nd, finlly, 5. there is lrge literture of pplictions of Petri nets to the modeling of rel distributed systems (see, e.g., [RR98b, Rei98]) nd the references therein). From now on, t n bstrct level, we tke the liberty of identifying distributed system with the Petri net which models it. Therefore, we shll consider specific clsses of Petri nets s specific clsses of distributed systems. Mny specifiction lnguges hve been proposed to describe rective, distributed systems, strting from the seminl work by Hore with CSP [Ho78, Ho85, Ros98], nd Milner with CCS [Mil80, Mil89, GV15]. These lnguges re usully clled process lgebrs, to reflect the lgebric nture of their syntctic nd semntic definitions. Mny process lgebrs hve been proposed in the literture: besides CSP nd CCS, lso ACP [BK84, BW90, BBR10] by Bergstr, Klop nd

10 1.1 The Alphbetiztion of Distributed Systems 3.0 b.0.b.0 + b..0 b b 0 b b.0.0 b b Fig. 1.1 Interleving lw: two isomorphic LTSs.0 b.0.b.0 + b..0 {} {b} {} {b} 0 b.0 {,b}.0 0 b.0.0 {b} {} {b} {} Fig. 1.2 Two step trnsition systems Beten (who coined the term process lgebr), LOTOS [BB87, BLV95] by Bolognesi nd Brinksm (who included bstrct dt types in process lgebr), CIRCAL [Mil85] by Milne (who introduced the step semntics for process lgebr), nd the π-clculus [MPW92, Mil99, Pr01, SW01] by Milner, Prrow nd Wlker (which models mobility), just to mention few (see [Be05] for historicl overview, nd [BPS01] for technicl survey on the mny fcets of process lgebrs). The semntics of these lnguges hve minly been given in n interleving style, either in terms of execution trces (e.g., CSP) or in terms of lbeled trnsition systems (e.g., CCS), but in no wy re prllelism (or co-occurrence of independent ctions) nd distribution modeled in these semntics. An exmple my help clrify the ide. The process composed of two prllel ctions nd b is denoted in CCS by the term.0 b.0, while the sequentil process performing these two ctions in either order is denoted by.b.0 + b..0. Inthe lbeled trnsition system semntics of [Mil89], these two CCS processes originte the isomorphic lbeled trnsition systems in Figure 1.1, so tht they re semnticlly equl. This is the essence of the so-clled interleving lw: (finite-stte) prllel process is semnticlly equivlent to (finite-stte) possibly nondeterministic, sequentil process. This is cler drwbck of the interleving semntics, s we will rgue further in Section 1.4. One my enrich the lbeling of the trnsition system by using, insted of single ctions, multisets of concurrently executble ctions: this is the so-clled step semntics (originlly introduced in [NT84, Mil85]), which is refined enough to model

11 4 1 Introduction.0 id id b.0.b.0 + b..0 ) b) b b b id id 0 b 0 Fig. 1.3 The Petri nets for.0 b.0 in ) nd.b.0 + b..0 in b), ccording to the DDM technique the potentil prllelism of process. According to the step semntics,.0 b.0 nd.b.0 + b..0 re not equivlent, s only the former cn perform trnsition lbeled with the set {,b}, denoting tht these two ctions cn be performed t the sme time, s outlined in Figure 1.2. Still, no informtion on stte distribution is ctully given. To chieve this, one hs to consider more detiled model, where the monolithic, globl stte of the system is decomposed into collection of locl sttes, such tht ech ction the system performs my be ctully due to subset of the locl sttes, only: this is indeed wht Petri nets cn model, where the globl stte, clled mrking, is multiset of locl sttes, clled plces. Strting from the seminl work by Degno, De Nicol, Montnri nd Olderog [DDM88, Old91], CCS nd CCSP (i.e., mixture of CCS nd CSP) hve been equipped with n opertionl Petri net semntics, so tht distribution is first-clss concept, nd prllelism nd nondeterminism re modeled differently. For instnce, ccording to this pproch, the Petri nets for.0 b.0 nd.b.0 + b..0, depicted in Figure 1.3, show visibly tht the former process is composed of two sequentil subprocesses, running possibly in prllel, while the ltter is just one sequentil process. To be more precise,.0 b.0 origintes distributed globl stte (i.e., mrking) composed of two locl sttes: plce.0 id, representing the sequentil process.0 decorted with informtion bout the context (it is the left component of prllel process), nd plce id b.0. The number of tokens in plce denotes the number of instnces of tht sequentil process, tht re vilble in the current globl stte; in our exmple, only one token is present in plce.0 id nd in plce id b.0. A trnsition (represented s lbeled box) is defined by triple (m 1,l,m 2 ), where m 1 is its pre-set, i.e., the mrking specifying the tokens to be consumed for its executbility, l is its lbel, nd m 2 is its post-set, i.e., the mrking specifying the token to be produced upon completion; in our exmple, the two trnsitions re ({.0 id},,{0 id}) nd ({id b.0}, b,{id 0}). Of course, these two trnsitions cn be performed in either order, independently of ech other, nd even in prllel: t the

12 1.1 The Alphbetiztion of Distributed Systems 5 ).0 id id.0 b).0 0 id id 0 Fig. 1.4 The nets for.0.0, ccording to the DDM technique in ) nd Goltz s technique in b) sme time, the two tokens in the initil mrking {.0 id, id b.0} cn perform their respective trnsitions nd end up in the finl mrking {0 id, id 0}. On the contrry,.b.0 + b..0 is modeled with mrking which is composed of one single token (i.e., one single instnce of.b.0 + b..0), which cn perform its trnsitions only sequentilly. This technique, sometimes clled the DDM technique, ws esily exploited lso to give net semntics to other lnguges [DGM88, BG89, GV11], witnessing the gret generlity nd wide pplicbility of the ide. However, this semntics hs one mjor drwbck: the exploited clss of Petri nets is limited to so-clled sfe Petri nets, i.e., nets where ech plce my contin t most one token, so tht identicl prllel subprocesses re mpped to distinct plces. Goltz [Gol88, Gol90] proposed denottionl unsfe Petri net semntics for the CCS subclculus without restriction; this ide ws lso followed by [GM90b] in n opertionl setting. To clrify the semntic difference between the DDM technique nd Goltz s, let us consider the CCS process term.0.0. Figure 1.4() shows its ssocited net ccording to the DDM technique, where the two identicl instnces of the sequentil process.0 re mpped to two distinct plces, becuse of the contextul decortion; Figure 1.4(b) shows the net obtined ccording to Goltz s pproch, which is n unsfe net with two tokens on the sme plce.0. Mny other reserchers studied the problem of relting process lgebrs nd Petri nets (see, e.g., [GV87, Tu89, GR94, MY94, GM95, Mey09, BBGM14] nd the references therein). Of prticulr interest is the work by Best, Devillers nd Koutny [BDK01, BDK02], who proposed rther rich lgebr of expressions, whose semntics is given, however, in terms of sfe nets only. All these ppers focus on the problem of defining suitble distributed, net-bsed semntics for some chosen process lgebr; hence, we cn lbel this line of reserch with the motto Petri Nets for Process Algebrs. The min im of this book is to pproch the reverse problem of finding the miniml set of process lgebric opertors tht re necessry in order to model ll nd only the Petri nets of certin clss, whence the title Process Algebrs for Petri Nets.

13 6 1 Introduction This book offers definitive nswer to this question. We single out six clsses of finite Petri nets, nd, for ech clss, we define corresponding process lgebr, such tht: ech term p of the process lgebr is given distributed semntics in terms of net Net(p) of tht clss (only nets of tht clss cn be modeled by the process lgebr); moreover, for ech net N(m 0 ) of tht clss (i.e., for ech net N with initil mrking m 0 ), we cn single out term p of the corresponding process lgebr, whose semntics is net isomorphic to N(m 0 ) (ll the nets of tht clss re represented by the process lgebr, up to net isomorphism); nd, finlly, ll the opertors of the process lgebr re necessry to get these results (the set of opertors, for ech proposed process lgebr, is irredundnt). Therefore, since clss of finite Petri nets is ment s clss of distributed systems, our contribution mounts to lphbetizing these six clsses of distributed systems, i.e., providing six lnguges, s simple s possible, ech one representing ll nd only the distributed systems of certin clss. 1.2 The Hierrchy Which clsses of Petri nets will be considered? And which corresponding process lgebrs will be singled out? Here we describe the hierrchy of the six clsses of nets nd lnguges, which is lso outlined in Figure 1.5, strting from the lest expressive one. All the clsses of nets, except the lst one, re subclsses of soclled Plce/Trnsition Petri nets (P/T nets, for short, see Chpter 3 for detils). 1. Sequentil Finite-Stte Mchines: A finite-stte mchine is finite Petri net whose trnsitions re strictly sequentil, i.e., the pre-set nd the post-set re singletons so tht ech trnsition is performed by single sequentil process tht evolves into single sequentil process; such net is sequentil if the initil mrking of the net is singleton; therefore, sequentil finite-stte mchine describes strictly sequentil system composed of one single sequentil process only. The corresponding process lgebr is clled SFM, nd is essentilly finitestte CCS [Mil89], whose opertors re the empty process 0, ction prefixing μ.p, choice p+q, nd constnts C, equipped with definition useful for describing recursive behvior. 2. Concurrent Finite-Stte Mchines: A finite-stte mchine is concurrent when the initil mrking is rbitrry, hence not necessrily singleton. In this wy, the distributed systems tht nets of this sort re ble to model re composed of collection of strictly sequentil processes tht work in prllel, but without ny form of synchroniztion. The corresponding process lgebr is clled CFM, nd enriches SFM with the binry opertor of synchronous (i.e., without synchroniztion cpbilities) prllel composition p q, to be used t the top level only.

14 1.2 The Hierrchy 7 Finite Nonpermissive Petri nets NPL Finite Plce/Trnsition Petri nets FNM Finite CCS nets FNC BPP nets BPP Concurrent FSM CFM Sequentil FSM SFM Fig. 1.5 The hierrchy of net clsses nd process lgebrs 3. BPP nets: The term BPP is the cronym of Bsic Prllel Processes, nd ws coined in [Chr93] to denote simple process lgebr, which essentilly extends CFM by modifying the ction prefixing opertor: in term μ.p, the ction μ my prefix not only sequentil process (s for CFM), but lso prllel process. The clss of finite nets tht such lnguge cn model re clled BPP nets, whose distinguishing feture is tht the pre-set of ech trnsition is singleton, while the post-set cn be rbitrry. 4. Finite CCS nets: If BPP is extended by enhncing its prllel composition opertor to llow for communiction (ccording to the binry, hndshke, synchroniztion discipline of CCS), nd by lso including the CCS restriction opertor (ν)p (in order to force synchroniztion within p), to be used t the top level only, then the resulting clculus, clled FNC, is essentilly finite-net CCS [GV15]. The clss of nets tht this lnguge cn represent re clled finite CCS nets: they re ll the finite Petri nets such tht the pre-set size of ech trnsition cn be t most two nd, if it is two, then the lbeling of this trnsition must be the invisible ction τ. 5. Finite P/T nets: The only constrint on this clss of Petri nets is tht they must be finite; in prticulr, the pre-set of trnsition cn be of rbitrry size nd with n rbitrry lbeling. The process lgebr FNC is to be extended with mechnism for multi-prty synchroniztion; this is chieved by introducing new prefixing opertor, clled strong prefixing in opposition to the norml one nd whose syntctic form is.p, which cn model tomic ctions; multi-prty synchroniztion is modeled s n tomic sequence of binry, CCS-like synchroniztions. The resulting process lgebr is clled FNM nd is slight simplifiction of finite-net Multi-CCS [GV15]. 6. Nonpermissive Petri nets: This clss is generliztion of finite P/T nets, where trnsition lso specifies set of plces to be tested for bsence of dditionl

15 8 1 Introduction tokens, clled the neg-set of the trnsition. In fct, differently from finite P/T nets, trnsition cn be executed if nd only if the required tokens in the pre-set re vilble, nd lso no dditionl token is present in ny plce belonging to the neg-set of the trnsition. Therefore, trnsition tht my be executble t given mrking m my be disbled t lrger mrking m, becuse some of the dditionl tokens of m my inhibit the executbility of the trnsition. Differently from finite P/T nets, Nonpermissive Petri nets, which re generliztion of P/T nets with inhibitor rcs [FA73, Hck76, Pet81, JK95, Bus02], re Turingcomplete formlism. The corresponding process lgebr, clled NPL, extends FNM with vrint of the strong prefixing opertor, D.p, whose strong prefix D expresses tht trnsition originting from p is ctully executble by D.p in prllel context, provided tht no instnce of constnt D is currently ctive in this prllel context; hence, D.p executes test for bsence of the testble constnt D, tomiclly with the ction p is ble to perform. Summing up, the set of process lgebric opertors tht we hve singled out, in order to represent these six clsses of nets, is relly smll: it comprises essentilly the CCS opertors, some of which re to be used in some constrined form, nd the strong prefixing opertor, originlly introduced in [GMM90, GM90], together with suitble synchroniztion discipline for tomic sequences; only for NPL we hve to introduce newly conceived opertor, which is, however, vrint of the strong prefixing opertor. 1.3 Structure of the Book Chpter 2 introduces the model of lbeled trnsition systems nd some behviorl equivlences over them; this prt of the chpter is just summry of Chpter 2 of [GV15]. Moreover, it includes lso description of the step trnsition systems model, equipped with some sensible behviorl equivlences. Chpter 3 gives n introduction to Petri nets. Most of the mteril is stndrd, but some originl contributions re lso outlined. First, the concept of stticlly rechble subnet is introduced becuse it will ply centrl role in the net semntics of the lnguges FNC, FNM nd NPL. In fct, the semntics of term p is not simply the net tht cn be dynmiclly reched from the initil mrking dec(p) corresponding to p, rther it is the net describing ll the potentil behviors of p, sif the number of tokens in dec(p) cn be incresed t will. For instnce, consider the FNC sequentil process p =.0 +.0; its dynmiclly rechble subnet, outlined in Figure 1.6(), shows the expected behvior tht p cn perform or its complementry ction. On the contrry, its stticlly rechble subnet in (b) describes dditionlly the potentil self-synchroniztion of p with nother copy of itself; in this wy, the net semntics for p nd for the prllel term p p differ only for the form of the initil mrking (one token for p nd two tokens for p p), but the underlying net is the sme. The stticlly rechble subnet is lwys lgorithmiclly computble, lso for Nonpermissive nets, even though they re Turing-complete

16 1.3 Structure of the Book ) b) 2 τ Fig. 1.6 The dynmiclly rechble subnet () nd the stticlly rechble subnet (b) for sequentil semntics Lbeled Trnsition Systems (LTSs) Process Algebr prllel semntics Step Trnsition Systems (STSs) distributed semntics Petri nets Fig. 1.7 Three semntics for ech process lgebr model of computtion, nd so for them, s the rechbility problem is undecidble, it is not lwys possible to compute the dynmiclly rechble subnet. As mtter of fct, nother originl contribution of this chpter is the introduction of the clss of Nonpermissive Petri nets ( proper extension of P/T nets with inhibitor rcs), equipped with nonstndrd behviorl equivlences, deviting from the trditionl definitions for P/T nets with inhibitor rcs. Chpters 4-8 present the six clculi: SFM in Chpter 4, CFM nd BPP in Chpter 5, FNC in Chpter 6, FNM in Chpter 7 nd NPL in Chpter 8. Ech clculus is equipped with three different semntics, s illustrted in Figure 1.7: n interleving (or sequentil) semntics by mens of lbeled trnsition systems, step (or prllel) semntics by mens of step trnsition systems, nd net (or distributed) semntics by mens of finite Petri nets. For ech process lgebr, if two terms re equivlent ccording to the net semntics, then they re equivlent in the step semntics; nd,

17 10 1 Introduction Finite P/T nets N(m 0 ) =r Net(p) trnsltion function FNM p = T FNM (N(m 0 )) distributed semntics Fig. 1.8 Grphicl description of the representbility theorem for finite P/T nets nd FNM similrly, if two terms re equivlent ccording to the step semntics, then they re equivlent in the interleving semntics. Therefore, these three semntics re ordered from the most bstrct (the interleving/sequentil one) to the most concrete (the net/distributed one). These five chpters ll hve the sme structure. First, the syntx of the clculus is given, followed by the interleving/sequentil semntics, defined in n opertionl style, ccording to Plotkin s SOS technique [Plo04b]. Then, the step/prllel semntics is introduced by mens of n obvious generliztion of the SOS technique; this prllel semntics is followed by the opertionl Petri net semntics, which is described in tight correspondence with the style of the trnsition system semntics, in order to show the similrities of the two. As mtter of fct, the SOS pproch cn lso esily be dpted to generte Petri nets insted of lbeled trnsition systems. This distributed semntics is followed by the representbility theorem: for ny mrked net N(m 0 ) of the clss, we cn single out term p of the corresponding lnguge, whose semntics is the net Net(p) isomorphic to N(m 0 ); in this sense, we cn sy tht p represents N(m 0 ), up to isomorphism. Figure 1.8 describes grphiclly this result for the cse of finite P/T nets nd FNM: given finite P/T mrked net N(m 0 ), the trnsltion function T FNM ( ) mps N(m 0 ) to the FNM process term p = T FNM (N(m 0 )); then, the distributed semntics ssocites Net(p) with p, such tht Net(p) = r N(m 0 ), where = r denotes (mrked) net isomorphism equivlence. The soundness of the net semntics w.r.t. the step semntics is lso proved; in fct, the step semntics is minly used s sort of snity check for the distributed semntics, in order to prove tht the distributed semntics preserves ll the intended prllel behvior. Finlly, n unsfe net semntics in denottionl style is introduced nd proved to be coherent with the opertionl net semntics (in most cses, the two semntics coincide). For the cses of FNC, FNM nd NPL, we were ble to define these denottionl semntics only becuse we chose to give semntics to process p by mens of its stticlly rechble subnet insted of its dynmiclly rechble subnet; this is n originl contribution of this book, too. Some repetitions in these chpters re inevitble, becuse ech one builds on top of the previous one; this is prticulrly true for Chpters 4 nd 5, nd for Chpters 6, 7 nd 8. However, s the Romns sid, repetit iuvnt! In fct, the presenttion style is rther didctic, s the expected udience is composed of Ph.D. students nd

18 1.4 Interleving vs True Concurrency 11 scholrs who re not well versed in the re of concurrency theory, or who re not experts in both Petri nets nd process lgebrs. The finl chpter describes side issues, relted to possible generliztions or vritions of the theory presented in this book. In prticulr, we hint how to extend our technique to clculi such s CCS, where the restriction opertor is not used t the top level only, but cn even occur inside the body of recursively defined constnts. 1.4 Interleving vs True Concurrency Trditionlly, the semntics of process lgebrs hve been given in interleving style, usully by mens of ction lbeled trnsition systems. There re some philosophicl nd prcticl resons for doing so, but we rgue tht none of them is relly convincing. Interction vs structure: It is often ssumed tht the only relevnt prt of the behvior of system is given by its interction cpbilities, i.e., the ctions the system performs, with which n observer cn interct. This rgument is bsed on the ssumption tht system, like vending mchine, is ctully blck box, nd tht the observer of the system cn only interct with this system by mens of the externlly visible buttons or slots, but without ny knowledge of the internl structure of the mchine. This ide is quite ppeling nd hs n obvious consequence tht n interleving model, showing which visible ctions cn be performed nd when, is more thn enough to completely describe the behvior of system. However, by modeling rective, distributed system with trnsition system, we re forced to mke one fundmentl bstrction on its structure: the sttes of the rective system re monolithic entities tht cnnot be inspected nd ech trnsition from, sy, stte q to stte q, with lbel, trnsforms the stte q s whole, even if only some components of the system re ctully involved in the execution of ction. As the structure of the system is bstrcted wy in interleving semntics, distributed systems with very different structures nd very different properties cn be equted. For instnce, deterministic, symmetric, fully distributed, dedlock-free solution to the well-known dining philosophers problem (originlly proposed by Dijkstr [Dij71], nd then elborted on by Hore [Ho85] in its current formultion) cn be provided in FNM (see Chpter 6 of [GV15]). Let p be the FNM solution to this problem; its interleving semntics is finite-stte lbeled trnsition system, which is lso the semntics of n SFM process q, i.e., of purely sequentil process; since p nd q re interleving equivlent, we my wrongly conclude tht q is lso deterministic, symmetric, fully distributed, dedlock-free solution to the dining philosophers problem! Of course, this is not the cse. Since the properties of interest for distributed system re often relted to the structure of the system, the semntics cnnot bstrct wy from this spect of the system. Truly concurrent semntics, such s step semntics or net semntics, my often offer more dequte behviorl equivlences, when properties of distributed systems re to be investigted.

19 12 1 Introduction Compositionlity: Most behviorl equivlences over lbeled trnsition systems re congruences w.r.t. the opertors of mny process lgebrs, notbly CCS, so tht compositionl resoning nd compositionl nlysis cn be performed profitbly nd esily. However, we will show tht interleving semntics is too bstrct semntics to be congruence for the prllel composition opertors of FNM nd NPL. These lnguges fford multi-prty communiction, linguistic mechnism tht cn be modeled in compositionl wy only by mens of truly concurrent semntics. Hence, if compositionlity is requisite of the semntics for FNM nd NPL, then truly concurrent model is necessry. Simplicity of the semntics definition: Another rgument in fvor of interleving semntics is the simplicity nd elegnce of its mthemticl formultion. However, the im of this book is to show, by mens of six cse studies, tht opertionl net semntics is often very similr nd only slightly more complex thn trnsition system semntics. For instnce, for SFM these two semntics re exctly the sme, while for FNC/FNM/NPL it is only necessry to hndle the bound nmes in suitble mnner in order to get simple definition of the mrkings nd of the net trnsitions. Moreover, ech opertionl net semntics is coupled with corresponding denottionl net semntics, which is lso not too complex. 1.5 Beyond Turing-Completeness In the theory of sequentil computtion, the computble entities re mthemticl functions from N to N, the set of nturl numbers. A formlism is Turing-complete if it cn compute ll the functions tht cn be computed by mens of Turing mchines [Tur36]. The fmous Church-Turing thesis sttes tht ny function tht is lgorithmiclly computble is ctully computble by Turing mchine. In the theory of concurrency nd distribution, the sitution is bit different. The problems tht re relevnt in distributed computing include the Turing-computble functions, but lso include other problems tht hve nothing to do with functions. In Section 3.5, we present problem in distributed computing, clled the Lst Mn Stnding problem (LMS, for short, originlly introduced in [VBG09]), tht cn be solved with dynmiclly cyclic, finite Nonpermissive Petri net ( clss of nets which is not Turing-complete), but tht cnnot be solved in CCS, well-known Turing-complete lnguge [Mil89, GV15] (see lso Section 9.3). Therefore, wht is the nlog of computble function for concurrency? And wht is the nlog of Turing-completeness for concurrency? New definitions re necessry. Here we give our own proposl s possible new foundtion for distributed computbility theory s generliztion of clssic, sequentil (or Turing) computbility theory. A process is semntic model, up to some behviorl equivlence. For instnce, if the chosen models re lbeled trnsition systems nd the chosen equivlence is (wek completed) trce equivlence (see Chpter 2 for detils), then process is nothing but forml lnguge [HMU01]; if, insted, the chosen models re Petri nets nd the chosen equivlence is net isomorphism, then process is Petri net, up

20 1.5 Beyond Turing-Completeness 13 NPL lnguges = recursively enumerble lnguges = CCS lnguges Context-dependent lnguges FNC lnguges = FNM lnguges = Petri net lnguges Context-free lnguges BPP lnguges SFM lnguges = CFM lnguges = regulr lnguges Fig. 1.9 The hierrchy of lnguges interleving semntics to net isomorphism. In other words, the notion of computble function on the nturl numbers for sequentil computtion is to be replced by model with n ssocited equivlence reltion, which we cll process, for concurrent computtion. A process lgebr is complete w.r.t. given clss of models if it cn represent ll the elements of tht clss, up to the chosen behviorl equivlence. A clss of models is Turing-complete if it cn compute ll the computble functions; if process lgebr is complete w.r.t. tht clss of models, then it is lso Turing-complete. Different process lgebrs usully hve different expressive power. In this book we present list of six, incresingly more nd more expressive, process lgebrs, ech one complete w.r.t. some clss of Petri nets, up to net isomorphism, sdescribed in Figure 1.5. The most expressive one is NPL, which is the only lnguge studied in this book to be Turing-complete, s it cn represent ll finite Nonpermissive Petri nets, nd which cn lso solve the LMS problem. However, there re other process lgebrs, such s CCS [Mil89, GV15] (only sketched in Section 9.3), tht re Turing-complete, but tht cn neither represent ll the finite Nonpermissive Petri nets, nor solve the LMS problem. Nonetheless, CCS cn represent mny infinite P/T nets, nd so NPL nd CCS re incomprble, t lest if the considered semntic equivlence is net isomorphism. It is interesting to note tht, if the chosen semntic models re lbeled trnsition systems nd the chosen behviorl equivlence is (wek completed) trce equivlence, then we remin within the ressuring confines of clssic, sequentil (or Turing) computbility theory, s illustrted in Figure 1.9. NPL, being Turing-complete, models ll the recursively enumerble lnguges, nd the sme holds for CCS; FNC nd FNM model the clss of Petri net lnguges (s shown in Sections nd