Reducing -regular Specifications to Safety Conditions Joint work - PowerPoint PPT Presentation

Reducing ω -regular Specifications to Safety Conditions Joint work with John Fearnley (University of Liverpool) Daniel Neider (RWTH Aachen University) Roman Rabinovich (TU Berlin) Martin Zimmermann Saarland University March 5th, 2014 AVACS Meeting, Oldenburg, Germany Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 1/13

ω -regular Specifications ω -regular expressions Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance ( Q , Σ , q 0 , ∆ , F ) with F ⊆ Q and q 0 q 1 q 2 · · · accepting ⇔ Inf ( q 0 q 1 q 2 · · · ) ∩ F � = ∅ Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance ( Q , Σ , q 0 , δ, F ) with F ⊆ 2 Q and q 0 q 1 q 2 · · · accepting ⇔ Inf ( q 0 q 1 q 2 · · · ) ∈ F Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance Many other acceptance conditions: parity, Rabin, Streett, .. Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance Many other acceptance conditions: parity, Rabin, Streett, .. Generality: Every acceptance condition that only depends on the states visited infinitely often is a Muller condition. Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance Many other acceptance conditions: parity, Rabin, Streett, .. Generality: Every acceptance condition that only depends on the states visited infinitely often is a Muller condition. Non-deterministic automata with safety acceptance Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance Many other acceptance conditions: parity, Rabin, Streett, .. Generality: Every acceptance condition that only depends on the states visited infinitely often is a Muller condition. Non-deterministic automata with safety acceptance ( Q , Σ , q 0 , ∆ , F ) with F ⊆ Q and q 0 q 1 q 2 · · · accepting ⇔ Occ ( q 0 q 1 q 2 · · · ) ⊆ F Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance Many other acceptance conditions: parity, Rabin, Streett, .. Generality: Every acceptance condition that only depends on the states visited infinitely often is a Muller condition. Non-deterministic automata with safety acceptance Weaker: not every ω -regular language is a safety condition. Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

ω -regular Specifications ω -regular expressions Monadic second-order logic with one successor Non-deterministic automata with B¨ uchi acceptance Deterministic automata with Muller acceptance Many other acceptance conditions: parity, Rabin, Streett, .. Generality: Every acceptance condition that only depends on the states visited infinitely often is a Muller condition. Non-deterministic automata with safety acceptance Weaker: not every ω -regular language is a safety condition. Is it nevertheless possible to turn every Muller condition into an equivalent safety condition? (under which equivalence?) Upside: simpler algorithms for safety conditions Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 2/13

Muller Games We study this question in a more general setting: infinite games. Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 3/13

Muller Games We study this question in a more general setting: infinite games. Running example 0 1 2 Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 3/13

Muller Games We study this question in a more general setting: infinite games. Running example F 0 = {{ 0 , 1 , 2 } , { 0 } , { 2 }} 0 1 2 F 1 = {{ 0 , 1 } , { 1 , 2 }} Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 3/13

Muller Games We study this question in a more general setting: infinite games. Running example F 0 = {{ 0 , 1 , 2 } , { 0 } , { 2 }} 0 1 2 F 1 = {{ 0 , 1 } , { 1 , 2 }} Formally: Muller game ( A , F 0 , F 1 ) with Arena A = ( V , V 0 , V 1 , E , v ) and partition ( F 0 , F 1 ) of 2 V . Player i wins play ρ iff Inf ( ρ ) ∈ F i . Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 3/13

Muller Games We study this question in a more general setting: infinite games. Running example F 0 = {{ 0 , 1 , 2 } , { 0 } , { 2 }} 0 1 2 F 1 = {{ 0 , 1 } , { 1 , 2 }} Formally: Muller game ( A , F 0 , F 1 ) with Arena A = ( V , V 0 , V 1 , E , v ) and partition ( F 0 , F 1 ) of 2 V . Player i wins play ρ iff Inf ( ρ ) ∈ F i . Emptiness of (non-deterministic) Muller automata and universality of deterministic Muller automata are one-player Muller games Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 3/13

Muller Games We study this question in a more general setting: infinite games. Running example F 0 = {{ 0 , 1 , 2 } , { 0 } , { 2 }} 0 1 2 F 1 = {{ 0 , 1 } , { 1 , 2 }} Formally: Muller game ( A , F 0 , F 1 ) with Arena A = ( V , V 0 , V 1 , E , v ) and partition ( F 0 , F 1 ) of 2 V . Player i wins play ρ iff Inf ( ρ ) ∈ F i . Emptiness of (non-deterministic) Muller automata and universality of deterministic Muller automata are one-player Muller games Our goal: give winner-preserving reduction from Muller to safety games. Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 3/13

Playing Muller Games in Finite Time Robert McNaughton: We believe that infinite games might have an interest for casual living-room recreation. But there is a problem: it takes a long time to play an infinite game! Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 4/13

Playing Muller Games in Finite Time Robert McNaughton: We believe that infinite games might have an interest for casual living-room recreation. But there is a problem: it takes a long time to play an infinite game! Thus: Scoring functions for Muller games. Use threshold score to obtain finite-duration variant. If threshold is large enough, obtain finite game with the same winning regions as infinite game. Question How large has the threshold to guarantee same winner? Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 4/13

Scores and Accumulators For F ⊆ V define Sc F : V + → N and Acc F : V + → 2 F . Intuition: Sc F ( w ): maximal k ∈ N such that F is visited k times since last vertex in V \ F (reset). Acc F ( w ): set A ⊂ F of vertices (from F ) seen since last increase or reset of Sc F . Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 5/13

Scores and Accumulators For F ⊆ V define Sc F : V + → N and Acc F : V + → 2 F . Intuition: Sc F ( w ): maximal k ∈ N such that F is visited k times since last vertex in V \ F (reset). Acc F ( w ): set A ⊂ F of vertices (from F ) seen since last increase or reset of Sc F . 0 0 1 1 0 0 1 2 w Sc { 0 } Acc { 0 } Sc { 0 , 1 } Acc { 0 , 1 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 5/13

Scores and Accumulators For F ⊆ V define Sc F : V + → N and Acc F : V + → 2 F . Intuition: Sc F ( w ): maximal k ∈ N such that F is visited k times since last vertex in V \ F (reset). Acc F ( w ): set A ⊂ F of vertices (from F ) seen since last increase or reset of Sc F . 0 0 1 1 0 0 1 2 w 1 Sc { 0 } ∅ Acc { 0 } Sc { 0 , 1 } Acc { 0 , 1 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Reducing ω -regular Specifications to Safety Conditions 5/13