Bisimulation Congruences in the Calculus of Looping Sequences Roberto Barbuti Andrea Maggiolo Schettini Paolo Milazzo Angelo Troina Dipartimento di Informatica, Universit` a di Pisa, Italy Tunis – ICTAC 2006 1/22
Introduction Formal models for systems of interactive components can be easily used or adapted for the modelling of biological phenomena Examples: Petri Nets, π –calculus, Mobile Ambients The modelling of biological systems allows: 1 the development of simulators 2 the verification of properties We defined the Calculus of Looping Sequences (CLS): a formalism to describe biochemical systems in cells In this talk: 1 we recall the definition of CLS 2 we present bisimulation relations for CLS 3 we show the CLS model of a gene regulation process in E. Coli 2/22
The Calculus of Looping Sequences (CLS) We assume an alphabet E . Terms T and Sequences S of CLS are given by the following grammar: � � � L ⌋ T � � � T ::= S S T | T � � � � S ::= ǫ a S · S where a is a generic element of E , and ǫ is the empty sequence. The operators are: S · S : Sequencing � � L S : Looping ( S is closed and it can rotate) T 1 ⌋ T 2 : Containment ( T 1 contains T 2 ) T | T : Parallel composition (juxtaposition) � � L ⌋ T . Actually, looping and containment form a single binary operator S 3/22
Example of Terms a c a c a f c g (i) (ii) (iii) d e d e b b b � � L ⌋ ǫ ( i ) a · b · c � � L ⌋ � � L ⌋ ǫ ( ii ) a · b · c d · e � � L ⌋ ( f · g | � � L ⌋ ǫ ) ( iii ) a · b · c d · e 4/22
Structural Congruence The Structural Congruence relations ≡ S and ≡ T are the least congruence relations on sequences and on terms, respectively, satisfying the following rules: S 1 · ( S 2 · S 3 ) ≡ S ( S 1 · S 2 ) · S 3 S · ǫ ≡ S ǫ · S ≡ S S T 1 | T 2 ≡ T T 2 | T 1 T 1 | ( T 2 | T 3 ) ≡ T ( T 1 | T 2 ) | T 3 � � L ⌋ ǫ ≡ T ǫ � � L ⌋ T ≡ T � � L ⌋ T T | ǫ ≡ T T S 1 · S 2 S 2 · S 1 ǫ We write ≡ for ≡ T . 5/22
Dinamics of the Calculus (1) Let T V be the set of terms which may contain variables of three kinds: term variables ( X , Y , Z , . . . ) sequence variables ( � x , � y , � z , . . . ) element variables ( x , y , z , . . . ) T σ denotes the term obtained by replacing any variable in T with the corresponding term, sequence or element. A Rewrite Rule is a pair ( T , T ′ ), denoted T �→ T ′ , where: T , T ′ ∈ T V variables in T ′ are a subset of those in T A rule T �→ T ′ can be applied to all terms T σ . Example: a · x · a �→ b · x · b can be applied to a · c · a (producing b · c · b ) cannot be applied to a · c · c · a 6/22
Bisimulations Bisimilarity is widely accepted as the finest extensional behavioral equivalence one may impose on systems. Two systems are bisimilar if they can perform step by step the same interactions with the environment. Properties of a system can be verified by assessing the bisimilarity with a system known to enjoy them. Bisimilarities need semantics based on labeled transition relations capturing the potential interactions with the environment. In process calculi, transitions are usually labeled with actions. In CLS labels are contexts in which rules can be applied. 7/22
Labeled Semantics (1) Contexts C are given by the following grammar: � � � � � L ⌋ C � � � C ::= � C | T T | C S where T ∈ T and S ∈ S . Context � is called the empty context . Parallel Contexts C P are given by the following grammar: � � � � C P ::= � C P | T T | C P . where T ∈ T . C [ T ] is context application and C [ C ′ ] is context composition. 8/22
Labeled Semantics (2) Given a set of rewrite rules R ⊆ ℜ , the labeled semantics of CLS is the labeled transition system given by the following inference rules: (rule appl) T �→ T ′ ∈ R T ′′ �≡ ǫ C [ T ′′ ] ≡ T σ σ ∈ Σ C ∈ C C T ′′ − → T ′ σ � C T − → T ′ (par) T − → T ′ C ∈ C P (cont) � � L ⌋ T � � L ⌋ T ′ � C → T ′ | T ′′ T | T ′′ − S − → S where the dual version of the (par) rule is omitted. Rule (rule appl) describes the (potential) application of a rule. T ′′ �≡ ǫ in the premise implies that C cannot provide completely the left hand side of the rewrite rule. � | b a | b Example: let R = a | b �→ c , we have a − − − → c , but ǫ � − − → . 9/22
Labeled Semantics (3) Given a set of rewrite rules R ⊆ ℜ , the labeled semantics of CLS is the labeled transition system given by the following inference rules: (rule appl) T �→ T ′ ∈ R T ′′ �≡ ǫ C [ T ′′ ] ≡ T σ σ ∈ Σ C ∈ C C T ′′ − → T ′ σ � C T − → T ′ (par) T − → T ′ C ∈ C P (cont) � � L ⌋ T � � L ⌋ T ′ � C → T ′ | T ′′ S − → S T | T ′′ − where the dual version of the (par) rule is omitted. Rule (cont) propagates � –labeled transitions from the inside to the outside of a looping sequence. Transition labeled with a non–empty context cannot be propagated. � � L ⌋ a � � | b � | b Example: let R = a | b �→ c , we have a − − − → c , but d − − → . 10/22
� Labeled Semantics (4) Given a set of rewrite rules R ⊆ ℜ , the labeled semantics of CLS is the labeled transition system given by the following inference rules: (rule appl) T �→ T ′ ∈ R T ′′ �≡ ǫ C [ T ′′ ] ≡ T σ σ ∈ Σ C ∈ C C T ′′ − → T ′ σ � C T − → T ′ (par) T − → T ′ C ∈ C P (cont) � � L ⌋ T � � L ⌋ T ′ � → T ′ | T ′′ C T | T ′′ − S − → S where the dual version of the (par) rule is omitted. Rule (par) propagates transitions labeled with parallel contexts in parallel components. ( a ) L ⌋ � Example: let R = ( a ) L ⌋ b �→ c , we have b − − − − − → c , but ( a ) L ⌋ � → because R cannot be applied ( a ) L ⌋ ( b | d ) b | d − − − − − 11/22
Bisimulations in CLS (1) A binary relation R on terms is a strong bisimulation if, given T 1 , T 2 such that T 1 RT 2 , the two following conditions hold: C C → T ′ ⇒ ∃ T ′ → T ′ 2 and T ′ 1 RT ′ T 1 − 1 = 2 s.t. T 2 − 2 C C → T ′ ⇒ ∃ T ′ → T ′ 1 and T ′ 2 RT ′ T 2 − 2 = 1 s.t. T 1 − 1 . The strong bisimilarity ∼ is the largest of such relations. A binary relation R on terms is a weak bisimulation if, given T 1 , T 2 such that T 1 RT 2 , the two following conditions hold: C C → T ′ ⇒ ∃ T ′ ⇒ T ′ 2 and T ′ 1 RT ′ T 1 − 1 = 2 s.t. T 2 = 2 C C → T ′ ⇒ ∃ T ′ ⇒ T ′ 1 and T ′ 2 RT ′ T 2 − 2 = 1 s.t. T 1 = 1 . The weak bisimilarity ≈ is the largest of such relations. Theorem: Strong and weak bisimilarities are congruences. 12/22
Bisimulations in CLS (2) Consider the following set of rewrite rules: R = { a | b �→ c d | b �→ e e �→ e c �→ e f �→ a } , , , , We have that a ∼ d , because � | b � → e � → e � a − − → c − − − → . . . � | b → e � → e � d − − − − → . . . and f ≈ d , because � | b � → e � � → e � f − → a − − → c − − − → . . . On the other hand, f �∼ e and f �≈ e . e � → e � → e � − − − → . . . 13/22
Bisimulations in CLS (3) Let us consider systems ( T , R ). . . A binary relation R is a strong bisimulation on systems if, given ( T 1 , R 1 ) and ( T 2 , R 2 ) such that ( T 1 , R 1 ) R ( T 2 , R 2 ): C C → T ′ ⇒ ∃ T ′ → T ′ 2 and ( T ′ 1 , R 1 ) R ( T ′ R 1 : T 1 − 1 = 2 s.t. R 2 : T 2 − 2 , R 2 ) C C → T ′ ⇒ ∃ T ′ → T ′ 1 and ( R 2 , T ′ 2 ) R ( R 1 , T ′ R 2 : T 2 − 2 = 1 s.t. R 1 : T 1 − 1 ). The strong bisimilarity on systems ∼ is the largest of such relations. A binary relation R is a weak bisimulation on systems if, given ( T 1 , R 1 ) and ( T 2 , R 2 ) such that ( T 1 , R 1 ) R ( T 2 , R 2 ): C C → T ′ ⇒ ∃ T ′ ⇒ T ′ 2 and ( T ′ 1 , R 1 ) R ( T ′ R 1 : T 1 − 1 = 2 s.t. R 2 : T 2 = 2 , R 2 ) C C → T ′ ⇒ ∃ T ′ ⇒ T ′ 1 and ( T ′ 2 , R 2 ) R ( T ′ R 2 : T 2 − 2 = 1 s.t. R 1 : T 1 = 1 , R 1 ) The weak bisimilarity on systems ≈ is the largest of such relations. Strong and weak bisimilarities on systems are NOT congruences. 14/22
Bisimulations in CLS (4) Consider the following sets of rewrite rules R 1 = { a | b �→ c } R 2 = { a | d �→ c , b | e �→ c } We have that � a , R 1 � ≈ � e , R 2 � because � | b � | b R 1 : a − − → c R 2 : e − − → c and � b , R 1 � ≈ � d , R 2 � , because � | a � | a R 1 : b − − → c R 2 : d − − → c but � a | b , R 1 � �≈ � e | d , R 2 � , because R 1 : a | b � R 2 : c | d � � − → c − → 15/22
The Lactose Operon in E.coli (1) i p o z y a DNA mRNA proteins R lac Repressor beta-gal. permease transacet. i p o z y a a) RNA R Polime- rase NO TRANSCRIPTION i p o z y a b) RNA Polime- rase R LACTOSE TRANSCRIPTION 16/22
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