Parametric Linear Temporal Logics Joint work with Peter Faymonville, Florian Horn, Wolfgang Thomas, and Nico Wallmeier Martin Zimmermann Saarland University March 10th, 2015 Aalborg University, Aalborg, Denmark Martin Zimmermann Saarland University Parametric Linear Temporal Logics 1/25
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. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. 2. LTL cannot express all ω -regular properties. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. Add F ≤ k for k ∈ N . 2. LTL cannot express all ω -regular properties. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. Add F ≤ k for k ∈ N . Not practical: how to determine appropriate k . 2. LTL cannot express all ω -regular properties. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. Add F ≤ k for k ∈ N . Not practical: how to determine appropriate k . Add F ≤ x for variable x . 2. LTL cannot express all ω -regular properties. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. Add F ≤ k for k ∈ N . Not practical: how to determine appropriate k . Add F ≤ x for variable x . Now: does there exist a valuation for x s.t. specification is satisfied? 2. LTL cannot express all ω -regular properties. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
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. Shortcomings: 1. LTL cannot express timing constraints. Add F ≤ k for k ∈ N . Not practical: how to determine appropriate k . Add F ≤ x for variable x . Now: does there exist a valuation for x s.t. specification is satisfied? 2. LTL cannot express all ω -regular properties. Many extensions that are equivalent to ω -regular languages: add regular expression-, grammar-, or automata-operators to LTL. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 2/25
Overview LTL Martin Zimmermann Saarland University Parametric Linear Temporal Logics 3/25
Overview PLTL LTL Martin Zimmermann Saarland University Parametric Linear Temporal Logics 3/25
Parametric LTL Alur et al. ’99: add parameterized operators to LTL ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ with x ∈ X , y ∈ Y ( X ∩ Y = ∅ ). Martin Zimmermann Saarland University Parametric Linear Temporal Logics 4/25
Parametric LTL Alur et al. ’99: add parameterized operators to LTL ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ with x ∈ X , y ∈ Y ( X ∩ Y = ∅ ). Semantics w.r.t. variable valuation α : X ∪ Y → N : As usual for LTL operators. ϕ = F ≤ x ϕ : ρ ( ρ, n , α ) | n n + α ( x ) ϕ ϕ ϕ ϕ ϕ = G ≤ y ϕ : ρ ( ρ, n , α ) | n n + α ( y ) Martin Zimmermann Saarland University Parametric Linear Temporal Logics 4/25
Parametric LTL Alur et al. ’99: add parameterized operators to LTL ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ≤ x ϕ | G ≤ y ϕ with x ∈ X , y ∈ Y ( X ∩ Y = ∅ ). Semantics w.r.t. variable valuation α : X ∪ Y → N : As usual for LTL operators. ϕ = F ≤ x ϕ : ρ ( ρ, n , α ) | n n + α ( x ) ϕ ϕ ϕ ϕ ϕ = G ≤ y ϕ : ρ ( ρ, n , α ) | n n + α ( y ) Fragments: PLTL F : no parameterized always operators G ≤ y . PLTL G : no parameterized eventually operators F ≤ x . Martin Zimmermann Saarland University Parametric Linear Temporal Logics 4/25
PLTL Games { q 0 } { p 0 } { q 0 , q 1 } { d } v 0 { q 1 } { p 1 } Martin Zimmermann Saarland University Parametric Linear Temporal Logics 5/25
PLTL Games { 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 ) : Player 0 wins. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 5/25
PLTL Games { 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 ) : Player 0 wins. ϕ 2 = FG d ∨ � i ∈{ 0 , 1 } G ( q i → F ≤ x i p i ) : Player 1 wins w.r.t. every α . Martin Zimmermann Saarland University Parametric Linear Temporal Logics 5/25
PLTL Games { q 0 } { p 0 } { q 0 , q 1 } { d } v 0 { q 1 } { p 1 } W i ( G ) = { α | Player i has winning strategy for G w.r.t. α } Martin Zimmermann Saarland University Parametric Linear Temporal Logics 5/25
PLTL Games { q 0 } { p 0 } { q 0 , q 1 } { d } v 0 { q 1 } { p 1 } 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 Saarland University Parametric Linear Temporal Logics 5/25
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 Saarland University Parametric Linear Temporal Logics 6/25
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 ’89) Solving LTL games is 2Exptime -complete. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 6/25
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 ’89) Solving LTL games is 2Exptime -complete. Adding parameterized operators does not increase complexity: Theorem (Z. ’11) All four decision problems are 2Exptime -complete. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 6/25
Proof Sketch (Emptiness) 1. Replacing G ≤ y ψ by ψ preserves emptiness (monotonicity). 2. Apply alternating color technique (Kupferman et al. ’06) : Add new proposition p and replace every F ≤ x ψ by ( p → p U ( ¬ p U ψ )) ∧ ( ¬ p → ¬ p U ( p U ψ )) ( ψ satisfied within one color change), obtain c ( ϕ ). Martin Zimmermann Saarland University Parametric Linear Temporal Logics 7/25
Proof Sketch (Emptiness) 1. Replacing G ≤ y ψ by ψ preserves emptiness (monotonicity). 2. Apply alternating color technique (Kupferman et al. ’06) : Add new proposition p and replace every F ≤ x ψ by ( p → p U ( ¬ p U ψ )) ∧ ( ¬ p → ¬ p U ( p U ψ )) ( ψ satisfied within one color change), obtain c ( ϕ ). Lemma ϕ and c ( ϕ ) “equivalent” on traces where distance between color changes is bounded. Martin Zimmermann Saarland University Parametric Linear Temporal Logics 7/25
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