Deciding Branching Bisimilarity between BPA and Finite-State Systems - PowerPoint PPT Presentation

Deciding Branching Bisimilarity between BPA and Finite-State Systems Hongfei Fu BASICS Laboratory Department of Computer Science Shanghai Jiao Tong University APLAS 2009 Paper Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 1 / 40

Outline Background 1 Definitions 2 The Bisimulation Base Technique 3 Computing The Bisimulation Base 4 Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 2 / 40

Background Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 3 / 40

Formal Verification formal specification: finite-state system real implementation: infinite-state system Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 4 / 40

Comparative semantics Bisimulation Semantics: strong bisimulation (Park) weak bisimulation (Milner) branching bisimulation (van Glabbeek and Weijland) Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 5 / 40

The Problem We Study Polynomial time algorithms deciding branching bisimilarity between: BPA (Basic Process Algebra) and finite-state systems (FS) Normed BPP (Basic Parallel Processes) and finite-state systems Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 6 / 40

Definitions Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 7 / 40

Notations Let V be a finite alphabet of symbols. Symbols of V are ranged over by X , Y , Z . . . . The set of words over V is denoted V ∗ . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 8 / 40

Notations We presume a set of actions Act τ . We always use Γ to refer to a FS State (Γ) to refer to the state set of Γ f , g , h . . . to range over State (Γ) Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 9 / 40

BPA Processes A BPA system is a tuple ( V , ∆) where V is a finite alphabet of symbols. a ∆ is a finite set of rules for which each rule has the form X − → α where X ∈ V , α ∈ V ∗ and a ∈ Act . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 10 / 40

BPA Processes A BPA system ( V , ∆) defines an LTS where states are elements of V ∗ . for α, β ∈ V ∗ , α a → γ ′ ∈ ∆ and β = γ ′ γ . a − → β if α = Y γ , Y − A state (word) α ∈ V ∗ is normed if α → ∗ ε . α ∈ V ∗ is unnormed if α is not normed. Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 11 / 40

BPA: An Example V = { I , Z } z i → II , I d i ∆ = { Z − → Z , Z − → IZ , I − − → ε } z i i i . . . Z IZ IIZ IIIZ d d d Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 12 / 40

Branching Bisimulation Under "Contraction" Let ( V , ∆) be a BPA system and Γ be a FS system. A binary relation R ⊆ V ∗ × State (Γ) is a branching bisimulation if whenever ( α, f ) ∈ R then for each a ∈ Act : α α α f f f R R R R R ↓ a ↓ τ ⇓ ↓ a or ↓ a α ′ f ′ α ′ α ′ f ′ R R ↓ a α ′′ The branching bisimilarity ≈ br is the largest branching bisimulation. Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 13 / 40

Bisimulation Base Technique Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 14 / 40

Description an effective technique developed by D. Caucal to decide bisimilarity concerning infinite-state systems a finite relation " bisimulation base " from which the whole bisimilarity relation can be effectively generated . We can decide the bisimulation problem if we can compute the corresponding bisimulation base. Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 15 / 40

Previous Application strong bisimulation over BPA/BPP S. Christensen, H. Hüttel, and C. Stirling 1992/ S. Christensen, Y. Hirshfeld, and F . Moller 1993 strong bisimulation over normed BPA/normed BPP Y. Hirshfeld, M. Jerrum and F . Moller, 1994/1996 Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 16 / 40

Previous Application weak bisimulation between FS and BPA/normed BPP ( † ) Antonín Kuˇ cera and Richard Mayr (2002) various bisimulations between FS and pushdown processes. Antonín Kuˇ cera and Richard Mayr (2004) Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 17 / 40

Our Application the application on the branching bisimilarity between a BPA system and a FS system. We follow and rely on the scheme of the previous work on weak bisimulation ( † ). Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 18 / 40

The Bisimulation Base between BPA and FS We fix a BPA system ( V , ∆) and a FS Γ . We construct the BPA system ( V ′ , ∆ ′ ) = ( V ∪ State (Γ) , ∆ ∪ Γ) Special attention on words of the form α f with α ∈ V ∗ , f ∈ State (Γ) . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 19 / 40

The Bisimulation Base between BPA and FS We also define: G Γ to be the ≈ br over Γ ∪ { ε } . Normed ( V ) = { X ∈ V | X is normed } . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 20 / 40

Well-formed Relations A relation K is well-formed if G Γ ⊆ K and K is a subset of the relation G defined by: G = (( Normed ( V ) · State (Γ)) × State (Γ)) ∪ ( V × State (Γ)) ∪ G Γ G is the largest well-formed relation. Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 21 / 40

The Bisimulation Base The bisimulation base B , is a well-formed relation as follows: B = { ( Yf , g ) ∈ Normed ( V ) · State (Γ) | Yf ≈ br g , Y ∈ Normed ( V ) } ∪{ ( Y , g ) ∈ V × State (Γ) | Y ≈ br g } ∪ G Γ Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 22 / 40

Closure Let K be a well-formed relation. The closure of K , denoted Cl ( K ) , is the least relation M such that: ( Yf , g ) ∈ K , ( α, f ) ∈ M ( Yf , g ) ∈ K , ( α h , f ) ∈ M K ⊆ M ( Y α, g ) ∈ M ( Y α h , g ) ∈ M ( α, g ) ∈ M , α is unnormed ( αβ, g ) , ( αβ h , g ) ∈ M for every β ∈ V ∗ and h ∈ State (Γ) Cl ( K ) will only contain pairs of two form: ( α, g ) and ( α f , g ) . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 23 / 40

The Idea Behind Closure The basic properties w.r.t sequential computation: Yf ≈ br g , α ≈ br f implies Y α ≈ br g . α ≈ br g , α is unnormed implies αβ ≈ br g , αβ h ≈ br g Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 24 / 40

Regular Property of the Closure Theorem Let K be a well-formed relation. For each g ∈ State (Γ) there is a finite automaton A K g constructible in polynomial time such that L ( A K g ) = { α | ( α, g ) ∈ Cl ( K ) } ∪ { α f | ( α f , g ) ∈ Cl ( K ) } ( α, g ) ∈ Cl ( K ) iff α ∈ L ( A K g ) . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 25 / 40

The Bisimulation Base: The Key Property Theorem Cl ( B ) = { ( α, g ) | α ≈ br g } ∪ { ( α f , g ) | α f ≈ br g } . α ≈ br g iff α ∈ A B g Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 26 / 40

Computing The Bisimulation Base Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 27 / 40

The Work Flow First we develop an expansion function Exp over well-formed relations such that Exp ( K ) ⊆ K for every K . Then we iteratively apply Exp to G : B 0 = G , B 1 = Exp ( B 0 ) ,. . . , B k + 1 = Exp ( B k ) , . . . Finally we obtain a fixed point of Exp which is exactly the bisimulation base Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 28 / 40

The Expansion Function: The Idea The idea naturally follows the definition of branching bisimulation. Let K be a well-formed relation. Roughly a pair, say, ( X , g ) expands in K by the following conditions: g g g X X X K K K ) ) K K ↓ a ↓ a ↓ a or ↓ τ ( ⇓ ( l l C C g ′ g ′ α ′ α ′ α ′ Cl ( K ) ) K ↓ a ( l C α ′′ Exp ( K ) is the set of pairs of K that expands in K . Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 29 / 40

Branching Bisimulation Under "Contraction" (van Glabbeek and P . Weijland) A binary relation R ⊆ V ∗ × State (Γ) is a branching bisimulation if whenever ( α, f ) ∈ R then for each a ∈ Act : α α α R f R f R f R R ↓ a ↓ a or ↓ τ ⇓ ↓ a α ′ f ′ α ′ α ′ f ′ R R ↓ a α ′′ The branching bisimilarity ≈ br is the largest branching bisimulation. Hongfei Fu (Shanghai Jiao Tong University) Deciding ≈ br between BPA and FS APLAS 2009 Paper 30 / 40