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Timed Logics .....

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  1. Logics & Preordersfrom logic to preorder – and backKim Guldstrand Larsen Paul Pettersson Mogens Nielsen BRICS@AalborgBRICS@Aarhus

  2. 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)

  3. Timed Modal Logic Kozen’83 Atomic Prop Action Modalities Boolean Connectives Recursion Variables

  4. 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

  5. Semantics formula state of timed automata timed asgn for formula clocks Semantics

  6. Derived Operators f holds between l and u Invariantly Timed Modal Mu-calculus is at least as expressive as TCTL Weak UNTIL Bounded UNTIL

  7. Symbolic Semantics formula location region over C and K Region-based Semantics THEOREM

  8. 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

  9. Timed Bimulation Wang’91, Cerans’92

  10. Timed Bisimulation Wang’91

  11. Timed Simulation

  12. Examples

  13. Towards Timed Bisimulation Algorithm Cerans’92 independent “product-construction”

  14. Towards Timed Bisimulation Algorithm Definition Theorem

  15. Timed Bisimulation Algorithm = Checking for TB-ness using Regions y 2 1 1 x AX,R0 AY,R3 AX,R1 a1 a2 AX,R2

  16. Characteristic Propertyfor finite state automata Larsen, Ingolfsdottir, Sifakis, 1987 Ingolfsdottir, Steffen, 1994 a1 m1 n mk ak

  17. Characteristic Propertyfor finite state automata Larsen, Ingolfsdottir, Sifakis, 1987 Ingolfsdottir, Steffen, 1994 a1 m1 n mk ak

  18. 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

  19. 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

  20. Timed Bisimulation as a formula

  21. 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

  22. Zone Semantics formula MC wrt Safety Logic is PSPACEcomplete location zone over C and K

  23. 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

  24. END