The modal -calculus Hierarchy on Restricted Classes of Transition - PowerPoint PPT Presentation

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Alessandro Facchini 1 Luca Alberucci 2 1 University of Lausanne and LaBRI, Bordeaux 2 IAM, University of Berne Logic Colloquium Berne, July 4th 2008

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Goal of our works Understand the expressive power of modal µ -calculus over different classes of models.

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction What is the modal µ -calculus

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction What is the modal µ -calculus the propositional modal µ -calculus = propositional modal logic + least and greatest fixpoint operators

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Expressive power Eventually ”p”: µ x . p ∨ ✸ x Allways ”p”: ν x . p ∧ � x Allways eventually ”p”: ν x . ( µ y . p ∨ ✸ y ) ∧ � x There is a branch such that infinitely often ”p”: ν x .µ y . ( p ∧ ✸ x ) ∨ ✸ y

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction The fixpoint alternation depth The fixpoint alternation of a formula is the number of non-trivial nestings of alternating least and greatest fixpoints.

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Example ϕ 1 := p ∨ ♦ q

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Example ϕ 2 := µ x . p ∨ ♦ x µ x ✛ . . . . . . . . . . . . . . . . . . ❄ . . . . . . . . . . ∨ x ✲ ✛ ❄ p ♦

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Example ϕ 3 := ν x .µ y . ( p ∧ ♦ x ) ∨ ♦ y ✲ ν x . . . . . . . . . . . . . . . . . . . . . . . ❄ . . . . . . . ✲ x µ y ✛ . . . . . . . . . . . . . . . . . . ❄ . . . . . . . . . . ♦ ∨ y ✛ ✲ ✛ ❄ p ✛ ∧ ♦

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Example ϕ 4 := µ x ( ν y ( p ∧ ♦ y ) ∨ � x ) µ x ✛ . . . . . . . . . . . . . . . . . . . ❄ . . . . . . . . . ∨ x ✻ ✲ ✛ ν y � ✲ . . . . . . . . . . . . . . . . . . . . ❄ . . . . . . . . y ∧ ✻ ✛ ✲ ♦ p

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction ∆ µ 3 ������� � � � � � � � Σ µ Π µ 2 2 � � � � � � � � � � � � � � � ∆ µ 2 ������� � � � � � � � Σ µ Π µ 1 1 � � � � � � � � � � � � � � � ∆ µ 1

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Is the semantical hierarchy strict? ∆ T µ 3 � � � � � � � � � � � � � � � � Σ T µ Π T µ 2 2 � � � � � � � � � � � � � � � � ∆ T µ 2 � � � � � � � � � � � � � � � � Σ T µ Π T µ 1 1 � � � � � � � � � � � � � � � � ∆ T µ 1

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Semantical complexity Bradfield (1996): Stictness of semantical modal µ -calculus hierarchy The semantical modal µ -calculus hierarchy is strict on the class of all transition systems. ⇒ For each n there is a formula ϕ with ad( ϕ ) = n such that for all formulae ψ with ad( ψ ) < n we do not have for all transition systems T : ( T | = ϕ ⇔ T | = ψ ) . Proof. Game formulae are complete for their corresponding level.

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Question What happens for restricted classes of transition systems? refl sym tr sym & tr tr & wf strict ? ? ? ? ? collapse ? ? ? ? ?

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems Introduction Overview The modal µ -calculus The Hierarchy on Transitive and Symmetric Transition Systems The Hierarchy on Transitive Transition Systems A final picture

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Syntax of µ -calculus ϕ :: ≡ p | ∼ p | ⊤ | ⊥ | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ✸ ϕ | � ϕ . . . . . . | µ x .ϕ | ν x .ϕ where p , x ∈ Prop and x occurs only positively.

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Semantics A transition system T is a triple (S , → T , λ T ) consisting of ◮ a set S of states , ◮ a binary relation → T ⊆ S × S called transition relation , ◮ the valuation λ : P → ℘ (S) assigning to each propositional variable p a subset λ ( p ) of S.

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Denotation and validity of a formula Given a transition system T = (S , → T , λ T ). We define � ϕ � T inductively such that ◮ � p � T = λ T ( p ) and � ∼ p � T = S \ λ T ( p ) ◮ � α ∧ β � T = � α � T ∩ � β � T and � α ∨ β � T = � α � T ∪ � β � T , ◮ � � α � T = { s ∈ S | ∀ s ′ ( s → T s ′ = ⇒ s ′ ∈ � α � T ) } , ◮ � ✸ α � T = { s ∈ S | ∃ s ′ ( s → T s ′ and s ′ ∈ � α � T ) } .

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Denotation and validity of a formula ◮ . . . �� { S ′ ⊆ S | S ′ ⊆ � ϕ ( x ) � T [ x �→ S ′ ] } ◮ � ν x .ϕ ( x ) � T = GFP ( � ϕ ( x ) � T ) �� { S ′ ⊆ S | � ϕ ( x ) � T [ x �→ S ′ ] ⊆ S ′ } ◮ � µ x .ϕ ( x ) � T = LFP ( � ϕ ( x ) � T )

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus An equivalence A formula is well-named if bound and free variables are pairwise distinct and if all bound variable occur only once and are guarded. Lemma Every formula ϕ is equivalent to a well-named formula nf( ϕ ) .

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Evaluation games ◮ The player are V (verifier) and F (falsifier). Consider w 0 ∈ S, V tries to show that w 0 � ϕ , while F tries to show that w 0 � ϕ .

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Evaluation games ◮ The player are V (verifier) and F (falsifier). Consider w 0 ∈ S, V tries to show that w 0 � ϕ , while F tries to show that w 0 � ϕ . ◮ the play starts at � ϕ, w 0 � ◮ the admissible moves are choices of subformulas and points of S which respect the following rules:

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Evaluation game for modal logic position player next position � p i , w � F iff p i ∈ λ ( w ) - �¬ p i , w � V iff p i ∈ λ ( w ) - � ψ ∨ φ, w � V chooses between � ψ, w � and � φ, w � V choice � ψ ∧ φ, w � F chooses between � ψ, w � and � φ, w � F choice V chooses a point w ′ s.t. wRw ′ � ψ, w ′ � �♦ ψ, w � F chooses a point w ′ s.t. wRw ′ � ψ, w ′ � � � ψ, w �

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Evaluation game for modal logic ◮ V wins iff F cannot move.

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Example ( ♦♦ p ∨ � p ) ∧ r ✄ ∧ ✛ ✲ { p , q , r } ∨ r ✛ ❄ ✛ ❄ { p } ∅ ♦ � ✲ ✛ ❄ ❄ ❄ p ❄ ∅ ∅ ∅ { p } ♦ . . . . . . . . . . . . p ❄

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Evaluation game for µ -calculus position player next position � p i , w � F iff p i ∈ λ ( w ) - �¬ p i , w � V iff p i ∈ λ ( w ) - � ψ ∨ φ, w � V chooses between � ψ, w � and � φ, w � V choice � ψ ∧ φ, w � F chooses between � ψ, w � and � φ, w � F choice V chooses a point w ′ s.t. wRw ′ �♦ ψ, w � � ψ, w ′ � F chooses a point w ′ s.t. wRw ′ � � ψ, w � � ψ, w ′ � � µ x .ψ, w � - � ψ, w � � ν x .ψ, w � � ψ, w � - � x , w � - � η x .ψ ( x ) , w �

The modal µ -calculus Hierarchy on Restricted Classes of Transition Systems The modal µ -calculus Evaluation game for µ -calculus ◮ . . . the game behind the µ -calculus is a parity game ◮ . . . “ µ means finite looping” ◮ . . . “ ν means infinite looping”