1. 1 Logics & Preorders from logic to preorder – and back Kim Guldstrand Larsen Paul PetterssonMogens Nielsen BRICS@Aalborg BRICS@Aarhus

2. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 2 Timed Logics..... zReal-time temporal logic (RTTL, Ostroff and Wonham 85) zMetric Temporal Logic (Koymans, 1990) zExplicit Clock Temporal Logic (Harel, Lichtenstein, Pnueli, 1990) zTimed Propositional Logic (Alur, Henzinger, 1991) zTimed Computational Tree Logic (Alur, Dill, 1989) zTimed Modal Mu-Calculus (Larsen, Laroussinie, Weise, 1995) zDuration Calculus (Chaochen, Hoare, Ravn, 1991)

3. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 3 Timed Modal Logic Atomic Prop Recursion Variables Action Modalities Boolean Connectives Kozen’83

4. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 4 Timed Modal Logic Atomic Prop Recursion Variables Action Modalities Boolean Connectives Formula Clock Constr Formula Clock Reset Delay Modalities Larsen, Laroussine, Weise, 1995 Larsen, Pettersson, Wang, 1995 Larsen, Holmer, Wang’91

5. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 5 Semantics state of timed automata timed asgn for formula clocks formula Semantics

6. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 6 Derived Operators  holds between l and u Invariantly Weak UNTIL Bounded UNTIL Timed Modal Mu-calculus is at least as expressive as TCTL

7. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 7 Symbolic Semantics locationregion over C and K formula Region-based Semantics THEOREM

8. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 8 Fundamental Results Given  does there exist an automaton A satisfying  ? Given  and given clock-set C and max constant M. Does there exist an automaton A over C and M satisfying  ? UNDECIDABLE (strong conjecture) Decidable Given  and automaton A does A satisfy  ? Decidable EXPTIME-complete (Aceto,Laroussinie’99)

9. 9 Timed Bimulation Wang’91, Cerans’92

10. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 10 Timed Bisimulation Wang’91

11. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 11 Timed Simulation

12. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 12 Examples

13. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 13 Towards Timed Bisimulation Algorithm independent “product-construction” Cerans’92

14. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 14 Definition Theorem Towards Timed Bisimulation Algorithm

15. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 15 Timed Bisimulation Algorithm = Checking for TB-ness using Regions x y AX,R 0 AX,R 1 AX,R 2 AY,R 3 a2 a1 1 1 2

16. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 16 Characteristic Property for finite state automata a1a1 akak n m1m1 mkmk Larsen, Ingolfsdottir, Sifakis, 1987 Ingolfsdottir, Steffen, 1994

17. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 17 Characteristic Property for finite state automata a1a1 akak n m1m1 mkmk Larsen, Ingolfsdottir, Sifakis, 1987 Ingolfsdottir, Steffen, 1994

18. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 18 Characteristic Property for timed automata a1a1 akak n m1m1 mkmk g1g1 r1r1 gkgk rkrk Inv(n) IDEA_ Automata clocks become formula clocks Larsen, Laroussinie, Weise, 1995

19. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 19 Characteristic Property for timed automata a1a1 akak n m1m1 mkmk g1g1 r1r1 gkgk rkrk Inv(n) IDEA_ Automata clocks become formula clocks Larsen, Laroussinie, Weise, 1995

20. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 20 Timed Bisimulation as a formula

21. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 21 Timed Safety Logic Back to Zones Atomic Prop Recursion Variables Action Modalities Boolean Connectives Formula Clock Constr Formula Clock Reset Delay Modalities Larsen, Pettersson, Wang, 1995

22. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 22 Zone Semantics location zone over C and K formula MC wrt Safety Logic is PSPACE complete

23. UCb Petri Net, June 2000Kim G. Larsen, Mogens Nielsen, Paul Pettersson 23 Characteristic Property/Simulation for deterministic timed automata a a n m1m1 mkmk g1g1 r1r1 gkgk rkrk Inv(n) Aceto, Burgueno,Bouyer, Larsen, 1998 g i and g j = Ø determinism

24. 24 END