Optimal Bounds in Parametric LTL Games Martin Zimmermann Universit¨ at des Saarlandes October 28th, 2013 AVACS Meeting Freiburg, Germany Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 1/13
Motivation Linear Temporal Logic ( LTL ) as specification language: Simple and variable-free syntax and intuitive semantics. Expressively equivalent to first-order logic on words. LTL model-checking routinely applied in industrial settings. But LTL cannot express timing constraints. Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/13
Motivation Linear Temporal Logic ( LTL ) as specification language: Simple and variable-free syntax and intuitive semantics. Expressively equivalent to first-order logic on words. LTL model-checking routinely applied in industrial settings. But LTL cannot express timing constraints. Possible remedies: Add F ≤ k for k ∈ N . Problem: finding “right” k impracticable. Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/13
Motivation Linear Temporal Logic ( LTL ) as specification language: Simple and variable-free syntax and intuitive semantics. Expressively equivalent to first-order logic on words. LTL model-checking routinely applied in industrial settings. But LTL cannot express timing constraints. Possible remedies: Add F ≤ k for k ∈ N . Problem: finding “right” k impracticable. Alur et. al, Kupferman et. al: add F ≤ x for variable x . Now: does there exist a value x such that F ≤ x ϕ holds? what is the best value x such that F ≤ x ϕ holds? In Model-Checking: adding variable time bounds does not increase complexity. Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 2/13
Infinite Games Arena A = ( V , V 0 , V 1 , E ): finite directed graph ( V , E ), V 0 ⊆ V positions of Player 0 (circles), V 1 = V \ V 0 positions of Player 1 (squares). 0 1 2 Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 3/13
Infinite Games Arena A = ( V , V 0 , V 1 , E ): finite directed graph ( V , E ), V 0 ⊆ V positions of Player 0 (circles), V 1 = V \ V 0 positions of Player 1 (squares). 0 1 2 Play: path ρ 0 ρ 1 · · · through A . Strategy for Player i : σ : V ∗ V i → V s.t. ( v , σ ( wv )) ∈ E . ρ 0 ρ 1 · · · consistent with σ : ρ n +1 = σ ( ρ 0 · · · ρ n ) for all n s.t. ρ n ∈ V i . Finite-state strategy: implemented by finite automaton with output. Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 3/13
PLTL: Syntax and Semantics Parametric LTL : p atomic proposition, x ∈ X , y ∈ Y ( X ∩ Y = ∅ ). ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/13
PLTL: Syntax and Semantics Parametric LTL : p atomic proposition, x ∈ X , y ∈ Y ( X ∩ Y = ∅ ). ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ Semantics w.r.t. variable valuation α : X ∪ Y → N : As usual for LTL operators. ϕ ( ρ, n , α ) | = F ≤ x ϕ : ρ n n + α ( x ) ϕ ϕ ϕ ϕ ϕ ( ρ, n , α ) | = G ≤ y ϕ : ρ n n + α ( y ) Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/13
PLTL: Syntax and Semantics Parametric LTL : p atomic proposition, x ∈ X , y ∈ Y ( X ∩ Y = ∅ ). ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ Semantics w.r.t. variable valuation α : X ∪ Y → N : As usual for LTL operators. ϕ ( ρ, n , α ) | = F ≤ x ϕ : ρ n n + α ( x ) ϕ ϕ ϕ ϕ ϕ ( ρ, n , α ) | = G ≤ y ϕ : ρ n n + α ( y ) Fragments: PLTL F : no parameterized always operators G ≤ y . PLTL G : no parameterized eventually operators F ≤ x . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 4/13
PLTL Games PLTL game: G = ( A , v 0 , ϕ ) with arena A (labeled by ℓ : V → 2 P ), initial vertex v 0 , and PLTL formula ϕ . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13
PLTL Games PLTL game: G = ( A , v 0 , ϕ ) with arena A (labeled by ℓ : V → 2 P ), initial vertex v 0 , and PLTL formula ϕ . Rules: all plays start in v 0 . Player 0 wins ρ 0 ρ 1 · · · w.r.t. α , if ( ℓ ( ρ 0 ) ℓ ( ρ 1 ) · · · , α ) | = ϕ . Player 1 wins ρ 0 ρ 1 · · · w.r.t. α , if ( ℓ ( ρ 0 ) ℓ ( ρ 1 ) · · · , α ) �| = ϕ . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13
PLTL Games PLTL game: G = ( A , v 0 , ϕ ) with arena A (labeled by ℓ : V → 2 P ), initial vertex v 0 , and PLTL formula ϕ . Rules: all plays start in v 0 . Player 0 wins ρ 0 ρ 1 · · · w.r.t. α , if ( ℓ ( ρ 0 ) ℓ ( ρ 1 ) · · · , α ) | = ϕ . Player 1 wins ρ 0 ρ 1 · · · w.r.t. α , if ( ℓ ( ρ 0 ) ℓ ( ρ 1 ) · · · , α ) �| = ϕ . σ is winning strategy for Player i w.r.t. α , if every consistent play is winning for Player i w.r.t. α . Winning valuations for Player i W i ( G ) = { α | Player i has winning strategy for G w.r.t. α } Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13
PLTL Games PLTL game: G = ( A , v 0 , ϕ ) with arena A (labeled by ℓ : V → 2 P ), initial vertex v 0 , and PLTL formula ϕ . Rules: all plays start in v 0 . Player 0 wins ρ 0 ρ 1 · · · w.r.t. α , if ( ℓ ( ρ 0 ) ℓ ( ρ 1 ) · · · , α ) | = ϕ . Player 1 wins ρ 0 ρ 1 · · · w.r.t. α , if ( ℓ ( ρ 0 ) ℓ ( ρ 1 ) · · · , α ) �| = ϕ . σ is winning strategy for Player i w.r.t. α , if every consistent play is winning for Player i w.r.t. α . Winning valuations for Player i W i ( G ) = { α | Player i has winning strategy for G w.r.t. α } Lemma Determinacy: W 0 ( G ) is the complement of W 1 ( G ) . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 5/13
An Example { q 0 } { p 0 } { q 0 , q 1 } { d } v 0 { q 1 } { p 1 } Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 6/13
An Example { q 0 } { p 0 } { q 0 , q 1 } { d } v 0 { q 1 } { p 1 } ϕ 1 = FG d ∨ � i ∈{ 0 , 1 } G ( q i → F p i ) : W 1 ( G 1 ) = ∅ . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 6/13
An Example { q 0 } { p 0 } { q 0 , q 1 } { d } v 0 { q 1 } { p 1 } ϕ 1 = FG d ∨ � i ∈{ 0 , 1 } G ( q i → F p i ) : W 1 ( G 1 ) = ∅ . ϕ 2 = FG d ∨ � i ∈{ 0 , 1 } G ( q i → F ≤ x i p i ) : W 0 ( G 2 ) = ∅ . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 6/13
More Example Properties Bounded B¨ uchi: GF ≤ x p Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 7/13
More Example Properties Bounded B¨ uchi: GF ≤ x p Finitary parity (Chatterjee, Henzinger, Horn): � � c ′ FG c → F ≤ x c odd c ′ > c c ′ even Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 7/13
More Example Properties Bounded B¨ uchi: GF ≤ x p Finitary parity (Chatterjee, Henzinger, Horn): � � c ′ FG c → F ≤ x c odd c ′ > c c ′ even Finitary Streett (CHH): k � FG ( R j → F ≤ x G j ) j =1 Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 7/13
Decision Problems Membership: given G , i ∈ { 0 , 1 } , and α , is α ∈ W i ( G )? Emptiness: given G and i ∈ { 0 , 1 } , is W i ( G ) empty? Finiteness: given G and i ∈ { 0 , 1 } , is W i ( G ) finite? Universality: given G and i ∈ { 0 , 1 } , is W i ( G ) universal? Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 8/13
Decision Problems Membership: given G , i ∈ { 0 , 1 } , and α , is α ∈ W i ( G )? Emptiness: given G and i ∈ { 0 , 1 } , is W i ( G ) empty? Finiteness: given G and i ∈ { 0 , 1 } , is W i ( G ) finite? Universality: given G and i ∈ { 0 , 1 } , is W i ( G ) universal? The benchmark: Theorem (Pnueli, Rosner 1989) Solving LTL games is 2Exptime -complete. Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 8/13
Decision Problems Membership: given G , i ∈ { 0 , 1 } , and α , is α ∈ W i ( G )? Emptiness: given G and i ∈ { 0 , 1 } , is W i ( G ) empty? Finiteness: given G and i ∈ { 0 , 1 } , is W i ( G ) finite? Universality: given G and i ∈ { 0 , 1 } , is W i ( G ) universal? The benchmark: Theorem (Pnueli, Rosner 1989) Solving LTL games is 2Exptime -complete. Adding parameterized operators does not increase complexity: Theorem All four decision problems are 2Exptime -complete. Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 8/13
Proof Idea Emptiness for PLTL F games, i.e., only F ≤ x in ϕ . 1. Duplicate arena, color one copy red, the other green. Player 0 can change between copies after every move. 2. Inductively replace every F ≤ x ψ by ( red → ( red U ( green U ψ ))) ∧ ( green → ( green U ( red U ψ ))) 3. Add conjunct GF red ∧ GF green to ϕ , obtain ϕ ′ . Martin Zimmermann Universit¨ at des Saarlandes Parametric LTL Games 9/13
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