Comparison of the Expressiveness of Timed Automata and Time Petri - - PowerPoint PPT Presentation

Comparison of the Expressiveness of Timed Automata and Time Petri Nets

• B. Bérard1, F. Cassez2, S. Haddad1, D. Lime3, O.H. Roux2

1LAMSADE, Paris

France

2IRCCyN, Nantes

France

3CISS, Aalborg

Denmark FORMATS’05 Uppsala, Sweden

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Outline of the talk

◮

Context & Motivation

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 2 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Outline of the talk

◮

Context & Motivation

◮

Timed Automata & Time Petri Nets

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 2 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Outline of the talk

◮

Context & Motivation

◮

Timed Automata & Time Petri Nets

◮

Expressiveness wrt Timed Bisimilarity

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 2 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Outline of the talk

◮

Context & Motivation

◮

Timed Automata & Time Petri Nets

◮

Expressiveness wrt Timed Bisimilarity

◮

Expressiveness wrt Timed Language Acceptance

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 2 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Outline of the talk

◮

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 2 / 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 3 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Timed Automata & Time Petri Nets

Timed Automata [Alur & Dill’94]

Timed Automata = Finite Automata + timing constraints given by clocks ℓ1 ℓ0 x < 1 a ; x < 1 b ; x ≤ 2 x := 0

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 4 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Timed Automata & Time Petri Nets

Timed Automata [Alur & Dill’94]

Timed Automata = Finite Automata + timing constraints given by clocks ℓ1 ℓ0 x < 1 a ; x < 1 b ; x ≤ 2 x := 0 A run: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · ·

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 4 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Timed Automata & Time Petri Nets

Timed Automata [Alur & Dill’94]

Timed Automata = Finite Automata + timing constraints given by clocks ℓ1 ℓ0 x < 1 a ; x < 1 b ; x ≤ 2 x := 0 A run: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · ·

Time Petri Nets [Merlin’74]

Time Petri Nets = Petri Nets + firing interval relative to markings P0 P1

• t0 : a; [0, 1[

t1 : b; [0, 2]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 4 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Timed Automata & Time Petri Nets

Timed Automata [Alur & Dill’94]

Timed Automata = Finite Automata + timing constraints given by clocks ℓ1 ℓ0 x < 1 a ; x < 1 b ; x ≤ 2 x := 0 A run: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · ·

Time Petri Nets [Merlin’74]

Time Petri Nets = Petri Nets + firing interval relative to markings P0 P1

• t0 : a; [0, 1[

t1 : b; [0, 2] A run: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P1, 2)

b

− → (P0, 0) · · ·

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 4 / 30

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) Emptyness (EP) Universality (UP) Language Inclusion Closure Properties Effect of ε-transition TCTL model checking

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Universality (UP) Language Inclusion Closure Properties Effect of ε-transition TCTL model checking

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] Closure Properties Effect of ε-transition TCTL model checking

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] Effect of ε-transition TCTL model checking

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] Effect of ε-transition ε-TA > TA [Bérard et al.’96] TCTL model checking

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] Effect of ε-transition ε-TA > TA [Bérard et al.’96] TCTL model checking Decidable [Alur & Dill’94]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Decidable [Berthomieu & Diaz’91] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] Effect of ε-transition ε-TA > TA [Bérard et al.’96] TCTL model checking Decidable [Alur & Dill’94]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Decidable [Berthomieu & Diaz’91] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] ?? Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] Effect of ε-transition ε-TA > TA [Bérard et al.’96] TCTL model checking Decidable [Alur & Dill’94]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Decidable [Berthomieu & Diaz’91] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] ?? Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] ?? Effect of ε-transition ε-TA > TA [Bérard et al.’96] TCTL model checking Decidable [Alur & Dill’94]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Decidable [Berthomieu & Diaz’91] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] ?? Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] ?? Effect of ε-transition ε-TA > TA [Bérard et al.’96] ?? TCTL model checking Decidable [Alur & Dill’94]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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) Emptyness (EP) PSPACE-Complete [Alur & Dill’94] Decidable [Berthomieu & Diaz’91] Universality (UP) Language Inclusion Undecidable [Alur & Dill’94] ?? Closure Properties Closed under ∩, ∪ but not under compl. [Alur & Dill’94] ?? Effect of ε-transition ε-TA > TA [Bérard et al.’96] ?? TCTL model checking Decidable [Alur & Dill’94] Decidable [C. & R., AVoCS’04]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 5 / 30

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

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

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

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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Our Contribution

Extended version of (bounded) TPNs: TPNε

• pen intervals, final and repeated markings, ε-transitions

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Our Contribution

Extended version of (bounded) TPNs: TPNε

• pen intervals, final and repeated markings, ε-transitions

TA are strictly more expressive than TPNε w.r.t. weak timed bisimilarity

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Our Contribution

Extended version of (bounded) TPNs: TPNε

• pen intervals, final and repeated markings, ε-transitions

TA are strictly more expressive than TPNε w.r.t. weak timed bisimilarity TAε and one-safe TPNε are equally expressive w.r.t. timed language acceptance

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Our Contribution

Extended version of (bounded) TPNs: TPNε

• pen intervals, final and repeated markings, ε-transitions

TA are strictly more expressive than TPNε w.r.t. weak timed bisimilarity TAε and one-safe TPNε are equally expressive w.r.t. timed language acceptance

Universal Problem is undecidable for (bounded and) one-safe TPNε (Language Inclusion is undecidable)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Our Contribution

Extended version of (bounded) TPNs: TPNε

• pen intervals, final and repeated markings, ε-transitions

TA are strictly more expressive than TPNε w.r.t. weak timed bisimilarity TAε and one-safe TPNε are equally expressive w.r.t. timed language acceptance

Universal Problem is undecidable for (bounded and) one-safe TPNε (Language Inclusion is undecidable) Bounded TPNε and one-safe TPNε are equally expressive (w.r.t. timed language acceptance)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Our Contribution

Extended version of (bounded) TPNs: TPNε

• pen intervals, final and repeated markings, ε-transitions

TA are strictly more expressive than TPNε w.r.t. weak timed bisimilarity TAε and one-safe TPNε are equally expressive w.r.t. timed language acceptance

Universal Problem is undecidable for (bounded and) one-safe TPNε (Language Inclusion is undecidable) Bounded TPNε and one-safe TPNε are equally expressive (w.r.t. timed language acceptance)

Timed Bisimilarity: B-TPNε(≤, ≥) (original class defined by Merlin) and TAε(≤, ≥) are equivalent w.r.t. timed bisimilarity

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 7 / 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 8 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · ·

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · · run 2: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

50

− − → (ℓ1, 50.78) · · ·

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · · run 2: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

50

− − → (ℓ1, 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · · run 2: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

50

− − → (ℓ1, 50.78) · · ·

Definition (Semantics of TA)

States: (ℓ, v) ∈ Q = L × (R≥0)X Discrete transition: (l, v)

a

− → (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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · · run 2: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

50

− − → (ℓ1, 50.78) · · ·

Definition (Semantics of TA)

States: (ℓ, v) ∈ Q = L × (R≥0)X Discrete transition: (l, v)

a

− → (l′, v′) Time transition: (l, v)

t

− → (l′, v′) iff

• l = l′ and v′ = v + t: the clocks are updated

The invariant of l holds for all v + t′, t′ ≤ t

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · · run 2: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

50

− − → (ℓ1, 50.78) · · ·

Definition (Semantics of TA)

States: (ℓ, v) ∈ Q = L × (R≥0)X Discrete transition: (l, v)

a

− → (l′, v′) Time transition: (l, v)

t

− → (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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Timed Automata

Timed Automata

Timed Automata

ℓ1 ℓ0 [x < 1] a ; x < 1 b ; x ≤ 2 x := 0 run 1: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

1.22

− − − → (ℓ1, 2)

b

− → (ℓ0, 0) · · · run 2: (ℓ0, 0)

0.78

− − − → (ℓ0, 0.78)

a

− → (ℓ1, 0.78)

50

− − → (ℓ1, 50.78) · · ·

Definition (Semantics of TA)

States: (ℓ, v) ∈ Q = L × (R≥0)X Discrete transition: (l, v)

a

− → (l′, v′) Time transition: (l, v)

t

− → (l′, v′) a TA generates a set of runs = alternating discrete and time steps semantics of a TA A is a Timed Transition System SA

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 9 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · ·

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · · run 2: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P0, 0) · · ·

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · · run 2: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P0, 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · · run 2: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P0, 0) · · ·

Definition (Semantics of Time Petri Nets )

States: (M, ν) with M a marking and ν : Enabled(M) → R≥0 Discrete transition: (M, ν)

a

− → (M′, ν′) iff ∃t : a, I in the s.t.      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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · · run 2: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P0, 0) · · ·

Definition (Semantics of Time Petri Nets )

States: (M, ν) with M a marking and ν : Enabled(M) → R≥0 Discrete transition: (M, ν)

a

− → (M′, ν′) Time transition: (M, ν)

d

− → (M′, ν′) iff

• 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · · run 2: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P0, 0) · · ·

Definition (Semantics of Time Petri Nets )

States: (M, ν) with M a marking and ν : Enabled(M) → R≥0 Discrete transition: (M, ν)

a

− → (M′, ν′) Time transition: (M, ν)

d

− → (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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Semantics of Time Petri Nets

Time Petri Net

Time Petri Nets

P0 P1 t0 : a; [0, 1[ t1 : b; [0, 2] run 1: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0.78)

1.5

− − → (P1, 1.5)

b

− → (P0, 0) · · · run 2: (P0, 0)

0.78

− − − → (P0, 0.78)

a

− → (P1, 0)

2

− → (P0, 0) · · ·

Definition (Semantics of Time Petri Nets )

States: (M, ν) with M a marking and ν : Enabled(M) → R≥0 Discrete transition: (M, ν)

a

− → (M′, ν′) Time transition: (M, ν)

d

− → (M′, ν′) A TPN generates a set of runs = alternating discrete and time steps Semantics of a TPN N = Timed Transition System SN

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 10 / 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

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 SA and SB s.t.: sA

0 ≡ sB 0 and

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

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 SA and SB s.t.: sA

0 ≡ sB 0 and

1 for each state s of SA there is a state q of SB s.t. s ≡ q and

if s

σ

− → s′ then q

σ

− → q′ and s′ ≡ q′ and s′

2 for each state q of SB . . . FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

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 SA and SB s.t.: sA

0 ≡ sB 0 and

1 for each state s of SA there is a state q of SB s.t. s ≡ q and

if s

σ

− → s′ then q

σ

− → q′ and s′ ≡ q′ and s′

2 for each state q of SB . . .

Weakly Timed Bisimilar: allows ε-moves

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

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 SA and SB s.t.: sA

0 ≡ sB 0 and

1 for each state s of SA there is a state q of SB s.t. s ≡ q and

if s

a

= ⇒ s′ then q

a

= ⇒ q′ and s′ ≡ q′ and s′

2 for each state q of SB . . .

Weakly Timed Bisimilar: allows ε-moves discrete step: s

a

= ⇒ s′ if s

ε∗

− − → a − → ε∗ − − → s′ time step: s

δ

= ⇒ s′ if s

ε∗

− − → δ1 − − → ε+ − − → · · ·

ε+

− − → δn − − → ε∗ − − → s′ and δi = δ

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

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 SA and SB s.t.: sA

0 ≡ sB 0 and

1 for each state s of SA there is a state q of SB s.t. s ≡ q and

if s

a

= ⇒ s′ then q

a

= ⇒ q′ and s′ ≡ q′ and s′

2 for each state q of SB . . .

Weakly Timed Bisimilar: allows ε-moves

Theorem ([C. & R., AVoCS’04])

Each bounded TPN (TPNε) is timed bisimilar to a TA (TAε).

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

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 SA and SB s.t.: sA

0 ≡ sB 0 and

1 for each state s of SA there is a state q of SB s.t. s ≡ q and

if s

a

= ⇒ s′ then q

a

= ⇒ q′ and s′ ≡ q′ and s′

2 for each state q of SB . . .

Weakly Timed Bisimilar: allows ε-moves

Theorem ([C. & R., AVoCS’04])

Each bounded TPN (TPNε) is timed bisimilar to a TA (TAε).

Converse: Each TA is timed bisimilar to a TPN ?

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 12 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1 (M0, 0)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1 (M0, 0) (M′

0, ν′ 0)

ε∗

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1 (M0, 0) (M′

0, ν′ 0)

ε∗ (M1, ν1) δ1

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1 (M0, 0) (M′

0, ν′ 0)

ε∗ (M1, ν1) δ1 (M′

1, ν′ 1)

ε+

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1 (M0, 0) (M′

0, ν′ 0)

ε∗ (M1, ν1) δ1 (M′

1, ν′ 1)

ε+ (M′

n−1, ν′ n−1)

(Mn, νn) δn

• • •

(M′

n, ν′ n)

ε∗

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) δ = 1 (M0, 0) (M′

0, ν′ 0)

ε∗ (M1, ν1) δ1 (M′

1, ν′ 1)

ε+ (M′

n−1, ν′ n−1)

(Mn, νn) δn

• • •

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) (ℓ0, x < 1) δ = 1 − δn δ = δn (M0, 0) (δ = δn) + ε∗ (M′

n−1, ν′ n−1)

(Mn, νn) δn

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) (ℓ0, x < 1) δ = 1 − δn δ = δn (M0, 0) (δ = δn) + ε∗ a (M′

n−1, ν′ n−1)

(Mn, νn) δn

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) (ℓ0, x < 1) δ = 1 − δn δ = δn (M0, 0) (δ = δn) + ε∗ a (M′

n−1, ν′ n−1)

(Mn, νn) δn ε∗a

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) (ℓ0, x < 1) δ = 1 − δn δ = δn (M0, 0) (δ = δn) + ε∗ a (M′

n−1, ν′ n−1)

(Mn, νn) δn ε∗a ε∗a

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Theorem (TA are strictly more expressive than TPNs)

Let A0 = ℓ0 ℓ1 a ; x < 1 There is no TPN weakly timed bisimilar to A0.

Proof.

1 Time elapsing cannot disable transitions in a TPN

(M, ν)

t1t2···tk

− − − − − → (M′, ν′) and (M, ν)

δ

− → (M′′, ν′′)

t1t2···tk

− − − − − →

2 Assume there is a TPN N weakly timed bisimilar to A0

(ℓ0, x = 0) (ℓ0, x = 1) (ℓ0, x < 1) δ = 1 − δn δ = δn (M0, 0) (δ = δn) + ε∗ a (M′

n−1, ν′ n−1)

(Mn, νn) δn ε∗a ε∗a

X

a

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 13 / 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 14 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Language Equivalence

Definition (Timed Word & Timed Language)

A timed word over Σ is a sequence w = (a0, δ0)(a1, δ1) · · · (an, δn) · · · with ai ∈ Σ; δi ∈ 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Language Equivalence

Definition (Timed Word & Timed Language)

A timed word over Σ is a sequence w = (a0, δ0)(a1, δ1) · · · (an, δn) · · · with ai ∈ Σ; δi ∈ 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Language Equivalence

Definition (Timed Word & Timed Language)

A timed word over Σ is a sequence w = (a0, δ0)(a1, δ1) · · · (an, δn) · · · with ai ∈ Σ; δi ∈ 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Language Equivalence

Definition (Timed Word & Timed Language)

A timed word over Σ is a sequence w = (a0, δ0)(a1, δ1) · · · (an, δn) · · · with ai ∈ Σ; δi ∈ 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0 FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0] [0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0] [0] tt

• FORMATS’05 (Uppsala, Sweden)

Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0] [0] tt

• ν(u) = x
• u : [0, 1[

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0] [0] tt

• ν(u) = x
• u : [0, 1[

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0] [0] tt

• ν(u) = x
• u : [0, 1[

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Encoding a TA into a Bounded TPN

ℓ1 ℓ0 a ; x < 1 b ; x ≤ 2 x := 0

Pℓ0

• Pℓ1

a b tt ν(t) = x

• t : [2, 2]

[0] [0] tt

• ν(u) = x
• u : [0, 1[

[0]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 16 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TA and TPN are equally Expressive

Theorem

One-safe TPNε and TAε are equally expressive w.r.t. Timed Language Acceptance.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 17 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TA and TPN are equally Expressive

Theorem

One-safe TPNε and TAε 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TA and TPN are equally Expressive

Theorem

One-safe TPNε and TAε 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TA and TPN are equally Expressive

Theorem

One-safe TPNε and TAε 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

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c Invariants are increasing: between two resets of a clock x, the sequence of invariants encountered from any location is of the form x ≤ c1 and later on x ≤ c2 with c1 ≤ c2 etc ℓ0 ℓ1 z ≤ c ; a ; {y}

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c Invariants are increasing: between two resets of a clock x, the sequence of invariants encountered from any location is of the form x ≤ c1 and later on x ≤ c2 with c1 ≤ c2 etc ℓ0 ℓ1 z ≤ c ; a ; {y} y ≤ c1 y ≤ c

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c Invariants are increasing: between two resets of a clock x, the sequence of invariants encountered from any location is of the form x ≤ c1 and later on x ≤ c2 with c1 ≤ c2 etc ℓ0 ℓ1 z ≤ c ; a ; {y} x ≤ c1

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c Invariants are increasing: between two resets of a clock x, the sequence of invariants encountered from any location is of the form x ≤ c1 and later on x ≤ c2 with c1 ≤ c2 etc ℓ0 ℓ1 z ≤ c ; a ; {y} x ≤ c1 x ≤ c2

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c Invariants are increasing: between two resets of a clock x, the sequence of invariants encountered from any location is of the form x ≤ c1 and later on x ≤ c2 with c1 ≤ c2 etc

Theorem (For Timed Bisimilarity)

B-TPNε(≤, ≥) and TAε(≤, ≥) are equally expressive.

Proof.

From B-TPNε(≤, ≥) to TAε(≤, ≥): [C. & R., AVoCS’04]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Back to Timed Bisimilarity

Definition (The class TAε(≤, ≥))

Guards are of the form x ≥ c or x ≤ c Invariants are increasing: between two resets of a clock x, the sequence of invariants encountered from any location is of the form x ≤ c1 and later on x ≤ c2 with c1 ≤ c2 etc

Theorem (For Timed Bisimilarity)

B-TPNε(≤, ≥) and TAε(≤, ≥) are equally expressive.

Proof.

From B-TPNε(≤, ≥) to TAε(≤, ≥): [C. & R., AVoCS’04] From TAε(≤, ≥) to B-TPNε(≤, ≥) Extension of the previous construction for Timed Language

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 18 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

New Results for TPNs

Theorem (One is enough)

One-safe B-TPNε and B-TPNε are equally expressive w.r.t. timed language acceptance.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 19 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

New Results for TPNs

Theorem (One is enough)

One-safe B-TPNε and B-TPNε are equally expressive w.r.t. timed language acceptance.

Theorem (Universal Problem Undecidable)

The universal problem is undecidable for B-TPNε.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 19 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

New Results for TPNs

Theorem (One is enough)

One-safe B-TPNε and B-TPNε are equally expressive w.r.t. timed language acceptance.

Theorem (Universal Problem Undecidable)

The universal problem is undecidable for B-TPNε.

Corollary (Language Inclusion Undecidable)

Language Inclusion in undecidable for B-TPNε.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 19 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

New Results for TPNs

Theorem (One is enough)

One-safe B-TPNε and B-TPNε are equally expressive w.r.t. timed language acceptance.

Theorem (Universal Problem Undecidable)

The universal problem is undecidable for B-TPNε.

Corollary (Language Inclusion Undecidable)

Language Inclusion in undecidable for B-TPNε.

Theorem

One-safe B-TPNε(≤, ≥) and B-TPNε(≤, ≥) are equally expressive w.r.t. weak timed bisimilarity.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 19 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ?

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ? Timed Bisimilarity

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ? Timed Bisimilarity TAε

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ? Timed Bisimilarity TAε TPNε

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ? Timed Bisimilarity TAε TPNε A0

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ? Timed Bisimilarity TAε TPNε A0 B-TPNε

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Current Picture

Timed Language Acceptance TAε = B-TPNε = 1-B-TPNε TA B-TPN Q:B-TPN = TA ? Timed Bisimilarity TAε TPNε A0 B-TPNε TA(≤, ≥) = B-TPNε(≤, ≥) = 1-B-TPNε(≤, ≥)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 20 / 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 21 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

Conclusion & Recent Work

Results: Expressive Power of TA vs. TPNs Timed Language Acceptance and Timed Bisimilarity Undecidability of the Universal Problem Equivalence between one-safe TPN and TPN Recent Results: more than one semantics for TPN [Bérard et al., ATVA’05] semantic definition of the class of TA that are timed bisimilar to TPN [Bérard et al., FSTTCS’05]

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 22 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

Event-Clock Aut.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

Event-Clock Aut. Event-Recording Aut.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

Event-Clock Aut. Event-Recording Aut. Event-Predicting Aut.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

Event-Clock Aut. Event-Recording Aut. Event-Predicting Aut.

• Det. Bound. TPN

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

Event-Clock Aut. Event-Recording Aut. Event-Predicting Aut.

• Det. Bound. TPN

Non Det. Bound. TPN ??

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

TPN and Event-Clock Automata [Alur et al.’99]

Non Det. TA

• Det. TA

Event-Clock Aut. Event-Recording Aut. Event-Predicting Aut.

• Det. Bound. TPN

Non Det. Bound. TPN ?? Waterloo Station •

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 23 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

References

• R. Alur and D. Dill.

A theory of timed automata. Theoretical Computer Science B, 126:183–235, 1994.

• R. Alur, L. Fix, and T. A. Henzinger.

Event-Clock Automata: A Determinizable Class of Timed Automata. Theoretical Computer Science B, 211:253-273, 1999.

• B. Bérard, F. Cassez, S. Haddad, O. H. Roux, and D. Lime.

Comparison of Different Semantics for Time Petri Nets. In Proceedings of the Third International Symposium on Automated Technology for Verification and Analysis (ATVA’2005), volume 3707 of Lecture Notes in Computer Science, Taipei, Taiwan, October 2005. Springer. To appear.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 24 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

References (cont.)

• B. Bérard, F. Cassez, S. Haddad, O. H. Roux, and D. Lime.

When are timed automata weakly timed bisimilar to timed Patri nets ?. In FSTTCS’05 Hyberadad, India, December 2005. Springer. To appear.

• B. Bérard, P. Gastin and A. Petit.

On the Power of Non-Observable Actions in Timed Automata. Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science (STACS’96), LNCS 1046, pages 257–268, 1996

• B. Berthomieu and M. Diaz.

Modeling and verification of time dependent systems using time Petri nets. IEEE Transactions on Software Engineering, 17(3):259–273, March 1991.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 25 / 30

Context TA & TPN Timed Bisimilarity Timed Language Conclusion

References (cont.)

• F. Cassez and O.H. Roux.

Structural Translation of Time Petri Nets into Timed Automata. In Michael Huth, editor, Workshop on Automated Verification of Critical Systems (AVoCS’04), Electronic Notes in Computer Science. Elsevier, August 2004. P.M. Merlin. A study of the recoverability of computing systems. PhD thesis, University of California, Irvine, CA, 1974.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 26 / 30

Timed Automata [Alur & Dill’94]

Back

A Timed Automaton A is a tuple (L, ℓ0, Act, X, inv, − →) where: L is a finite set of locations ℓ0 is the initial location X is a finite set of clocks Act is a finite set of actions − → is a set of transitions of the form ℓ

g , a , R

− − − − → ℓ′ with:

ℓ, ℓ′ ∈ L, a ∈ Act a guard g which is a clock constraint over X a reset set R which is the set of clocks to be reset to 0

Clock constraints are boolean combinations of x ∼ k with x ∈ C and k ∈ Z and ∼∈ {≤, <}.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 27 / 30

Semantics of Timed Automata

Back

Let A = (L, ℓ0, Act, X, inv, − →) be a Timed Automaton. A state (ℓ, v) of A is in L × RX

≥0

The semantics of A is a Timed Transition System SA = (Q, q0, Act ∪ R≥0, − →) with: Q = L × RX

≥0

q0 = (ℓ0, 0) − → consists in: discrete transition: (ℓ, v)

a

→ (ℓ′, v′) ⇐ ⇒        ∃ ℓ

g , a , r

− − − − → ℓ′ ∈ A v | = g v′ = v[r ← 0] v′ | = inv(ℓ′) delay transition: (ℓ, v) d → (ℓ, v + d) ⇐ ⇒ d ∈ R≥0 ∧ v + d | = inv(ℓ)

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 28 / 30

Time Petri Nets

Back

A Time Petri Net N is a tuple (P, T, Σε, •(.), (.)•, M0, Λ, I) where: P = {p1, p2, · · · , pm} is a finite set of places T = {t1, t2, · · · , tn} is a finite set of transitions and P ∩ T = ∅; Σ is a finite set of actions

• (.) ∈ (NP)T is the backward incidence mapping; (.)• ∈ (NP)T is the

forward incidence mapping; M0 ∈ NP is the initial marking; Λ : T → Σε is the labeling function; I : T → I(Q≥0) associates with each transition a firing interval;

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 29 / 30

Semantics of Time Petri Nets

Back

Let N = (P, T, Σε, •(.), (.)•, M0, Λ, I) be a Time Petri Net. A state (M, ν) of N is a pair with in NP and ν ∈ REnabled(M)

≥0

. ADM(N) is the set of states of N. The semantics of N is a Timed Transition System SN = (Q, {q0}, T, →): Q = ADM(N), q0 = (M0, 0), F ′ = {(M, ν) | M ∈ F} and − →∈ Q × (T ∪ R≥0) × Q is the transition relations:

the discrete transition relation is defined ∀t ∈ T by: (M, ν)

Λ(t)

− − − → (M′, ν′) iff          t ∈ Enabled(M) ∧ M′ = M − •t + t• ν(t) ∈ I(t), ∀t ∈ REnabled(M′)

≥0

, ν′(t) =

• 0 if ↑enabled(t′, M, t),

ν(t) otherwise.

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 30 / 30

Semantics of Time Petri Nets

Back

Let N = (P, T, Σε, •(.), (.)•, M0, Λ, I) be a Time Petri Net. A state (M, ν) of N is a pair with in NP and ν ∈ REnabled(M)

≥0

. ADM(N) is the set of states of N. The semantics of N is a Timed Transition System SN = (Q, {q0}, T, →): Q = ADM(N), q0 = (M0, 0), F ′ = {(M, ν) | M ∈ F} and − →∈ Q × (T ∪ R≥0) × Q is the transition relations:

the discrete transition relation is defined ∀t ∈ T by: continuous transition relation is defined ∀d ∈ R≥0 by: (M, ν)

d

− → (M, ν′) iff

• ν′ = ν + d

∀t ∈ Enabled(M), ν′(t) ∈ I(t)↓

FORMATS’05 (Uppsala, Sweden) Expressiveness of TPNs vs. TA 30 / 30