Optimal Bounds in Parametric LTL Games Martin Zimmermann RWTH Aachen University November 18th, 2010 Gasics Meeting Fall 2010 Paris, France Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 1/16
Motivation Parametric temporal logic ( PLTL , [Alur et. al., ’99] ): LTL with F ≤ x , G ≤ y . x , y variables ranging over N . Semantics w.r.t. variable valuation. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 2/16
Motivation Parametric temporal logic ( PLTL , [Alur et. al., ’99] ): LTL with F ≤ x , G ≤ y . x , y variables ranging over N . Semantics w.r.t. variable valuation. Results: Gasics Meeting Aachen (2009): determining whether Player 0 wins a PLTL game w.r.t. some, infinitely many, or all variable valuations is 2EXPTIME -complete. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 2/16
Motivation Parametric temporal logic ( PLTL , [Alur et. al., ’99] ): LTL with F ≤ x , G ≤ y . x , y variables ranging over N . Semantics w.r.t. variable valuation. Results: Gasics Meeting Aachen (2009): determining whether Player 0 wins a PLTL game w.r.t. some, infinitely many, or all variable valuations is 2EXPTIME -complete. Today: determining optimal variable valuations that let Player 0 win a PLTL game can be computed in doubly-exponential time. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 2/16
Outline 1. Introduction 2. Results 3. Proof Sketch 4. Conclusion Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 3/16
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 4/16
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 4/16
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 ) | = F ≤ x ϕ : ρ i + α ( x ) i ϕ ϕ ϕ ϕ ϕ ( ρ, i , α ) | = G ≤ y ϕ : ρ i + α ( y ) i Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 4/16
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 ) | = F ≤ x ϕ : ρ i + α ( x ) i ϕ ϕ ϕ ϕ ϕ ( ρ, i , α ) | = G ≤ y ϕ : ρ i + α ( y ) i The operators U ≤ x , R ≤ y , F > y , G > x , U > y , and R > x (with the obvious semantics) are syntactic sugar, and will be ignored. Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 4/16
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 5/16
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 5/16
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 . Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 5/16
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 6/16
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 6/16
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 6/16
Outline 1. Introduction 2. Results 3. Proof Sketch 4. Conclusion Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 7/16
Solving PLTL Games 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 8/16
Solving PLTL Games 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 8/16
Solving PLTL Games 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 The following problems are 2EXPTIME -complete: Given G and i: 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 8/16
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 9/16
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 9/16
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 9/16
Outline 1. Introduction 2. Results 3. Proof Sketch 4. Conclusion Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 10/16
A First Idea Duality and monotonicity: it suffices to determine min α ∈W 0 G F max x ∈ var ( ϕ F ) α ( x ). Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 11/16
A First Idea Duality and monotonicity: it suffices to determine min α ∈W 0 G F max x ∈ var ( ϕ F ) α ( x ). Lemma There exists a k ∈ O ( |A| · 2 2 | ϕ F | ) such that W 0 ⇒ x �→ k ∈ W 0 G F � = ∅ ⇐ G F ⇐ ⇒ min x ∈ var ( ϕ F ) α ( x ) ≤ k . max α ∈W 0 G F Martin Zimmermann RWTH Aachen University Optimal Bounds in Parametric LTL Games 11/16
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