Undecidability of Model Checking in Brane Logic Giorgio Bacci - PowerPoint PPT Presentation

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Undecidability of Model Checking in Brane Logic Giorgio Bacci Marino Miculan Department of Mathematics and Computer Science University of Udine, ITALY DCM 2007 1 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Talk Outline + Summary of the Calculus and Logic + Proof of model checking undecidability calculus with replication logic with adjoints and quantifiers + Conclusions 2 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions (Basic) Brane Calculus [Cardelli ’04] Intended to be a model of biological membranes k | σ h P i | P m Q | ! P systems P , Q ::= nests of membranes branes σ, τ ::= 0 | σ | τ | a .σ | ! σ combination of actions actions a , b ::= . . . (not now) σ P σ|τ P h i h i σ P σ P membrane membrane τ patches contents 3 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Structural Equivalence ≡ Systems Membranes P m Q ≡ Q m P σ | τ ≡ τ | σ Fluidity P m ( Q m R ) ≡ ( P m Q ) m R σ | ( τ | ρ ) ≡ ( σ | τ ) | ρ P m k ≡ P σ | 0 ≡ σ Plenitude ! P ≡ P m ! P ! σ ≡ σ | ! σ etc. etc. Congruence P ≡ Q ⇒ P m R ≡ Q m R σ ≡ τ ⇒ σ | ρ ≡ τ | ρ P ≡ Q ⇒ ! P ≡ ! Q σ ≡ τ ⇒ ! σ ≡ ! τ 4 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Brane Reactions } (PEP semantics) . . . J n | J I n ( σ ) | K n | K I n | G ( σ ) phago J , exo K , pino G actions J I (ρ).τ τ n J .σ σ n Q Q ρ P σ ʻ τ ʻ P σ ʻ τ ʻ phago σ K I τ .τ n .σ K Q Q n τ ʻ τ ʻ P P exo σ ʻ σ ʻ (ρ).τ G τ P ρ P σ pino σ 5 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Brane Logic [CMSB ’06]: motivations Logics allow to express formally the properties of biological systems, usually written in natural language. System specification and verification (possibly automatic): check whether a given system P satisfies a given property A System synthesis: find a system which satisfies a given property A (synthetic biology) System characterization: find the formula which characterizes the behaviour of the system P Model validation: predict a property which should hold in a system and mount an experiment to verify it (predictive biology) 6 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Brane Logic: syntax There are two interacting logics: system formulas A , B ::= T | ¬A | A ∨ B (classical propositional fragment) k (void system) M h A i | A @ M (compartment, compartment adjoint) like Ambient A m B | A ⊲ B (spatial composition, composition adjoint) Logic N A | m A (eventually modality, somewhere modality) but . . . ∀ x . A (quantification over names) brane formulas M , N ::= T | ¬M | M ∨ N (classical propositional fragment) 0 (void membrane) a kind of M|N | M ◮ N (spatial composition, composition adjoint) Hennessy-Milner ) α * M (action modality) logic 7 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Brane Logic: satisfaction � Spatial connectives and their adjoints. . . (properly of spatial calculi) ∃ P ′ , P ′′ . P ≡ P ′ m P ′′ ∧ P ′ � A ∧ P ′′ � B P � A m B � ∃ P ′ , σ. P ≡ σ h P ′ i ∧ P ′ � A ∧ σ � M P � M h A i � ∀ σ.σ � M ⇒ σ h P i � A P � A @ M � ∀ P ′ . P ′ � A ⇒ P m P ′ � B P � A ⊲ B � (guarantee) . . . both temporal and spatial modalities (bi-modal logic) ∃ P ′ : Π . P } ∗ P ′ ∧ P ′ � A P � N A � ∃ P ′ : Π . P ↓ ∗ P ′ ∧ P ′ � A P � m A � 8 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Undecidability of model checking Given P and A , is P � A ? Two sources of undecidability: if processes have unbound replication ( ! P ), model 1 checking is undecidable Solution: consider only finite calculi (without replications) or admit only guarded replications [Busi-Zavattaro ’04] if the logic contain guarantee ( ⊲ ) and quantifiers, model 2 checking the finite state Brane Calulus is also undecidable. In [CMSB ’06] a model checking algorithm for finite calculus and ⊲ -free logic 9 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Undecidability of model checking Given P and A , is P � A ? Two sources of undecidability: if processes have unbound replication ( ! P ), model 1 checking is undecidable Solution: consider only finite calculi (without replications) or admit only guarded replications [Busi-Zavattaro ’04] if the logic contain guarantee ( ⊲ ) and quantifiers, model 2 checking the finite state Brane Calulus is also undecidable. In [CMSB ’06] a model checking algorithm for finite calculus and ⊲ -free logic 9 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Undecidability in presence of replication The proof is done by reduction of a undecidable problem: Proof Outline encode in Brane Calculus the Post Corrispondence Problem give a formula that holds iff PCP as a solution 10 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Encoding PCP Post Corrispondence Problem Instance: a finite set of pairs of words { ( α 1 , β 1 ) , . . . , ( α n , β n ) } Question: there exist a sequence i 0 , i 1 , . . . , i k (1 ≤ i j ≤ n for all 0 ≤ j ≤ k ) such that α i 0 · . . . · α i k = β i 0 · . . . · β i k Encoding idea: start from two empty words W 1 , W 2 non-deterministically choose a pair from the instace to concatenate to W 1 and W 2 compare the two words and repeat. . . 11 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Encoding PCP: strings . . . we use membranes as string constructors String a 1 a 2 ... a k “a 1 “ “a 2 “ ... “a k “ action names used as symbols of the alphabet double layer systems are immersed in a fluid, so order to preserve tonality does not matter membrane nesting preserves the ordering 12 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Encoding PCP: concatenation & comparison “a 1 “ concatenation ... “a 1 “ “a n “ ... “b 1 “ “a n “ ... * “b 1 “ “b m “ ... “b m “ m a t c h “a 2 “ “a“ ... comparison “a 2 “ “a n “ “a“ ... “a n “ “b 2 “ “b 2 “ * ... “b m “ ... “b m “ 13 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Undecidability in presence of replication Two replication constructors: ( ! P ≡ P m ! P ) replication on systems replication on branes ( ! σ ≡ σ | ! σ ) We have to treat them separately. . . PCP S PCP m 14 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Encoding PCP on systems: first definition Word 1 ( ǫ ) m Word 2 ( ǫ ) m � PCP S Concatenate m Compare ! Concatenate ( α 1 , β 1 ) m . . . m ! Concatenate ( α n , β n ) � Concatenate ! Consume ( a ) m ! Consume ( b ) � Compare 15 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Encoding PCP on systems: first definition Word 1 ( ǫ ) m Word 2 ( ǫ ) m ! � PCP S G Concatenate m Compare N O ! Concatenate ( α 1 , β 1 ) m . . . m ! Concatenate ( α n , β n ) R � Concatenate W ! Consume ( a ) m ! Consume ( b ) � Compare if comparison is interleaved with concatenation? 15 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Synchronizing jobs. . . t i o n s o - a c m c a t c h mate I mate start . 0 start P W 1 W 2 mate start Q the two words are n e a b r m e m p r o e c t i v e t enveloped in a protective membrane 0 no hope for match mate W 1 start P Q W 2 inactive! 16 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Encoding PCP on systems: final definition formally. . . PCP S � mate I start h Word 1 ( ǫ ) m Word 2 ( ǫ ) m End im Concatenate m Compare Concatenate � ! Concatenate ( α 1 , β 1 ) m . . . m ! Concatenate ( α n , β n ) Compare � ! Consume ( a ) m ! Consume ( b ) 17 / 26

Outline Calculus + Logic Undecidability with Replication Guarantee + Quantifiers Conclusions Undecidability (systems replication) if PCP S satisfy the the formula A the PCP as a solution A � N ( nonempty ( w 1 ) ∧ N ( empty ( w 1 ) ∧ empty ( w 2 ))) A contains only propositional connectives, temporal and spatial modalities and the compartment connective. No need of quantifiers or adjoint connectives Theorem The model checking problem for Brane Calculi with replication on systems against the Brane Logic is undecidable. 18 / 26