Decidability of branching bisimulation on normed commutative context - - PowerPoint PPT Presentation

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Decidability of branching bisimulation on normed commutative context - - PowerPoint PPT Presentation

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Decidability of branching bisimulation on normed commutative context - free graphs W ojciech Czerwi ski Piotr Hofman S awomir Lasota University of W arsaw Problem Problem Given: infinite graph G, two vertices u, v Problem Given:

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Decidability of branching bisimulation on normed commutative context-free graphs

W

  • jciech Czerwiński

Piotr Hofman Sławomir Lasota University of W arsaw

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Problem

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Problem

Given: infinite graph G, two vertices u, v

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Problem

Given: infinite graph G, two vertices u, v (G - normed commutative context-free graph)

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Problem

Given: infinite graph G, two vertices u, v Question: is u equivalent to v? (G - normed commutative context-free graph)

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Problem

Given: infinite graph G, two vertices u, v Question: is u equivalent to v? (G - normed commutative context-free graph) (branching bisimulation equivalent)

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Problem

Given: infinite graph G, two vertices u, v Question: is u equivalent to v? Our result: the problem is decidable (G - normed commutative context-free graph) (branching bisimulation equivalent)

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Outline

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Outline

  • Commutative context-free graphs
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Outline

  • Commutative context-free graphs
  • Bisimulation equivalence
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Outline

  • Commutative context-free graphs
  • Bisimulation equivalence
  • Ideas of the proof
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context-free grammars Commutative

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context-free grammars

X XCBC a

Commutative

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context-free grammars

X XCBC Greibach normal form a

Commutative

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context-free grammars

X XCBC Greibach normal form a

Commutative

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context-free grammars

X XCBC Greibach normal form a B C b c X BC a ε ε

Commutative

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context-free grammars

X XCBC Greibach normal form a B C b c X BC a ε ε

Commutative

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context-free grammars

X Greibach normal form a B C b c X BC a ε ε

Commutative

XBC2

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context-free grammars

X Greibach normal form a B C b c X BC a ε ε

Commutative

X XBC2

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context-free grammars

X Greibach normal form a B C b c X BC a ε ε

Commutative

X XBC2 a XBC2

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context-free grammars

X Greibach normal form a B C b c X BC a ε ε

Commutative

X XBC2 a b XC2 XBC2

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context-free grammars

X Greibach normal form a B C b c X BC a ε ε

Commutative

X XBC2 a b XC2 a BC3 XBC2

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Commutative context-free graphs

X a B C b c X BC a ε ε XBC2

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Commutative context-free graphs

X3B7C4

X a B C b c X BC a ε ε XBC2

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Commutative context-free graphs

X3B7C4 X3B8C6

a X a B C b c X BC a ε ε XBC2

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Commutative context-free graphs

X3B7C4 X2B8C5

a

X3B8C6

a X a B C b c X BC a ε ε XBC2

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Commutative context-free graphs

X3B7C4 X2B8C5

a

X3B8C6

a

X3B6C4

b X a B C b c X BC a ε ε XBC2

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Commutative context-free graphs

X3B7C4 X2B8C5

a

X3B8C6

a

X3B6C4

b

X3B7C3

c X a B C b c X BC a ε ε XBC2

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Commutative context-free graphs

X3B7C4 X2B8C5

a

X3B8C6

a

X3B6C4

b

X3B7C3

c X a B C b c X BC a ε ε XBC2 Multisets over {X, B, C}

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Commutative context-free graphs

X3B7C4 X2B8C5

a

X3B8C6

a

X3B6C4

b

X3B7C3

c V ertices: points in N3 X a B C b c X BC a ε ε XBC2 Multisets over {X, B, C}

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Commutative context-free graphs

X3B7C4 X2B8C5

a

X3B8C6

a

X3B6C4

b

X3B7C3

c V ertices: points in N3 X a B C b c X BC a ε ε XBC2 Special kind of Petri Net Multisets over {X, B, C}

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Normed grammars

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Normed grammars

X a B C b c X BC a ε ε XBC2

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Normed grammars

variable X normed ⇔ L(X) ≠ ∅ X a B C b c X BC a ε ε XBC2

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Normed grammars

G normed ⇔ every variable normed variable X normed ⇔ L(X) ≠ ∅ X a B C b c X BC a ε ε XBC2

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Normed grammars

G normed ⇔ every variable normed variable X normed ⇔ L(X) ≠ ∅ X a B C b c X BC a ε ε XBC2 Y YB a

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Normed grammars

G normed ⇔ every variable normed variable X normed ⇔ L(X) ≠ ∅ Our assumption: grammar G is normed X a B C b c X BC a ε ε XBC2 Y YB a

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Normed grammars

G normed ⇔ every variable normed variable X normed ⇔ L(X) ≠ ∅ Our assumption: grammar G is normed X a B C b c X BC a ε ε XBC2 Y YB a

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Process rewrite systems

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Process rewrite systems

commutative context-free graphs (BPP)

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets

⊆

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets context-free graphs (BPA)

⊆

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets context-free graphs (BPA)

⊆

FS

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets context-free graphs (BPA)

⊆

FS

⊆

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets context-free graphs (BPA)

⊆

FS

⊆ ⊆

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets context-free graphs (BPA) PDA

⊆

FS

⊆ ⊆

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Process rewrite systems

commutative context-free graphs (BPP) Petri Nets context-free graphs (BPA) PDA

⊆ ⊆

FS

⊆ ⊆

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Strong bisimilarity

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Strong bisimilarity

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a

Strong bisimilarity

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a a

Strong bisimilarity

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a a

Strong bisimilarity

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W eak bisimilarity

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W eak bisimilarity

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W eak bisimilarity

a

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W eak bisimilarity

a

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W eak bisimilarity

a

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W eak bisimilarity

a a

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W eak bisimilarity

a a

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W eak bisimilarity

a a

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W eak bisimilarity

a a

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Long-standing open problem

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Long-standing open problem

  • Given: commutative context-free graph G,

two vertices u, v

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Long-standing open problem

  • Given: commutative context-free graph G,

two vertices u, v

  • Question: are u and v weak bisimilar?
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Long-standing open problem

  • Given: commutative context-free graph G,

two vertices u, v

  • Question: are u and v weak bisimilar?
  • Open since 90-ties
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Long-standing open problem

  • Given: commutative context-free graph G,

two vertices u, v

  • Question: are u and v weak bisimilar?
  • Open since 90-ties
  • W

e did not solve this

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Long-standing open problem

  • Given: commutative context-free graph G,

two vertices u, v

  • Question: are u and v weak bisimilar?
  • Open since 90-ties
  • W

e did not solve this

  • W

e solve a slight modification (branching bisimulation)

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Branching bisimilarity

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Branching bisimilarity

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Branching bisimilarity

a

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Branching bisimilarity

a a

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Branching bisimilarity

a a

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Branching bisimilarity

a a

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Branching bisimilarity

a a

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Main result

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Main result

Branching bisimulation on commutative context-free graphs is decidable

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Algorithm for strong bisimulation

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Algorithm for strong bisimulation

  • Key property: image finiteness
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Algorithm for strong bisimulation

  • Key property: image finiteness

finite Duplicator

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Algorithm for strong bisimulation

  • Key property: image finiteness
  • Two semi-decision procedures:

finite Duplicator

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Algorithm for strong bisimulation

  • Key property: image finiteness
  • Two semi-decision procedures:
  • one looking for positive witness

finite Duplicator

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Algorithm for strong bisimulation

  • Key property: image finiteness
  • Two semi-decision procedures:
  • one looking for positive witness
  • one looking for negative witness

finite Duplicator

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Positive semi-procedure

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Positive semi-procedure

  • Every congruence in Nk is semilinear
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Positive semi-procedure

  • Every congruence in Nk is semilinear
  • Strong bisimilarity is a congruence
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Positive semi-procedure

  • Every congruence in Nk is semilinear
  • Strong bisimilarity is a congruence
  • Procedure: enumerate all semilinear relations
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Positive semi-procedure

  • Every congruence in Nk is semilinear
  • Strong bisimilarity is a congruence
  • Procedure: enumerate all semilinear relations
  • W
  • rks for weak and branching bisimulation as

well

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Negative semi-procedure

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Negative semi-procedure

  • Search for a Spoiler’s winning

strategy

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Negative semi-procedure

  • Search for a Spoiler’s winning

strategy Spoiler Duplicator Spoiler . . . . . . . .

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Negative semi-procedure

  • Search for a Spoiler’s winning

strategy

  • Difficulty: it has to be finite

Spoiler Duplicator Spoiler . . . . . . . .

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Negative semi-procedure

  • Search for a Spoiler’s winning

strategy

  • Difficulty: it has to be finite
  • Strategy is winning: every path

finite Spoiler Duplicator Spoiler . . . . . . . .

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Negative semi-procedure

  • Search for a Spoiler’s winning

strategy

  • Difficulty: it has to be finite
  • Strategy is winning: every path

finite

  • T

ree is finitely-branching Spoiler Duplicator Spoiler . . . . . . . .

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Negative semi-procedure

  • Search for a Spoiler’s winning

strategy

  • Difficulty: it has to be finite
  • Strategy is winning: every path

finite

  • T

ree is finitely-branching

  • By König’s Lemma the tree is

finite Spoiler Duplicator Spoiler . . . . . . . .

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Idea of the algorithm

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Idea of the algorithm

  • Branching bisimilarity is essentially image

finite

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Idea of the algorithm

  • Branching bisimilarity is essentially image

finite

  • It is enough to inspect only boundedly many

possible Duplicator’s responses

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Small response property

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Small response property

α β

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Small response property

a α β α’

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Small response property

a a α β α’ β’

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Small response property

a a α β α’ β’

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Small response property

a a α β α’ β’ size(β’) ≤ C size(α’)

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Small response property

a a α β α’ β’ size(β’) ≤ C size(α’) C efectively computed

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Branching vs weak

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Branching vs weak

Everything except one case works fine

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Further work

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Further work

  • Complexity of the problem
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Further work

  • Complexity of the problem
  • W

eak bisimulation

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Further work

  • Complexity of the problem
  • W

eak bisimulation

  • Context-free graphs (BPA)
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Further work

  • Complexity of the problem
  • W

eak bisimulation

  • Context-free graphs (BPA)

Thank you!