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Infinite State Model-Checking of Propositional Dynamic Logics Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart August 25, 2006 Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

Pushdown systems A pushdown system is a tuple S = ( P , Γ , ∆), where P is a finite set of control states , Γ is a finite stack alphabet , ∆ is a set of rewriting rules , where either p γ ֌ p ′ p γ ֌ p ′ γ ′ γ. or Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

Pushdown systems A pushdown system is a tuple S = ( P , Γ , ∆), where P is a finite set of control states , Γ is a finite stack alphabet , ∆ is a set of rewriting rules , where either p γ ֌ p ′ p γ ֌ p ′ γ ′ γ. or The pushdown graph G ( S ) has as nodes: P Γ ∗ edges: pw → p ′ w ′ if there is a rewriting rule in ∆ that can be applied to the prefixes accordingly. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

Model-checking pushdown systems INPUT: A pushdown system S , a configuration c , and a logical formula ϕ . QUESTION: ( G ( S ) , c ) | = ϕ ? Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

Model-checking pushdown systems INPUT: A pushdown system S , a configuration c , and a logical formula ϕ . QUESTION: ( G ( S ) , c ) | = ϕ ? Related results: MSO: decidable (non-elementary) [Muller/Schupp 96] µ -calculus: EXP-complete [Walukiewicz 96, Kupfermann/Vardi 00] CTL: EXP-complete [Walukiewicz 00] EF: PSPACE-complete [Esparza et al. 97, Walukiewicz 00] Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Syntax Fix some countable set A of atomic programs. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Syntax Fix some countable set A of atomic programs. Formulas ϕ and programs π of PDL ∩ are given by the following grammar, where a ∈ A : ::= true | ¬ ϕ | ϕ 1 ∨ ϕ 2 | � π � ϕ ϕ a | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 ◦ π 2 | π ∗ | ϕ ? π ::= Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Syntax Fix some countable set A of atomic programs. Formulas ϕ and programs π of PDL ∩ are given by the following grammar, where a ∈ A : ::= true | ¬ ϕ | ϕ 1 ∨ ϕ 2 | � π � ϕ ϕ a | π 1 ∪ π 2 | π 1 ∩ π 2 | π 1 ◦ π 2 | π ∗ | ϕ ? π ::= Abbreviation: [ π ] ϕ = ¬� π �¬ ϕ Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Semantics A Kripke structure is a tuple K = ( X , {→ a | a ∈ A } ), where X is a set of states , and → a ⊆ X × X is a binary relation for each a ∈ A . Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Semantics A Kripke structure is a tuple K = ( X , {→ a | a ∈ A } ), where X is a set of states , and → a ⊆ X × X is a binary relation for each a ∈ A . Define [ [ π ] ] K ⊆ X × X and [ [ ϕ ] ] K ⊆ X inductively: [ [ a ] ] K = → a [ [ true ] ] K = X [ [ ϕ ?] ] K = { ( x , x ) | x ∈ [ [ ϕ ] ] K } [ [ ¬ ϕ ] ] K = X \ [ [ ϕ ] ] K [ π ∗ ] ] ∗ [ ] K = [ [ π ] [ [ ϕ 1 ∨ ϕ 2 ] ] K = [ [ ϕ 1 ] ] K ∪ [ [ ϕ 2 ] ] K K [ [ π 1 op π 2 ] ] K = [ [ π 1 ] ] K op [ [ π 2 ] ] K where op ∈ {∪ , ∩ , ◦} [ [ � π � ϕ ] ] K = { x | ∃ y : ( x , y ) ∈ [ [ π ] ] K ∧ y ∈ [ [ ϕ ] ] K } Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ : An example The formula � ( a ◦ b ∗ ◦ a ) ∩ true ? � true enforces a cycle that begins with an a -labeled edge, followed by an arbitrary sequence of b -labeled edges, and ends with an a -labeled edge. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ : A non-trivial example Let K = ( X , {→ a | a ∈ Σ } ) be a deterministic Kripke structure. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ : A non-trivial example Let K = ( X , {→ a | a ∈ Σ } ) be a deterministic Kripke structure. We call a state x ∈ X a recovery state if, wherever we can get from x , we can always move back to x . Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ : A non-trivial example Let K = ( X , {→ a | a ∈ Σ } ) be a deterministic Kripke structure. We call a state x ∈ X a recovery state if, wherever we can get from x , we can always move back to x . A node x ∈ X is a recovery state if and only if � � = [Σ ∗ ] � � a � true ⇒ � true ? ∩ a ◦ Σ ∗ � true (K , x ) | . a ∈ Σ Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ : A non-trivial example Let K = ( X , {→ a | a ∈ Σ } ) be a deterministic Kripke structure. We call a state x ∈ X a recovery state if, wherever we can get from x , we can always move back to x . A node x ∈ X is a recovery state if and only if � � = [Σ ∗ ] � � a � true ⇒ � true ? ∩ a ◦ Σ ∗ � true (K , x ) | . a ∈ Σ The recovery state property cannot be expressed in the modal µ -calculus. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Properties and difficulties PDL ∩ does not have the tree model property, e.g. � a ∩ true ? � true enforces a • Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Properties and difficulties PDL ∩ does not have the tree model property, e.g. � a ∩ true ? � true enforces a • is therefore not bisimulation invariant. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Properties and difficulties PDL ∩ does not have the tree model property, e.g. � a ∩ true ? � true enforces a • is therefore not bisimulation invariant. does not have the finite model property. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

The logic PDL ∩ : Properties and difficulties PDL ∩ does not have the tree model property, e.g. � a ∩ true ? � true enforces a • is therefore not bisimulation invariant. does not have the finite model property. satisfiability is 2EXP-complete [Danecki 84, Lange/Lutz 2005]. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

Complexity results of the model-checking problem Basic process Pushdown Pref.-recogn. algebras systems systems EXP- P-complete data complete EF PSPACE-complete expression PDL \ ? EXP- combined complete data P-complete EXP-complete PDL expression combined PSPACE-hard EXP-complete data in EXP PDL ∩ expression 2EXP-complete PDL ∩ \ ? combined Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ over pushdown systems is in 2EXP (i) A two-way alternating parity ω -tree automaton (TWAPTA) T is an automaton, that Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ over pushdown systems is in 2EXP (i) A two-way alternating parity ω -tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ over pushdown systems is in 2EXP (i) A two-way alternating parity ω -tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, may use alternation, Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ over pushdown systems is in 2EXP (i) A two-way alternating parity ω -tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, may use alternation, can either move to some child, move to the parent node, or stay in the same node, and Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

PDL ∩ over pushdown systems is in 2EXP (i) A two-way alternating parity ω -tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, may use alternation, can either move to some child, move to the parent node, or stay in the same node, and uses a parity acceptance condition. Stefan G¨ oller and Markus Lohrey Universit¨ at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics