Modal Logics for Timed Control Patricia Bouyer 1 , Franck Cassez 2 and François Laroussinie 1 1 LSV, ENS-Cachan 2 IRCCyN, Nantes France CONCUR’05 San Francisco, CA
Control of Timed Systems Controllability with L ν Outline of the talk ◮ Control of Timed Systems ◮ Controllability with L ν CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 2 / 30
Control of Timed Systems Controllability with L ν Outline ◮ Control of Timed Systems ◮ Controllability with L ν CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 3 / 30
Control of Timed Systems Controllability with L ν Model Checking and Control Problems � (not bad) φ S CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30
Control of Timed Systems Controllability with L ν Model Checking and Control Problems � (not bad) | φ S = Model Checking Problem Does S satisfy φ ? CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30
Control of Timed Systems Controllability with L ν Model Checking and Control Problems � (not bad) c φ S Model Checking Problem Does S satisfy φ ? CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30
Control of Timed Systems Controllability with L ν Model Checking and Control Problems � (not bad) X φ S Model Checking Problem Does S satisfy φ ? Control Problem Can S be restricted to satisfy φ ? CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30
Control of Timed Systems Controllability with L ν Model Checking and Control Problems � c � (not bad) c � | φ S C = Model Checking Problem Does S satisfy φ ? Control Problem Can S be restricted to satisfy φ ? Is there a Controller C s.t. ( S � C ) | = φ ? CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30
Control of Timed Systems Controllability with L ν Model for Timed Systems: Timed Automata TA = Finite Automata + clocks Timed Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 5 / 30
Control of Timed Systems Controllability with L ν Model for Timed Systems: Timed Automata TA = Finite Automata + clocks Timed Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Semantics = runs = sequences of dense-time and discrete steps 1 . 1 c 1 2 . 1 c 2 − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) ρ : ( ℓ 0 , 0 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 5 / 30
Control of Timed Systems Controllability with L ν Model for Control: Timed Game Automata TGA = TA + controllable and uncontrollable actions Actions partitioned as Act c = { c 1 , c 2 , c 3 } Act u = { u } [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u ℓ 0 ℓ 1 x := 0 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Control Objective = subset of the runs of a TGA Safety objective “Avoid the Bad state” CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 6 / 30
Control of Timed Systems Controllability with L ν Solving Timed Games (1/2) [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A general controller is defined by a strategy f if ρ is a run from the initial state: f ( ρ ) = do a controllable action or do nothing CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30
Control of Timed Systems Controllability with L ν Solving Timed Games (1/2) [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A general controller is defined by a strategy f A Partial Strategy f f ( each run ending in ℓ 0 , x < 2 ) = do nothing f ( each run ending in ℓ 0 , x = 2 ) = c 1 CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30
Control of Timed Systems Controllability with L ν Solving Timed Games (1/2) [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A general controller is defined by a strategy f A strategy restricts the set of runs of the TGA CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30
Control of Timed Systems Controllability with L ν Solving Timed Games (1/2) [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A general controller is defined by a strategy f A strategy restricts the set of runs of the TGA ( G � f ) = G controlled by strategy f CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30
Control of Timed Systems Controllability with L ν Solving Timed Games (1/2) [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A general controller is defined by a strategy f A strategy restricts the set of runs of the TGA ( G � f ) = G controlled by strategy f Given φ a control objective, s a state, The strategy f is winning from s if s | = φ in ( G � f ) The state s is winning if there is a winning strategy f s from s CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30
Control of Timed Systems Controllability with L ν Solving Timed Games (2/2) Input: a TGA G and a control objective φ Problem: is there a strategy f s.t. ( G � f ) | = φ ? Solution: compute the set of winning states define a controllable predecessors operator 1 compute a fixed point that gives the set of winning states 2 check whether the initial state is winning 3 Fundamental Results for Timed Control [Maler et al., 95, De Alfaro et al., 01] Control Problem is EXPTIME-Complete for TA and reachability objectives Controller Synthesis is effective Memoryless strategies are sufficient to win CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 8 / 30
Control of Timed Systems Controllability with L ν Our Contribution Control objective in L ν (safety and bounded liveness) CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30
Control of Timed Systems Controllability with L ν Our Contribution Control objective in L ν (safety and bounded liveness) Reduction of the Control Problem for ( TA , L ν ) to a Model-Checking Problem for ( TA , L c ν ) : there is a strategy f s.t. ( G � f ) | = φ ⇐ ⇒ G | = φ CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30
Control of Timed Systems Controllability with L ν Our Contribution Control objective in L ν (safety and bounded liveness) Reduction of the Control Problem for ( TA , L ν ) to a Model-Checking Problem for ( TA , L c ν ) : there is a strategy f s.t. ( G � f ) | = φ ⇐ ⇒ G | = φ Properties of the new logic L c ν Expressiveness Model Checking over TA Compositionality CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30
Control of Timed Systems Controllability with L ν Our Contribution Control objective in L ν (safety and bounded liveness) Reduction of the Control Problem for ( TA , L ν ) to a Model-Checking Problem for ( TA , L c ν ) : there is a strategy f s.t. ( G � f ) | = φ ⇐ ⇒ G | = φ Properties of the new logic L c ν Expressiveness Model Checking over TA Compositionality Implementation The tool CMC [Laroussinie et al., 98] CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30
Control of Timed Systems Controllability with L ν Outline ◮ Control of Timed Systems ◮ Controllability with L ν CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 10 / 30
Control of Timed Systems Controllability with L ν The Timed Modal Logic L ν Atomic propositions + and, or CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30
Control of Timed Systems Controllability with L ν The Timed Modal Logic L ν Atomic propositions + and, or Discrete step properties: � a � ϕ , [ a ] ϕ , a an action CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30
Control of Timed Systems Controllability with L ν The Timed Modal Logic L ν Atomic propositions + and, or Discrete step properties: � a � ϕ , [ a ] ϕ , a an action Time step properties: � δ � ϕ , [ δ ] ϕ CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30
Control of Timed Systems Controllability with L ν The Timed Modal Logic L ν Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: � a � ϕ , [ a ] ϕ , a an action Time step properties: � δ � ϕ , [ δ ] ϕ Time guarded properties: x in ϕ with x a formula clock CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30
Control of Timed Systems Controllability with L ν The Timed Modal Logic L ν Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: � a � ϕ , [ a ] ϕ , a an action Time step properties: � δ � ϕ , [ δ ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z = ν ϕ ( Z ) CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30
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