Membrane Computing: Power, Efficiency, Applications (A Quick Introduction) Gheorghe P˘ aun Romanian Academy, Bucharest, RGNC, Sevilla University, Spain george.paun@imar.ro, gpaun@us.es Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 1
Summary: • generalities • the basic idea • examples • classes of P systems • types of results • types of applications – applications in biology – modeling/simulating ecosystems – Nishida’s membrane algorithms – MC and economics; numerical P systems Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 2
Goal: abstracting computing models/ideas from the structure and functioning of living cells (and from their organization in tissues, organs, organisms) hence not producing models for biologists (although, this is now a tendency) result: • distributed, parallel computing model • compartmentalization by means of membranes • basic data structure: multisets (but also strings; recently, numerical variables) Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 3
WHY? – the cell exists! (challenge for mathematics) – biology needs new models (discrete, algorithmic; system biology, the whole cell modelling/simulating) – computer science can learn (e.g., parallelism, coordination, data structure, architecture, operations, strategies) – computing in vitro/in vivo (“the cell is the smallest computer”) – distributed extension of molecular computing – a posteriori: power, efficiency (“solving” NP-complete problems) – a posteriori: applications in biology, computer graphics, linguistics, economics, etc. – nice mathematical/computer science problems Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 4
References: • Gh. P˘ aun, Computing with Membranes. Journal of Computer and System Sciences , 61, 1 (2000), 108–143, and Turku Center for Computer Science- TUCS Report No 208, 1998 (www.tucs.fi) ISI: “fast breaking paper”, “emerging research front in CS” (2003) http://esi-topics.com • Gh. P˘ aun, Membrane Computing. An Introduction , Springer, 2002 • G. Ciobanu, Gh. P˘ aun, M.J. P´ erez-Jim´ enez, eds., Applications of Membrane Computing , Springer, 2006 • forthcoming Handbook of Membrane Computing , OUP • Website: http://ppage.psystems.eu (Yearly events: BWMC (February), WMC (summer), TAPS/WAPS (fall)) Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 5
SOFTWARE AND APPLICATIONS: http://www.dcs.shef.ac.uk/ ∼ marian/PSimulatorWeb/P Systems applications.htm www.cbmc.it – PSim2.X simulator Verona (Vincenzo Manca: vincenzo.manca@univr.it ) Sheffield (Marian Gheorghe: M.Gheorghe@dcs.shef.ac.uk ) Sevilla (Mario P´ erez-Jim´ enez: marper@us.es ) Milano (Giancarlo Mauri: mauri@disco.unimib.it ) Nottingham, Trento, Nagoya, Leiden, Vienna, Evry, Ia¸ si Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 6
FRAMEWORK: Natural computing Biology Models Implementation ( in vivo/vitro ) ( in info ) Neural ✲ ❳❳❳❳❳❳❳❳❳❳❳❳ Brain computing ③ Electronic media ✶ ✏ ✏✏✏✏✏✏✏✏✏✏✏ ( in silico ) Evolutionary ✲ Evolution ✄✄ ✗ computing ✄ ? ✄ ✄ ✄ ✄ DNA(molecular) Bio-media DNA ✄ ✲ ✲ ✄ computing (molecules) ( in vitro, in vivo? ) ✄ ❍❍❍❍❍❍❍ ✣ ✡ ✄ ✡ ✄ ✡ ✄ ❍ ❥ ✡ ? ✄ ✡ ✄ Membrane ✡ ✲ Cell computing Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 7
FRAMEWORK: Natural computing Biology Models Implementation ( in vivo/vitro ) ( in info ) Neural ✲ ❳❳❳❳❳❳❳❳❳❳❳❳ Brain computing ③ Electronic media ✶ ✏ ✏✏✏✏✏✏✏✏✏✏✏ ( in silico ) Evolutionary ✲ Evolution ✄✄ ✗ computing ✄ ? ✄ ✄ ✄ ✄ DNA(molecular) Bio-media DNA ✄ ✲ ✲ ✄ computing (molecules) ( in vitro, in vivo? ) ✄ ❍❍❍❍❍❍❍ ✣ ✡ ✄ ✡ ✄ ✡ ✄ ❥ ❍ ✡ ? ✄ ✲ ✡ ✄ Membrane ✡ ✲ Cell computing Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 8
Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 9
WHAT IS A CELL? (for a mathematician) • membranes, separating “inside” from “outside” (hence protected compartments, “reactors”) • chemicals in solution (hence multisets) • biochemistry (hence parallelism, nondeterminism, decentralization) • enzymatic activity/control • selective passage of chemicals across membranes • etc. Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 10
Importance of membranes for biology:. . . MARCUS: Life = DNA software + membrane hardware Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 11
THE BASIC IDEA skin membrane ✬ ✩ ✚ 1 ✚ ✚ ❂ elementary ✬ ✩ 2 membrane ✬ ✩ 4 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ★ ✥ ✙ ✟ 3 ✫ ✪ ✧ ✦ region ✫ ✪ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✬ ✩ ✘ ✘ ✘ ✘ 5 ✘ ✘ ✬ ✩ ✘ ✘ ✘ ✘ ✘ ✘ ✾ ✘ ✘ 6 environment ✫ ✪ ✫ ✪ ✫ ✪ Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 12
✬ ✩ 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. Power, Efficiency, Applications Bertinoro 2008 13
✬ ✩ 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. Power, Efficiency, Applications Bertinoro 2008 14
Functioning (basic ingredients): • nondeterministic choice of rules and objects • maximal parallelism • transition, computation, halting • internal output, external output, traces Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 15
✬ ✩ EXAMPLES 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. Power, Efficiency, Applications Bertinoro 2008 16
✬ ✩ 1 ✬ ✩ 2 c a → b 1 b 2 cb 1 → cb ′ 1 b 2 → b 2 e cb 1 → cb ′ 1 δ ✫ ✪ b 1 → b 1 e → e out ✫ ✪ → n 2 ), with catalyst, dissolution, nondeterminism, external The same function ( n − output Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 17
Generative mode : { n 2 | n ≥ 1 } 0 af ✬ ✩ 1 ab ′ ff 1 ✬ ✩ 2 . . . . . . ✬ ✩ 3 ab ′ m f 2 m ✬ ✩ m ≥ 0 4 a f b ′ m +1 f 2 m +1 m + 1 δ b m +1 f 2 m m + 2 a → ab ′ ✫ ✪ b m +1 f 2 m − 1 e m +1 m + 3 a → b ′ δ in 4 f → ff . . . . . . . . . ✫ ✪ e m +1 b m +1 f 2 2 m + 1 in 4 e m +1 b m +1 f 2 m + 2 b ′ → b in 4 e m +1 b m +1 aδ 2 m + 3 b → be in 4 in 4 m + 1 times HALT! ❩❩❩❩❩ ✑ ff → f > f → aδ ✑ ✑ ✫ ✪ ✑ ✑ ✫ ✪ ⑦ ❩ ✑ ✰ ( m + 1) × ( m + 1) N (Π) = { n 2 | n ≥ 1 } Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 18
SIMULATING A REGISTER MACHINE M = ( m, B, l 0 , l h , R ) E = { a r | 1 ≤ r ≤ m } ∪ { l, l ′ , l ′′ , l ′′′ , l iv | l ∈ B } ✬ ✩ 1 l 0 � ( l 1 , out ; a r l 2 , in ) for l 1 : ( add ( r ) , l 2 , l 3 ) ( l 1 , out ; a r l 3 , in ) ( l 1 , out ; l ′ 1 l ′′ 1 , in ) ( l ′ 1 a r , out ; l ′′′ 1 , in ) ( l ′′ 1 , out ; l iv 1 , in ) for l 1 : ( sub ( r ) , l 2 , l 3 ) ( l iv l ′′′ 1 , out ; l 2 , in ) ( l iv l ′ 1 , out ; l 3 , in ) ✫ ✪ ( l h , out ) Symport/antiport rules (of weight 2) Gh. P˘ aun, Membrane Computing. Power, Efficiency, Applications Bertinoro 2008 19
Types of rules: u → v with targets in v (possibly conditional: promoters or inhibitors) particular cases: ca → cu (catalytic) a → u (non-cooperative) ( ab, in ) , ( ab, out ) – symport (in general, ( x, in ) , ( x, out ) ) ( a, in ; b, out ) – antiport (in general, ( u, in ; v, out ) ) 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. Power, Efficiency, Applications Bertinoro 2008 20
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. Power, Efficiency, Applications Bertinoro 2008 21
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