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Comparison of the Expressiveness of Timed Automata and Time Petri Nets B. Brard 1 , F. Cassez 2 , S. Haddad 1 , D. Lime 3 , O.H. Roux 2 1 LAMSADE, Paris 2 IRCCyN, Nantes 3 CISS, Aalborg France France Denmark FORMATS05 Uppsala, Sweden

  1. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Fundamental Problems for Timed Automata We consider (Bounded) TPN introduced by [Merlin’74] Problem Timed Automata B-Time Petri Nets Reachability (RP) PSPACE-Complete Decidable Emptyness (EP) [Alur & Dill’94] [Berthomieu & Diaz’91] Universality (UP) Undecidable ?? Language Inclusion [Alur & Dill’94] Closed under ∩ , ∪ Closure Properties but not under compl. ?? [Alur & Dill’94] ε -TA > TA Effect of ε -transition ?? [Bérard et al.’96] Decidable TCTL model checking [Alur & Dill’94] FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

  2. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Fundamental Problems for Timed Automata We consider (Bounded) TPN introduced by [Merlin’74] Problem Timed Automata B-Time Petri Nets Reachability (RP) PSPACE-Complete Decidable Emptyness (EP) [Alur & Dill’94] [Berthomieu & Diaz’91] Universality (UP) Undecidable ?? Language Inclusion [Alur & Dill’94] Closed under ∩ , ∪ Closure Properties but not under compl. ?? [Alur & Dill’94] ε -TA > TA Effect of ε -transition ?? [Bérard et al.’96] Decidable Decidable TCTL model checking [Alur & Dill’94] [C. & R., AVoCS’04] FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

  3. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Motivation For Studying the Problems for TPNs Universality Problem checking a TPN against a spec. given by a TPN L ( A ) ⊆ L ( B ) if undecidable then Language Inclusion Problem undecidable FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 6 / 30

  4. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Motivation For Studying the Problems for TPNs Universality Problem checking a TPN against a spec. given by a TPN L ( A ) ⊆ L ( B ) if undecidable then Language Inclusion Problem undecidable Closure Properties TA not closed under complement find subclasses of TA enjoying nice closure properties TPN is such a class ? FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 6 / 30

  5. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Motivation For Studying the Problems for TPNs Universality Problem checking a TPN against a spec. given by a TPN L ( A ) ⊆ L ( B ) if undecidable then Language Inclusion Problem undecidable Closure Properties TA not closed under complement find subclasses of TA enjoying nice closure properties TPN is such a class ? Time Petri Nets Properties e.g. every Bounded-PN is equivalent to a one-safe PN What about TPN ? FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 6 / 30

  6. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Motivation For Studying the Problems for TPNs Universality Problem checking a TPN against a spec. given by a TPN L ( A ) ⊆ L ( B ) if undecidable then Language Inclusion Problem undecidable Closure Properties TA not closed under complement find subclasses of TA enjoying nice closure properties TPN is such a class ? Time Petri Nets Properties e.g. every Bounded-PN is equivalent to a one-safe PN What about TPN ? TA or TPN as a specification language ? precise comparison of expressive power FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 6 / 30

  7. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Our Contribution Extended version of (bounded) TPNs: TPN ε open intervals, final and repeated markings, ε -transitions FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

  8. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Our Contribution Extended version of (bounded) TPNs: TPN ε open intervals, final and repeated markings, ε -transitions TA are strictly more expressive than TPN ε w.r.t. weak timed bisimilarity FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

  9. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Our Contribution Extended version of (bounded) TPNs: TPN ε open intervals, final and repeated markings, ε -transitions TA are strictly more expressive than TPN ε w.r.t. weak timed bisimilarity TA ε and one-safe TPN ε are equally expressive w.r.t. timed language acceptance FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

  10. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Our Contribution Extended version of (bounded) TPNs: TPN ε open intervals, final and repeated markings, ε -transitions TA are strictly more expressive than TPN ε w.r.t. weak timed bisimilarity TA ε and one-safe TPN ε are equally expressive w.r.t. timed language acceptance Universal Problem is undecidable for (bounded and) one-safe TPN ε (Language Inclusion is undecidable) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

  11. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Our Contribution Extended version of (bounded) TPNs: TPN ε open intervals, final and repeated markings, ε -transitions TA are strictly more expressive than TPN ε w.r.t. weak timed bisimilarity TA ε and one-safe TPN ε are equally expressive w.r.t. timed language acceptance Universal Problem is undecidable for (bounded and) one-safe TPN ε (Language Inclusion is undecidable) Bounded TPN ε and one-safe TPN ε are equally expressive (w.r.t. timed language acceptance) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

  12. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Our Contribution Extended version of (bounded) TPNs: TPN ε open intervals, final and repeated markings, ε -transitions TA are strictly more expressive than TPN ε w.r.t. weak timed bisimilarity TA ε and one-safe TPN ε are equally expressive w.r.t. timed language acceptance Universal Problem is undecidable for (bounded and) one-safe TPN ε (Language Inclusion is undecidable) Bounded TPN ε and one-safe TPN ε are equally expressive (w.r.t. timed language acceptance) Timed Bisimilarity: B- TPN ε ( ≤ , ≥ ) (original class defined by Merlin) and TA ε ( ≤ , ≥ ) are equivalent w.r.t. timed bisimilarity FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

  13. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Outline Context & Motivation ◮ Timed Automata & Time Petri Nets ◮ Expressiveness wrt Timed Bisimilarity ◮ Expressiveness wrt Timed Language Acceptance ◮ Conclusion ◮ FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 8 / 30

  14. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata [ x < 1 ] a ; x < 1 ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  15. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  16. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 0 . 78 a run 2: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → b ; x ≤ 2 x := 0 50 ( ℓ 1 , 0 . 78 ) − − → ( ℓ 1 , 50 . 78 ) · · · FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  17. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 0 . 78 a run 2: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → b ; x ≤ 2 x := 0 50 ( ℓ 1 , 0 . 78 ) − − → ( ℓ 1 , 50 . 78 ) · · · Definition (Semantics of TA) States: ( ℓ, v ) ∈ Q = L × ( R ≥ 0 ) X FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  18. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 0 . 78 a run 2: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → b ; x ≤ 2 x := 0 50 ( ℓ 1 , 0 . 78 ) − − → ( ℓ 1 , 50 . 78 ) · · · Definition (Semantics of TA) States: ( ℓ, v ) ∈ Q = L × ( R ≥ 0 ) X a → ( l ′ , v ′ ) Discrete transition: ( l , v ) − iff there is a transition ( l , g , a , R , l ′ ) in A s.t.  the guard is true in ( l , v )   v ′ is v with the clocks in R equal to zero The invariant of l ′ holds for v ′   FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  19. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 0 . 78 a run 2: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → b ; x ≤ 2 x := 0 50 ( ℓ 1 , 0 . 78 ) − − → ( ℓ 1 , 50 . 78 ) · · · Definition (Semantics of TA) States: ( ℓ, v ) ∈ Q = L × ( R ≥ 0 ) X a → ( l ′ , v ′ ) Discrete transition: ( l , v ) − t → ( l ′ , v ′ ) Time transition: ( l , v ) − l = l ′ and v ′ = v + t: the clocks are updated � iff The invariant of l holds for all v + t ′ , t ′ ≤ t FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  20. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 0 . 78 a run 2: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → b ; x ≤ 2 x := 0 50 ( ℓ 1 , 0 . 78 ) − − → ( ℓ 1 , 50 . 78 ) · · · Definition (Semantics of TA) States: ( ℓ, v ) ∈ Q = L × ( R ≥ 0 ) X a → ( l ′ , v ′ ) Discrete transition: ( l , v ) − t → ( l ′ , v ′ ) Time transition: ( l , v ) − a TA generates a set of runs = alternating discrete and time steps FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  21. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Timed Automata Timed Automata Timed Automata 0 . 78 a run 1: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → [ x < 1 ] a ; x < 1 1 . 22 b ( ℓ 1 , 0 . 78 ) − − − → ( ℓ 1 , 2 ) − → ( ℓ 0 , 0 ) · · · ℓ 0 ℓ 1 0 . 78 a run 2: ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 0 . 78 ) − → b ; x ≤ 2 x := 0 50 ( ℓ 1 , 0 . 78 ) − − → ( ℓ 1 , 50 . 78 ) · · · Definition (Semantics of TA) States: ( ℓ, v ) ∈ Q = L × ( R ≥ 0 ) X a → ( l ′ , v ′ ) Discrete transition: ( l , v ) − t → ( l ′ , v ′ ) Time transition: ( l , v ) − a TA generates a set of runs = alternating discrete and time steps semantics of a TA A is a Timed Transition System S A FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

  22. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets t 0 : a ; [ 0 , 1 [ P 0 P 1 t 1 : b ; [ 0 , 2 ] FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  23. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · t 1 : b ; [ 0 , 2 ] FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  24. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · 0 . 78 a run 2: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → 2 t 1 : b ; [ 0 , 2 ] ( P 1 , 0 ) − → ( P 0 , 0 ) · · · FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  25. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · 0 . 78 a run 2: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → 2 t 1 : b ; [ 0 , 2 ] ( P 1 , 0 ) − → ( P 0 , 0 ) · · · Definition (Semantics of Time Petri Nets ) States: ( M , ν ) with M a marking and ν : Enabled ( M ) �→ R ≥ 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  26. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · 0 . 78 a run 2: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → 2 t 1 : b ; [ 0 , 2 ] ( P 1 , 0 ) − → ( P 0 , 0 ) · · · Definition (Semantics of Time Petri Nets ) States: ( M , ν ) with M a marking and ν : Enabled ( M ) �→ R ≥ 0 a → ( M ′ , ν ′ ) iff ∃ t : a , I in the s.t. Discrete transition: ( M , ν ) − t is enabled in M and M ′ = M − • t + t •    ν ( t ) is in I (the interval associated with t) ν ′ ( t ′ ) = 0 if t ′ is enabled when firing t, ν ′ ( t ′ ) = ν ( t ′ ) otherwise   FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  27. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · 0 . 78 a run 2: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → 2 t 1 : b ; [ 0 , 2 ] ( P 1 , 0 ) − → ( P 0 , 0 ) · · · Definition (Semantics of Time Petri Nets ) States: ( M , ν ) with M a marking and ν : Enabled ( M ) �→ R ≥ 0 a → ( M ′ , ν ′ ) Discrete transition: ( M , ν ) − d → ( M ′ , ν ′ ) iff Time transition: ( M , ν ) − M = M ′ and ν ′ = ν + d (clocks of enabled transitions updated � For all enabled t, for all d ′ ≤ d, ν ( t ) ∈ I ( t ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  28. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · 0 . 78 a run 2: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → 2 t 1 : b ; [ 0 , 2 ] ( P 1 , 0 ) − → ( P 0 , 0 ) · · · Definition (Semantics of Time Petri Nets ) States: ( M , ν ) with M a marking and ν : Enabled ( M ) �→ R ≥ 0 a → ( M ′ , ν ′ ) Discrete transition: ( M , ν ) − d → ( M ′ , ν ′ ) Time transition: ( M , ν ) − A TPN generates a set of runs = alternating discrete and time steps FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  29. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Semantics of Time Petri Nets Time Petri Net Time Petri Nets 0 . 78 a t 0 : a ; [ 0 , 1 [ run 1: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → P 0 P 1 1 . 5 b ( P 1 , 0 . 78 ) − − → ( P 1 , 1 . 5 ) − → ( P 0 , 0 ) · · · 0 . 78 a run 2: ( P 0 , 0 ) − − − → ( P 0 , 0 . 78 ) − → 2 t 1 : b ; [ 0 , 2 ] ( P 1 , 0 ) − → ( P 0 , 0 ) · · · Definition (Semantics of Time Petri Nets ) States: ( M , ν ) with M a marking and ν : Enabled ( M ) �→ R ≥ 0 a → ( M ′ , ν ′ ) Discrete transition: ( M , ν ) − d → ( M ′ , ν ′ ) Time transition: ( M , ν ) − A TPN generates a set of runs = alternating discrete and time steps Semantics of a TPN N = Timed Transition System S N FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

  30. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Outline Context & Motivation ◮ Timed Automata & Time Petri Nets ◮ Expressiveness wrt Timed Bisimilarity ◮ Expressiveness wrt Timed Language Acceptance ◮ Conclusion ◮ FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 11 / 30

  31. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Timed Bisimilarity Definition (Weak Timed Bisimilarity) Two TTS A and B are timed bisimilar if there is an equivalence relation ≡ on the states of S A and S B s.t.: s A 0 ≡ s B 0 and FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

  32. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Timed Bisimilarity Definition (Weak Timed Bisimilarity) Two TTS A and B are timed bisimilar if there is an equivalence relation ≡ on the states of S A and S B s.t.: s A 0 ≡ s B 0 and 1 for each state s of S A there is a state q of S B s.t. s ≡ q and → s ′ then q σ σ → q ′ and s ′ ≡ q ′ and s ′ if s − − 2 for each state q of S B . . . FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

  33. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Timed Bisimilarity Definition (Weak Timed Bisimilarity) Two TTS A and B are timed bisimilar if there is an equivalence relation ≡ on the states of S A and S B s.t.: s A 0 ≡ s B 0 and 1 for each state s of S A there is a state q of S B s.t. s ≡ q and → s ′ then q σ σ → q ′ and s ′ ≡ q ′ and s ′ if s − − 2 for each state q of S B . . . Weakly Timed Bisimilar: allows ε -moves FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

  34. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Timed Bisimilarity Definition (Weak Timed Bisimilarity) Two TTS A and B are weakly timed bisimilar if there is an equivalence relation ≡ on the states of S A and S B s.t.: s A 0 ≡ s B 0 and 1 for each state s of S A there is a state q of S B s.t. s ≡ q and a ⇒ s ′ then q a ⇒ q ′ and s ′ ≡ q ′ and s ′ if s = = 2 for each state q of S B . . . Weakly Timed Bisimilar: allows ε -moves ε ∗ → ε ∗ ⇒ s ′ if s a → a → s ′ discrete step: s = − − − − − → s ′ and � δ i = δ → ε + ε + ε ∗ → ε ∗ δ ⇒ s ′ if s → δ 1 → δ n − − − − − − → · · · − − − − − − time step: s = FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

  35. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Timed Bisimilarity Definition (Weak Timed Bisimilarity) Two TTS A and B are weakly timed bisimilar if there is an equivalence relation ≡ on the states of S A and S B s.t.: s A 0 ≡ s B 0 and 1 for each state s of S A there is a state q of S B s.t. s ≡ q and ⇒ s ′ then q a a ⇒ q ′ and s ′ ≡ q ′ and s ′ if s = = 2 for each state q of S B . . . Weakly Timed Bisimilar: allows ε -moves Theorem ([C. & R., AVoCS’04]) Each bounded TPN (TPN ε ) is timed bisimilar to a TA (TA ε ). FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

  36. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Timed Bisimilarity Definition (Weak Timed Bisimilarity) Two TTS A and B are weakly timed bisimilar if there is an equivalence relation ≡ on the states of S A and S B s.t.: s A 0 ≡ s B 0 and 1 for each state s of S A there is a state q of S B s.t. s ≡ q and ⇒ s ′ then q a a ⇒ q ′ and s ′ ≡ q ′ and s ′ if s = = 2 for each state q of S B . . . Weakly Timed Bisimilar: allows ε -moves Theorem ([C. & R., AVoCS’04]) Each bounded TPN (TPN ε ) is timed bisimilar to a TA (TA ε ). Converse: Each TA is timed bisimilar to a TPN ? FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

  37. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) a ; x < 1 Let A 0 = ℓ 0 ℓ 1 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  38. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  39. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  40. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  41. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  42. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  43. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  44. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  45. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ε ∗ ( M ′ 0 , ν ′ 0 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  46. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M 1 , ν 1 ) ε ∗ δ 1 ( M ′ 0 , ν ′ 0 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  47. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M 1 , ν 1 ) ε ∗ ε + δ 1 ( M ′ 0 , ν ′ ( M ′ 1 , ν ′ 0 ) 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  48. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M 1 , ν 1 ) ( M n , ν n ) ε ∗ ε + • • • ε ∗ δ 1 δ n ( M ′ 0 , ν ′ ( M ′ 1 , ν ′ ( M ′ n − 1 , ν ′ ( M ′ n , ν ′ 0 ) 1 ) n − 1 ) n ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  49. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 ( ℓ 0 , x = 0 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M 1 , ν 1 ) ( M n , ν n ) ε ∗ ε + • • • δ 1 δ n ( M ′ 0 , ν ′ ( M ′ 1 , ν ′ ( M ′ n − 1 , ν ′ 0 ) 1 ) n − 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  50. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 − δ n δ = δ n ( ℓ 0 , x = 0 ) ( ℓ 0 , x < 1 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M n , ν n ) ( δ = δ n ) + ε ∗ δ n ( M ′ n − 1 , ν ′ n − 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  51. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 − δ n δ = δ n ( ℓ 0 , x = 0 ) ( ℓ 0 , x < 1 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M n , ν n ) ( δ = δ n ) + ε ∗ a δ n ( M ′ n − 1 , ν ′ n − 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  52. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 − δ n δ = δ n ( ℓ 0 , x = 0 ) ( ℓ 0 , x < 1 ) ( ℓ 0 , x = 1 ) ( M 0 , 0 ) ( M n , ν n ) ( δ = δ n ) + ε ∗ a δ n ε ∗ a ( M ′ n − 1 , ν ′ n − 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  53. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 − δ n δ = δ n ( ℓ 0 , x = 0 ) ( ℓ 0 , x < 1 ) ( ℓ 0 , x = 1 ) ε ∗ a ( M 0 , 0 ) ( M n , ν n ) ( δ = δ n ) + ε ∗ a δ n ε ∗ a ( M ′ n − 1 , ν ′ n − 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  54. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Theorem (TA are strictly more expressive than TPNs) There is no TPN weakly timed a ; x < 1 Let A 0 = ℓ 0 ℓ 1 bisimilar to A 0 . Proof. 1 Time elapsing cannot disable transitions in a TPN t 1 t 2 ··· t k t 1 t 2 ··· t k δ → ( M ′ , ν ′ ) and ( M , ν ) → ( M ′′ , ν ′′ ) ( M , ν ) − − − − − − − − − − − → 2 Assume there is a TPN N weakly timed bisimilar to A 0 δ = 1 − δ n δ = δ n a X ( ℓ 0 , x = 0 ) ( ℓ 0 , x < 1 ) ( ℓ 0 , x = 1 ) ε ∗ a ( M 0 , 0 ) ( M n , ν n ) ( δ = δ n ) + ε ∗ a δ n ε ∗ a ( M ′ n − 1 , ν ′ n − 1 ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

  55. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Outline Context & Motivation ◮ Timed Automata & Time Petri Nets ◮ Expressiveness wrt Timed Bisimilarity ◮ Expressiveness wrt Timed Language Acceptance ◮ Conclusion ◮ FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 14 / 30

  56. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Language Equivalence Definition (Timed Word & Timed Language) A timed word over Σ is a sequence w = ( a 0 , δ 0 )( a 1 , δ 1 ) · · · ( a n , δ n ) · · · with a i ∈ Σ ; δ i ∈ R ≥ 0 . A timed language is a set of timed words. A TTS A with final and repeated states accepts a timed language L ( A ) . FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 15 / 30

  57. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Language Equivalence Definition (Timed Word & Timed Language) A timed word over Σ is a sequence w = ( a 0 , δ 0 )( a 1 , δ 1 ) · · · ( a n , δ n ) · · · with a i ∈ Σ ; δ i ∈ R ≥ 0 . A timed language is a set of timed words. A TTS A with final and repeated states accepts a timed language L ( A ) . Definition (Language Equivalence) Two TTS A and B are equivalent w.r.t. Timed Language Acceptance if L ( A ) = L ( B ) . FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 15 / 30

  58. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Language Equivalence Definition (Timed Word & Timed Language) A timed word over Σ is a sequence w = ( a 0 , δ 0 )( a 1 , δ 1 ) · · · ( a n , δ n ) · · · with a i ∈ Σ ; δ i ∈ R ≥ 0 . A timed language is a set of timed words. A TTS A with final and repeated states accepts a timed language L ( A ) . Definition (Language Equivalence) Two TTS A and B are equivalent w.r.t. Timed Language Acceptance if L ( A ) = L ( B ) . Theorem ([C. & R., AVoCS’04]) A timed language accepted by a bounded TPN is accepted by a TA. FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 15 / 30

  59. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Language Equivalence Definition (Timed Word & Timed Language) A timed word over Σ is a sequence w = ( a 0 , δ 0 )( a 1 , δ 1 ) · · · ( a n , δ n ) · · · with a i ∈ Σ ; δ i ∈ R ≥ 0 . A timed language is a set of timed words. A TTS A with final and repeated states accepts a timed language L ( A ) . Definition (Language Equivalence) Two TTS A and B are equivalent w.r.t. Timed Language Acceptance if L ( A ) = L ( B ) . Theorem ([C. & R., AVoCS’04]) A timed language accepted by a bounded TPN is accepted by a TA. Each language accepted by a TA is accepted by a TPN ? FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 15 / 30

  60. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a ; x < 1 ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  61. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a ; x < 1 ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  62. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN • P ℓ 1 P ℓ 0 a ; x < 1 ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  63. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • P ℓ 1 P ℓ 0 a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  64. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • P ℓ 1 P ℓ 0 tt a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  65. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • ν ( t ) = x • t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  66. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  67. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  68. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  69. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  70. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt [ 0 ] a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  71. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a • tt • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt [ 0 ] a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  72. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a u : [ 0 , 1 [ ν ( u ) = x • tt • • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt [ 0 ] a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  73. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a u : [ 0 , 1 [ [ 0 ] ν ( u ) = x • tt • • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt [ 0 ] a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  74. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a u : [ 0 , 1 [ [ 0 ] ν ( u ) = x • tt • • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt [ 0 ] a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  75. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Encoding a TA into a Bounded TPN a u : [ 0 , 1 [ [ 0 ] ν ( u ) = x • tt • • ν ( t ) = x • [ 0 ] t : [ 2 , 2 ] P ℓ 1 P ℓ 0 tt [ 0 ] a ; x < 1 b ℓ 0 ℓ 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

  76. Context TA & TPN Timed Bisimilarity Timed Language Conclusion TA and TPN are equally Expressive Theorem One-safe TPN ε and TA ε are equally expressive w.r.t. Timed Language Acceptance. FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 17 / 30

  77. Context TA & TPN Timed Bisimilarity Timed Language Conclusion TA and TPN are equally Expressive Theorem One-safe TPN ε and TA ε are equally expressive w.r.t. Timed Language Acceptance. Sketch. A a TA; N the TPN as described previously. To prove L ( A ) = L ( N ) we prove FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 17 / 30

  78. Context TA & TPN Timed Bisimilarity Timed Language Conclusion TA and TPN are equally Expressive Theorem One-safe TPN ε and TA ε are equally expressive w.r.t. Timed Language Acceptance. Sketch. A a TA; N the TPN as described previously. To prove L ( A ) = L ( N ) we prove 1 N simulates A which entails L ( A ) ⊆ L ( N ) Define a proper simulation relation FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 17 / 30

  79. Context TA & TPN Timed Bisimilarity Timed Language Conclusion TA and TPN are equally Expressive Theorem One-safe TPN ε and TA ε are equally expressive w.r.t. Timed Language Acceptance. Sketch. A a TA; N the TPN as described previously. To prove L ( A ) = L ( N ) we prove 1 N simulates A which entails L ( A ) ⊆ L ( N ) Define a proper simulation relation 2 for L ( N ) ⊆ L ( A ) : Design A ′ s.t. L ( A ) = L ( A ′ ) and prove that A ′ simulates N FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 17 / 30

  80. Context TA & TPN Timed Bisimilarity Timed Language Conclusion Back to Timed Bisimilarity Definition (The class TA ε ( ≤ , ≥ ) ) FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

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