Membrane Computing at Twelve Years Gheorghe P˘ aun Romanian Academy, Bucure¸ sti, RGNC, Sevilla University, Spain george.paun@imar.ro, gpaun@us.es Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 1
Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 2
Everything started 12 years ago, in Turku... Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 3
Great environment... ...with really big hats... Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 4
...also the Magician was around Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 5
...still, not too satisfied (with DNA computing) Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 6
Let’s go to the cell! Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 7
...what a jungle! Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 8
BUT, WE (THE MATHEMATICIANS) CAN SIMPLIFY: skin membrane ✬ ✩ ✚ 1 ✚ ✚ ❂ elementary ✬ ✩ 2 membrane ✬ ✩ 4 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ★ ✥ ✙ ✟ 3 ✫ ✪ ✧ ✦ region ✫ ✪ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✬ ✩ ✘ ✘ ✘ ✘ 5 ✘ ✘ ✬ ✩ ✘ ✘ ✘ ✘ ✘ ✘ ✾ ✘ ✘ 6 environment ✫ ✪ ✫ ✪ ✫ ✪ Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 9
BUT, WE (THE MATHEMATICIANS) CAN SIMPLIFY: ✬ ✩ 1 a b ✬ ✩ 2 ✬ ✩ 4 t b a ★ ✥ 3 t b b b ✫ ✪ ✧ ✦ ✫ ✪ ✬ ✩ 5 a ✬ ✩ b 6 c ✫ ✪ c b ✫ ✪ ✫ ✪ Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 10
BUT, WE (THE MATHEMATICIANS) CAN SIMPLIFY: ✬ ✩ 1 a ab → dd out e in 5 b ✬ ✩ 2 ✬ ✩ 4 t → t t → t ′ δ t b a ★ ✥ 3 t b b b ✫ ✪ ✧ ✦ ✫ ✪ d → a in 4 b out ca → cb ✬ ✩ 5 a ✬ ✩ b 6 c ✫ ✪ c b ✫ ✪ ✫ ✪ Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 11
Functioning (basic ingredients): • nondeterministic choice of rules and objects • maximal parallelism • transition, computation, halting • internal output, external output Result: Cell-like P system Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 12
Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 13
Handbook of Membrane Computing Editors: Gheorghe P˘ aun (Bucharest, Romania) Grzegorz Rozenberg (Leiden, The Netherlands) Arto Salomaa (Turku, Finland) Advisory Board: E. Csuhaj-Varj´ u (Budapest, Hungary) R. Freund (Vienna, Austria) M. Gheorghe (Sheffield, UK) O.H. Ibarra (Santa Barbara, USA) V. Manca (Verona, Italy) G. Mauri (Milan, Italy) M.J. P´ erez-Jim´ enez (Seville, Spain) Oxford University Press, 2010 Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 14
Contents Preface 1. An overview of membrane computing Gh. P˘ aun, G. Rozenberg 2. Cell biology for membrane computing D. Besozzi, I.I. Ardelean, G. Rozenberg 3. Computability elements for membrane computing Gh. P˘ aun, G. Rozenberg, A. Salomaa 4. Catalytic P systems R. Freund, O.H. Ibarra, A. P˘ aun, P. Sosik, H.-C. Yen 5. Communication membrane computing systems R. Freund, Y. Rogozhin, A. Alhazov Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 15
6. P automata E. Csuhaj-Varj´ u, M. Oswald, G. Vaszil 7. P systems with string objects C. Ferretti, G. Mauri, C. Zandron 8. Splicing P systems S. Verlan, P. Frisco 9. Tissue and population P systems F. Bernardini, M. Gheorghe 10. Conformon P systems P. Frisco 11. Active membranes Gh. P˘ aun 12. Complexity – Membrane division, membrane creation M.J. P´ erez-Jim´ enez, A. Riscos-N´ u˜ nez, ´ A. Romero-Jim´ enez, D. Woods Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 16
13. Spiking neural P systems O.H. Ibarra, A. Leporati, A. P˘ aun, S. Woodworth 14. P systems with objects on membranes M. Cavaliere, S.N. Krishna, A. P˘ aun, Gh. P˘ aun 15. Software for membrane computing D. D´ ıaz-Pernil, C. Graciani, M.A. Guti´ errez-Naranjo, I. P´ erez-Hurtado, M.J. P´ erez-Jim´ enez 16. Fundamentals of metabolic P systems V. Manca 17. Principles of metabolic P dynamics V. Manca 18. Probabilistic/stochastic models P. Cazzaniga, M. Gheorghe, N. Krasnogor, G. Mauri, D. Pescini, F.J. Romero-Campero Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 17
19. Membrane algorithms T. Nishida, T. Shiotani, Y. Takahashi 20. Petri nets and membrane computing J. Kleijn, M. Koutny 21. Semantics of the membrane systems G. Ciobanu 22. Membrane computing and computer science R. Ceterchi, D. Sburlan 23. Other developments 23.1. P Colonies A. Kelemenov´ a 23.2. Time in membrane computing M. Cavaliere, D. Sburlan 23.3. Membrane computing and self-assembly M. Gheorghe, N. Krasnogor Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 18
23.4. Membrane computing and X-machines P. Kefalas, I. Stamatopoulou, M. Gheorghe, G. Eleftherakis 23.5. Quantum inspired (UREM) P systems A. Leporati 23.6. Membrane computing and economics Gh. P˘ aun, R. P˘ aun 23.7. Other topics Gh. P˘ aun, G. Rozenberg Selective bibliography Index of notions Index of authors Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 19
Introducing MC through 12 basic ideas: 1. Cell-like P system Π = ( O, µ, w 1 , . . . , w m , R 1 , . . . , R m , i o ) , where: • O = alphabet of objects • µ = (labeled) membrane structure of degree m • w i = strings/multisets over O • R i = sets of evolution rules typical form ab → ( a, here )( c, in 2 )( c, out ) • i o = the output membrane Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 20
EXAMPLE ✬ ✩ 1 c a → b 1 b 2 cb 1 → cb ′ 1 b 2 → b 2 e in | b 1 ✬ ✩ 2 ✫ ✪ ✫ ✪ → n 2 Computing system: n − (catalyst, promoter, determinism, internal output) Input (in membrane 1): a n Output (in membrane 2): e n 2 Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 21
Computational power (Universality) Families NOP m ( α, tar ) , α ∈ { coo, ncoo, cat } ∪ { cat i | i ≥ 1 } , m ≥ 1 or m = ∗ . Lemma 1. (collapsing hierarchy) NOP ∗ ( α, tar ) = NOP m ( α, tar, ) for all α ∈ { ncoo, cat, coo } and m ≥ 2 . Theorem 1. NOP ∗ ( ncoo, tar ) = NOP 1 ( ncoo ) = NCF . Proof: use Lemma 1 and CD grammar systems Theorem 2. NOP ∗ ( coo, tar ) = NOP m ( coo, tar ) = NRE , for all m ≥ 1 . Theorem 3. [Sosik: 8], [Sosik, Freund: 6], [Freund, Kari, Sosik, Oswald: 2] NOP 2 ( cat 2 , tar ) = NRE Conjecture NRE − NOP ∗ ( cat 1 ) � = ∅ Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 22
2. String objects: ...processed by string operations: • rewriting • splicing (DNA computing) • other DNA-inspired operations More complex objects, e.g., arrays Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 23
3. Computing by communication: symport-antiport ( ab, in ) , ( ab, out ) – symport (in general, ( x, in ) , ( x, out ) ) ( a, in ; b, out ) – antiport (in general, ( u, in ; v, out ) ) ( max( | x | , | y | ) = weight) System Π = ( O, µ, w 1 , . . . , w m , E, R 1 , . . . , R m , i o ) , where E ⊆ O is the set of objects which appear in the environment in arbitrarily many copies Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 24
Families NOP m ( sym p , anti q ) Power: (universality) Theorem 4. 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 ) More general rules: u ] i v → u ′ ] i v ′ – boundary (Manca, Bernardini) ab → a tar 1 b tar 2 – communication (Sosik) ab → a tar 1 b tar 2 c come a → a tar Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 25
4. Active membranes: a [ ] i → [ b ] i go in [ a ] i → b [ ] i go out [ a ] i → b membrane dissolution a → [ b ] i membrane creation [ a ] i → [ b ] j [ c ] k membrane division [ a ] i [ b ] j → [ c ] k membrane merging [ a ] i [ ] j → [[ b ] i ] j endocytosis [[ a ] i ] j → [ b ] i [ ] j exocytosis [ u ] i → [ ] i [ u ] @ j gemmation [ Q ] i → [ O − Q ] j [ Q ] k separation and others Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 26
5. tissue-like P systems - membranes in the nodes of a graph population P systems 6. using P systems in the accepting mode P automata 7. trace languages 8. numerical P systems Basic idea: numerical variables in regions, evolving by “production functions”, whose value is distributed according to “repartition protocols”; dynamical systems approach (sequences of configurations), but also computing device (the set of values of a specified variable). Gh. P˘ aun, Membrane computing at twelve years CMC11, Jena 2010 27
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