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Logics & Preorders from logic to preorder – and back Kim Guldstrand Larsen Paul Pettersson Mogens Nielsen BRICS@Aalborg BRICS@Aarhus. Timed Logics. Real-time temporal logic (RTTL, Ostroff and Wonham 85) Metric Temporal Logic (Koymans, 1990)

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## Timed Logics .....

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**Logics & Preordersfrom logic to preorder**– and backKim Guldstrand Larsen Paul Pettersson Mogens Nielsen BRICS@AalborgBRICS@Aarhus**Timed Logics .....**• Real-time temporal logic (RTTL, Ostroff and Wonham 85) • Metric Temporal Logic (Koymans, 1990) • Explicit Clock Temporal Logic (Harel, Lichtenstein, Pnueli, 1990) • Timed Propositional Logic (Alur, Henzinger, 1991) • Timed Computational Tree Logic (Alur, Dill, 1989) • Timed Modal Mu-Calculus(Larsen, Laroussinie, Weise, 1995) • Duration Calculus (Chaochen, Hoare, Ravn, 1991)**Timed Modal Logic**Kozen’83 Atomic Prop Action Modalities Boolean Connectives Recursion Variables**Timed Modal Logic**Larsen, Holmer, Wang’91 Larsen, Laroussine, Weise, 1995 Larsen, Pettersson, Wang, 1995 Delay Modalities Atomic Prop Action Modalities Boolean Connectives Formula Clock Constr Recursion Variables Formula Clock Reset**Semantics**formula state of timed automata timed asgn for formula clocks Semantics**Derived Operators**f holds between l and u Invariantly Timed Modal Mu-calculus is at least as expressive as TCTL Weak UNTIL Bounded UNTIL**Symbolic Semantics**formula location region over C and K Region-based Semantics THEOREM**Fundamental Results**Decidable EXPTIME-complete (Aceto,Laroussinie’99) Given f and automaton A does A satisfy f ? Given f does there exist an automaton A satisfying f ? UNDECIDABLE (strong conjecture) Given f and given clock-set C and max constant M. Does there exist an automaton A over C and M satisfying f ? Decidable**Timed Bimulation**Wang’91, Cerans’92**Timed Bisimulation**Wang’91**Towards Timed Bisimulation Algorithm**Cerans’92 independent “product-construction”**Towards Timed Bisimulation Algorithm**Definition Theorem**Timed Bisimulation Algorithm = Checking for TB-ness using**Regions y 2 1 1 x AX,R0 AY,R3 AX,R1 a1 a2 AX,R2**Characteristic Propertyfor finite state automata**Larsen, Ingolfsdottir, Sifakis, 1987 Ingolfsdottir, Steffen, 1994 a1 m1 n mk ak**Characteristic Propertyfor finite state automata**Larsen, Ingolfsdottir, Sifakis, 1987 Ingolfsdottir, Steffen, 1994 a1 m1 n mk ak**Characteristic Propertyfor timed automata**Larsen, Laroussinie, Weise, 1995 r1 a1 m1 g1 n Inv(n) gk mk ak rk IDEA_ Automata clocks become formula clocks**Characteristic Propertyfor timed automata**Larsen, Laroussinie, Weise, 1995 r1 a1 m1 g1 n Inv(n) gk mk ak rk IDEA_ Automata clocks become formula clocks**Timed Safety LogicBack to Zones**Larsen, Pettersson, Wang, 1995 Delay Modalities Atomic Prop Action Modalities Boolean Connectives Formula Clock Constr Recursion Variables Formula Clock Reset**Zone Semantics**formula MC wrt Safety Logic is PSPACEcomplete location zone over C and K**Characteristic Property/Simulationfor deterministic timed**automata Aceto, Burgueno,Bouyer, Larsen, 1998 r1 a m1 g1 n Inv(n) gk mk a rk gi and gj = Ø determinism

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