Optimal Bounds in Parametric LTL Games Martin Zimmermann RWTH - PowerPoint PPT Presentation

Optimal Bounds in Parametric LTL Games Martin Zimmermann RWTH Aachen University June 16th, 2011 GandALF 2011 Minori, Italy Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 1/17

Motivation LTL as specification language in formal verification. Advantages: compact, variable-free syntax, intuitive semantics, successfully employed in model checking tools. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 2/17

Motivation LTL as specification language in formal verification. Advantages: compact, variable-free syntax, intuitive semantics, successfully employed in model checking tools. However, LTL lacks capabilities to express timing constraints. There are many extensions of LTL that deal with this. We consider Parametric LTL (Alur, Etessami, La Torre, Peled ’99) Prompt LTL (Kupferman, Piterman, Vardi ’07) Here: infinite games with winning conditions in parametric LTL. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 2/17

Outline 1. Introduction 2. Decision Problems 3. Optimization Problems 4. Conclusion Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 3/17

Parametric LTL LTL : ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 4/17

Parametric LTL PLTL : ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ where x ∈ X and y ∈ Y are variables ranging over N . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 4/17

Parametric LTL PLTL : ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ where x ∈ X and y ∈ Y are variables ranging over N . Semantics defined w.r.t. variable valuation α : X ∪ Y → N : ϕ ( ρ, i , α ) | ρ = F ≤ x ϕ : i + α ( x ) i ϕ ϕ ϕ ϕ ϕ = G ≤ y ϕ : ρ ( ρ, i , α ) | i + α ( y ) i Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 4/17

Parametric LTL PLTL : ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ where x ∈ X and y ∈ Y are variables ranging over N . Semantics defined w.r.t. variable valuation α : X ∪ Y → N : ϕ ( ρ, i , α ) | ρ = F ≤ x ϕ : i + α ( x ) i ϕ ϕ ϕ ϕ ϕ = G ≤ y ϕ : ρ ( ρ, i , α ) | i + α ( y ) i PROMPT − LTL : var ( ϕ ) = { x } ⊆ X . The operators U ≤ x , R ≤ y , F > y , G > x , U > y , and R > x (with the expected semantics) are syntactic sugar, and will be ignored. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 4/17

Infinite Games An arena A = ( V , V 0 , V 1 , E , v 0 , l ) consists of p q , r a finite, directed graph ( V , E ), v 1 v 3 p , q a partition { V 0 , V 1 } of V , v 0 an initial vertex v 0 , v 2 v 4 a labeling l : V → 2 P for some set P of ∅ r atomic propositions. Winning conditions are expressed by a PLTL formula ϕ over P . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 5/17

Infinite Games An arena A = ( V , V 0 , V 1 , E , v 0 , l ) consists of p q , r a finite, directed graph ( V , E ), v 1 v 3 p , q a partition { V 0 , V 1 } of V , v 0 an initial vertex v 0 , v 2 v 4 a labeling l : V → 2 P for some set P of ∅ r atomic propositions. Winning conditions are expressed by a PLTL formula ϕ over P . Play: path ρ 0 ρ 1 ρ 2 . . . through ( V , E ) starting in v 0 . ρ 0 ρ 1 ρ 2 . . . winning for Player 0 w.r.t. variable valuation α : ( ρ 0 ρ 1 ρ 2 . . . , 0 , α ) | = ϕ . Otherwise winning for Player 1. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 5/17

Infinite Games An arena A = ( V , V 0 , V 1 , E , v 0 , l ) consists of p q , r a finite, directed graph ( V , E ), v 1 v 3 p , q a partition { V 0 , V 1 } of V , v 0 an initial vertex v 0 , v 2 v 4 a labeling l : V → 2 P for some set P of ∅ r atomic propositions. Winning conditions are expressed by a PLTL formula ϕ over P . Play: path ρ 0 ρ 1 ρ 2 . . . through ( V , E ) starting in v 0 . ρ 0 ρ 1 ρ 2 . . . winning for Player 0 w.r.t. variable valuation α : ( ρ 0 ρ 1 ρ 2 . . . , 0 , α ) | = ϕ . Otherwise winning for Player 1. Strategy for Player i : σ : V ∗ V i → V s.t. ( v , σ ( wv )) ∈ E . Winning strategy for Player i w.r.t. α : every play that is consistent with σ is won by Player i w.r.t. α . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 5/17

PLTL Games: Examples Winning condition FG ≤ y p . Player 0’s goal: eventually satisfy p for at least α ( y ) steps. p p p p ≥ α ( y ) Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 6/17

PLTL Games: Examples Winning condition FG ≤ y p . Player 0’s goal: eventually satisfy p for at least α ( y ) steps. p p p p ≥ α ( y ) Winning condition G ( q → F ≤ x p ). Player 0’s goal: uniformly bound the waiting times between requests q and responses p by α ( x ). q p q p ≤ α ( x ) ≤ α ( x ) Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 6/17

PLTL Games: Examples Winning condition FG ≤ y p . Player 0’s goal: eventually satisfy p for at least α ( y ) steps. p p p p ≥ α ( y ) Winning condition G ( q → F ≤ x p ). Player 0’s goal: uniformly bound the waiting times between requests q and responses p by α ( x ). q p q p ≤ α ( x ) ≤ α ( x ) Note: both winning conditions induce an optimization problem: maximize α ( y ) respectively minimize α ( x ). Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 6/17

Outline 1. Introduction 2. Decision Problems 3. Optimization Problems 4. Conclusion Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 7/17

Previous Work Theorem (Pnueli, Rosner ’89) Determining the winner of an LTL game is 2EXPTIME -complete. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 8/17

Previous Work Theorem (Pnueli, Rosner ’89) Determining the winner of an LTL game is 2EXPTIME -complete. The set of winning valuations for Player i in a PLTL game G is W i G = { α | Player i has winning strategy for G w.r.t. α } . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 8/17

Previous Work Theorem (Pnueli, Rosner ’89) Determining the winner of an LTL game is 2EXPTIME -complete. The set of winning valuations for Player i in a PLTL game G is W i G = { α | Player i has winning strategy for G w.r.t. α } . Theorem (Kupferman, Piterman, Vardi ’07) The following problem is 2EXPTIME -complete: Given a PROMPT − LTL game G , is W 0 G non-empty? Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 8/17

Solving PLTL Games Useful properties of PLTL : Duality: F ≤ x ϕ ≡ ¬ G ≤ x ¬ ϕ . Monotonicity: α ( x ) ≤ β ( x ) and α ( y ) ≥ β ( y ). ( ρ, i , α ) | = F ≤ x ϕ ⇒ ( ρ, i , β ) | = F ≤ x ϕ . ( ρ, i , α ) | = G ≤ y ϕ ⇒ ( ρ, i , β ) | = G ≤ y ϕ . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 9/17

Solving PLTL Games Useful properties of PLTL : Duality: F ≤ x ϕ ≡ ¬ G ≤ x ¬ ϕ . Monotonicity: α ( x ) ≤ β ( x ) and α ( y ) ≥ β ( y ). ( ρ, i , α ) | = F ≤ x ϕ ⇒ ( ρ, i , β ) | = F ≤ x ϕ . ( ρ, i , α ) | = G ≤ y ϕ ⇒ ( ρ, i , β ) | = G ≤ y ϕ . Application: Theorem The following problems are 2EXPTIME -complete: Given PLTL game G and i ∈ { 0 , 1 } . i) Is W i G non-empty? ii) Is W i G infinite? iii) Is W i G universal? Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 9/17

Outline 1. Introduction 2. Decision Problems 3. Optimization Problems 4. Conclusion Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 10/17

Finding Optimal Bounds If ϕ contains only F ≤ x respectively only G ≤ y , then solving games is an optimization problem: which is the best valuation in W 0 G ? Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 11/17

Finding Optimal Bounds If ϕ contains only F ≤ x respectively only G ≤ y , then solving games is an optimization problem: which is the best valuation in W 0 G ? Theorem Let ϕ F be G ≤ y -free and ϕ G be F ≤ x -free, let G F = ( A , ϕ F ) and G G = ( A , ϕ G ) . The following values can be computed in doubly-exponential time: min α ∈W 0 G F max x ∈ var ( ϕ F ) α ( x ) . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 11/17

Finding Optimal Bounds If ϕ contains only F ≤ x respectively only G ≤ y , then solving games is an optimization problem: which is the best valuation in W 0 G ? Theorem Let ϕ F be G ≤ y -free and ϕ G be F ≤ x -free, let G F = ( A , ϕ F ) and G G = ( A , ϕ G ) . The following values can be computed in doubly-exponential time: min α ∈W 0 G F max x ∈ var ( ϕ F ) α ( x ) . min α ∈W 0 G F min x ∈ var ( ϕ F ) α ( x ) . max α ∈W 0 G G max y ∈ var ( ϕ G ) α ( y ) . max α ∈W 0 G G min y ∈ var ( ϕ G ) α ( y ) . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 11/17