An Overview of Membrane Computing

Transcription

1 An Overview of Membrane Computing Krishna Shankara Narayanan Department of Computer Science & Engineering Indian Institute of Technology Bombay

2 Membrane Computing The paradigmatic idea of membrane computing is to see whether we can mimic the living cell - its structure and functioning Gheorghe Păun

3 The Living Cell What a jungle!

4 Abstraction of the Living Cell

5 Abstraction of the Living Cell elementary membrane region non-elementary membrane skin membrane

6 Abstraction of the Living Cell

7 Abstraction of the Living Cell f a i b c d h a a a e g e f

8 Abstraction of the Living Cell i f a b c d h a a a e a bb b (c, out) i δ h efgδ g e f

9 Abstraction of the Living Cell f g bb c c d efg e f bb bb bb e a bb b (c, out) i δ h efgδ

10 Basic Ingredients non-deterministic choice of rules, objects maximal parallelism transitions, computation, halting internal output, external output Result : Cell-like abstract system : P System

11 Basic Ideas in Membrane Computing

12 Symbol Objects (as in the example) Π = (O,µ, w 1,..., w m, R 1,..., R m, i 0 ) µ : membrane structure w i : multisets of objects in membrane i R i : rules of the form a (u, in)(v, out)w or ca c(u, in)(v, out)w, c : catalyst i 0 : output membrane

13 Example : Symbol Objects Using catalysts, promoters. Input : a n ; Output : e n2 c aa a b 1 b 2 cb 1 cb 1 b 2 b 2 (e, in)/b 1

14 Example : Symbol Objects Using catalysts, promoters. Input : a n ; Output : e n2 c b 1 b 2 b 1 b 2 a b 1 b 2 cb 1 cb 1 b 2 b 2 (e, in)/b 1

15 Example : Symbol Objects Using catalysts, promoters. Input : a n ; Output : e n2 c b 1 b 2b 1 b 2 a b 1 b 2 cb 1 cb 1 b 2 b 2 (e, in)/b 1 ee

16 Example : Symbol Objects Using catalysts, promoters. Input : a n ; Output : e n2 c b 1 b 2b 1 b 2 a b 1 b 2 cb 1 cb 1 b 2 b 2 (e, in)/b 1 eeee

17 Computational Power Families NOP m (α), α {coo, ncoo, cat} {cat i i 1}, m 1 or m = Lemma (collapsing hierarchy) NOP (α) = NOP m (α), α {coo, ncoo, cat} and m 2. Theorem NOP (ncoo) = NOP 1 (ncoo) = NCF Theorem NOP 2 (cat 2 ) = NRE Conjecture: NRE NOP (cat 1 )

18 String Objects rewriting rules inspired by splicing and (other) DNA operations More complex objects : arrays, pictures

19 Computing by communication : symport/antiport (x, in);(x, out) : symport; (x, in; y, out) : antiport max( x, y ) : weight System : Π = (O,µ, w 1,..., w m, E, R 1,..., R m, i 0 ) E O is the set of objects that appear in the environment in arbitrarily many copies Families NOP m (symp p, anti q ) Theorem NRE = NOP 1 (sym 0, anti 2 ) = NOP 2 (sym 2, anti 0 ) = NOP 1 (sym 3, anti 0 ) = NOP 3 (sym 1, anti 1 )

20 Active Membranes a[ ] i [ b] i go in [ a] i b[ ] i go out [ a] i b dissolution a [ b] i membrane creation [ a] i [ b] j [ c] k membrane division [ b] j [ c] k [ bc] l membrane merging [ u] i [ ] i [ u] j gemmation [ Q] i [ O Q] j [ Q] k separation Applications: Efficient solutions of intractable problems (illustration coming up later)

21 Some More Variants Tissue-like systems : membranes in the nodes of a graph P automata Systems with objects on membranes : inspired by brane calculi Spiking neural systems P colonies : set of cells of a bounded capacity, with minimal object processing rules Metabolic Systems Mobile Membranes

22 Results Characterization of Turing completeness (RE, NRE, PsRE) Comparison with Chomsky hierarchy, L-systems Polynomial time solutions to NP-complete/PSPACE-somplete problems (using an exponential workspace created in a biological way) Other types of mathematical results (normal forms, hierarchies, determinism versus non-determinism, complexity) Connections with ambient calculus, petri nets, X-machines, quantum computing, brane calculus Simulations and implementations Applications

23 Turing Completeness

24 Mobile Membranes Endocytosis a

25 Mobile Membranes Endocytosis b

26 Mobile Membranes Exocytosis a

27 Mobile Membranes Exocytosis b

28 Mobile Membranes Forced Endocytosis a

29 Mobile Membranes Forced Endocytosis b

30 Mobile Membranes Forced Exocytosis a

31 Mobile Membranes Forced Exocytosis b

32 Mobile Membranes [ a] h [ ] m [[w] h ] m endocytosis (endo) [[a] h ] m [ w] h [ ] m exocytosis (exo) [ ] h [ a] m [[ ] h w] m forced endocytosis (fendo) [ a[ ] h ] m [ ] h [ w] m forced exocytosis (fexo)

33 Mobile Membranes are Turing Complete Families NEM n (α), α {exo, endo, fendo, fexo} Theorem NEM 5 (endo, exo, fendo, fexo) = NRE Proof Idea: Simulate a register machine with 3 registers Register machine : (n, P, i, h) n : number of registers P : program with labeled instructions of the form (add(r), k, l) or (sub(r), k, l) i : initial instruction; h: final instruction Theorem If L V, card(v) = k, L RE, then a 3-register machine M exists such that for every w V we have w L if and only if M halts when starting with val k+1 (w) in its first register; in the halting step, all registers of the machine are empty.

34 Proof Details Initial configuration r 1 r 2 r 3

35 Proof Details Generation of Initial contents of register 1: r 3 r 3 r 1 r 2 r 1 r 1 c r 3 r 3 l1

36 Proof Details Generation of Initial contents of register 1: r 1 c x r 2 l 1

37 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 empty. r 1 r 2 l

38 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 empty. r 2 L r 1

39 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 empty. r 2 L r 1

40 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 empty. r 1 r 2 k

41 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 empty. r 1 r 2 k

42 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 non-empty. r 1 c y r 2 l

43 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 non-empty. r 2 L r 1 c y

44 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 non-empty. r 1 c y 1 r 2 L

45 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 non-empty. r 1 c y 1 r 2 L

46 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 non-empty. r 1 c y 1 r 2 j

47 Simulation of a decrement instruction l : (sub(1), j, k). Register 1 non-empty. r 1 c y 1 r 2 j

48 An Application : Uniform Solution to SAT

49 Solving SAT ϕ = C 1 C 2 C m in CNF Variables {x 1,..., x n } C i of the form y 1 y 2 y r, r n, y i = x i or x i Propose a polynomial time uniform solution : for all formula instances of size (n, m), takes time O(m + n) Uses operations endo, exo, div

50 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} c 0 a 1 a 2 c

51 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} t 1 a 2 c 1 f 1 a 2 c

52 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} c 2 t 1 t 2 t 1 f 2 c f 1 t 2 f 1 f 2

53 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} t 1 t 2 c 3 t 1 f 2 c f 1 t 2 f 1 f 2

54 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} c 4 t 1 t 2 t 1 f 2 c f 1 f 2 f 1 t 2

55 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} c 4 yes t 1 t 2 t 1 f 2 f 1 t 2 f 1 f 2

56 Example ϕ = (x 1 x 2 ) ( x 1 x 2 ). Solution : {TT, FF} d yes t 1 t 2 t 1 f 2 f 1 t 2 f 1 f 2

57 Applications biology, medicine, ecosystems computer science linguistics (modeling framework/parsing) optimization (membrane algorithms) economics

58 A typical application in biology/medicine M. J. Pérez-Jiménez, F. J. Romero-Campero. A Study of the Robustness of the EGFR Signalling Cascade Using Continuous Membrane Systems. In Mechanisms, Symbols, and Models Underlying Cognition. First International Work-Conference on the Interplay between Natural and Artificial Computation, IWINAC proteins, 160 reactions/rules reaction rates from literature results as in experiments

59 Implementation Efforts Currently underway at E. Keinan lab at Technion, Haifa. Success would mean a huge boost to the area, and the development of wet computers

60 Open problems, research directions borderlines: universality/non-universality, efficiency/non-efficiency local problems : the power of 1 catalyst, the role of polarizations, dissolution, etc general problems : uniform versus semi-uniform, deterministic versus non-deterministic, complexity classes etc semantics (operational, behavioural) user friendly, flexible, efficient software for bio-applications implementations (electronic, bio-lab)

61 State of the art Handbook of Membrane Computing, Oxford University Press, 2010.

62 Across the world Originated in Turku, Finland in Nov Major active groups Bucharest and Iaşi, Romania Budapest, Hungary Caltech and Santa Barbara, USA Edinburgh and Sheffield, UK Leiden, The Netherlands Madrid and Seville, Spain Milan and Verona, Italy Chennai and Mumbai, India Opava, Czech Republic Toyama and Waseda, Japan Vienna, Austria

63 THANKYOU