Massively Parallel Algorithm for Evolution Rules Application in Transition P System Luís Fernández Fernando Arroyo Jorge A. Tejedor Juan Castellanos Grupo de Computación Natural Universidad Politécnica de Madrid

Seventh Workshop on Membrane Computing, WMC'07 INTRODUCTION PROBLEM: Algorithm for Application Rules in Transition P System ω = a ωa b ωb c ωc d ωd e ωe r 1 : a r1a b r1b c r1c d r1d e r1e -> … r 2 : a r2a b r2b c r2c d r2d e r2e -> … r 3 : a r3a b r3b c r3c d r3d e r3e -> … r 1 k1 r 2 k2 r 3 k3 ω’ = a ω’a b ω’b c ω’c d ω’d e ω’e Multiset Active Rules Set OBJECTIVE: Massively Parallel Algorithm Membrane r 1 : a r1a b r1b c r1c d r1d e r1e -> … r 2 : a r2a b r2b c r2c d r2d e r2e -> … r 3 : a r3a b r3b c r3c d r3d e r3e -> … Multiset Active Rules Set Membrane

Seventh Workshop on Membrane Computing, WMC'07 RELATED WORKS ( I ) ω … ω … ω … ω … ω … ω … ω … Ciobanu et al [WMC’2003] Rules Parallel applicability checking Concurrent, NO PARALLEL, rules application ¿? r1r2r3mr1r2r3m O ( | w | ) ¿?

Seventh Workshop on Membrane Computing, WMC'07 RELATED WORKS ( I ) ω … ω … ω … ω … ω … ω … ω … Fernandez et al [BIOCOMP’2006] Sequential, NO PARALLEL, rules application m O ( log 2 | w | ) ¿?

Seventh Workshop on Membrane Computing, WMC'07 MASSIVELY PARALLEL ALGORITHM ( I ) ω … ω … ω … ω … Fernandez et al. [WMC’2006] Rules Parallel applicability checking Rules Parallel application Mutual Exclusion over rules indexes ¿? r1r2r3mr1r2r3m

Seventh Workshop on Membrane Computing, WMC'07 MASSIVELY PARALLEL ALGORITHM ( II ) Phase 1. Membrane initialization. Phase 2. Evolution rules initialization. Phase 3. Multiset proposition Phase 4. Proposed multiset addition Phase 5. Collision Management. A. By excess. B. By defect. Phase 6. Symbols consume. Phase 7. Checking halt rules. Phase 8. Checking halt membrane. ¿ω’> ω? ¿ω’=Ø? ω ← ω -ω’ Phases

Seventh Workshop on Membrane Computing, WMC'07 MASSIVELY PARALLEL ALGORITHM ( III ) BEGIN [1] Membrane initialization REPEAT [5] Collision management - By excess. - By defect. UNTIL NOT Collision; [6] Symbols consume [8] Checking halt membrane UNTIL End END BEGIN [2] Rules initialization REPEAT [3] Multisets proposition [4] Proposed multisets addition UNTIL NOT Collision [7] Checking halt rules UNTIL End END Processes Membrane Process Rule Process

Seventh Workshop on Membrane Computing, WMC'07 MASSIVELY PARALLEL ALGORITHM ( IV ) Synchronization

Seventh Workshop on Membrane Computing, WMC'07 MASSIVELY PARALLEL ALGORITHM ( V ) O ( log 2 R ·log 2 | ω | ) Efficiency

Seventh Workshop on Membrane Computing, WMC'07 CONCLUSIONS The Massively Parallel Algorithm is “PARALLEL” Parallel rules execution in the 50 % of the phases Simultaneity in the application of rules Fined-grained critical sections. Empirical results exhibits better behaviour than other parallel algorithms. It gives a real chance to parallel implementation of Transition P systems. In Hardware Architectures specifically designed it is possible to obtain a 100 % parallelism degree when the conditions are appropriates.