Comparison of the expressiveness of Arc, Place and Transition Time - - PowerPoint PPT Presentation

Motivation Definition Expressiveness Conclusion

Comparison of the expressiveness of Arc, Place and Transition Time Petri Nets

Marc Boyer (1) and Olivier (H.) Roux (2)

(1) IRIT, Toulouse, France - (2) IRCCyN, Nantes, France

june, 28th, 2007 - ICATPN’07

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 1 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions

◮

Comparing model expressiveness

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 2 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions

◮

Comparing model expressiveness

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 3 / 39

Motivation Definition Expressiveness Conclusion

Why this study?

In 30 years, several models developed: intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] each model has its own application domain people are attached to their model when failing to model: user skill or model power?

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 4 / 39

Motivation Definition Expressiveness Conclusion

Why this study?

In 30 years, several models developed: intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] each model has its own application domain people are attached to their model when failing to model: user skill or model power? Previous studies several studies on power of some models: decidability of reachability, covering some comparisons have been done: [CA99], [BV00] but

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 4 / 39

Motivation Definition Expressiveness Conclusion

Why this study?

In 30 years, several models developed: intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] each model has its own application domain people are attached to their model when failing to model: user skill or model power? Previous studies several studies on power of some models: decidability of reachability, covering some comparisons have been done: [CA99], [BV00] but some results are contradictory

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 4 / 39

Motivation Definition Expressiveness Conclusion

Comparing what ? With which criterion ?

Comparing: Petri nets with time constraint with intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] with weak and strong semantics

• ne bounded ( ⇐

⇒ k-bounded with mono-server semantics) with large and strict intervals bounds ([a, b], [a, b[, ]a, b], ]a, b[), cf [BV00]

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 5 / 39

Motivation Definition Expressiveness Conclusion

Comparing what ? With which criterion ?

Comparing: Petri nets with time constraint with intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] with weak and strong semantics

• ne bounded ( ⇐

⇒ k-bounded with mono-server semantics) with large and strict intervals bounds ([a, b], [a, b[, ]a, b], ]a, b[), cf [BV00] Criterion: weak timed bisimulation “weak bisimulation” is “the same visible actions are possible in both models”

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 5 / 39

Motivation Definition Expressiveness Conclusion

Comparing what ? With which criterion ?

Comparing: Petri nets with time constraint with intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] with weak and strong semantics

• ne bounded ( ⇐

⇒ k-bounded with mono-server semantics) with large and strict intervals bounds ([a, b], [a, b[, ]a, b], ]a, b[), cf [BV00] Criterion: weak timed bisimulation “weak bisimulation” is “the same visible actions are possible in both models” “weak timed bisimulation” is “the same visible actions are possible in both models at the same time”

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 5 / 39

Motivation Definition Expressiveness Conclusion

Comparing what ? With which criterion ?

Comparing: Petri nets with time constraint with intervals on transitions [Mer74, BD91], places [KDCD96] or arcs [Han93, AN01, dFRA00] with weak and strong semantics

• ne bounded ( ⇐

⇒ k-bounded with mono-server semantics) with large and strict intervals bounds ([a, b], [a, b[, ]a, b], ]a, b[), cf [BV00] Criterion: weak timed bisimulation “weak bisimulation” is “the same visible actions are possible in both models” “weak timed bisimulation” is “the same visible actions are possible in both models at the same time” bisimulation implies language equality

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 5 / 39

Motivation Definition Expressiveness Conclusion

Bisimulation

b c a a a b c b c a a ε c b a a c

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 6 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 7 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 8 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): basis

Example (Basic: net from L. Gallon) p1 p3 p2 t1 t3 t2

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 9 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): basis

Example (Basic: net from L. Gallon) p1 p3 p2 t1 t3 t2

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 9 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): basis

Example (Basic: net from L. Gallon) p1 p3 p2 t1 [0, 4] t3 [5, 6] t2 [3, 4]

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 9 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): basis

Example (Basic: net from L. Gallon) p1 p3 p2 t1 [0, 4] t3 [5, 6] t2 [3, 4]

• {p1, p3}

ν(t1) = 0 ν(t3) = 0

4

→ {p1, p3} ν(t1) = 4 ν(t3) = 4

t1

→ {p2, p3} ν(t2) = 0 ν(t3) = 4

1

→ {p2, p3} ν(t2) = 1 ν(t3) = 5

t3

→ · · ·

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 9 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Priority) p1 p2 t1 [1, 2] t2 [3, 4] t3 [3, 3]

• Strong semantics (T-TPN): t3 is never fired, t1 and t2 always are

Weak semantics (T-TPN): all t1, t2 and t3 can be fired, or not

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 10 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Priority) p1 p2 t1 [1, 2] t2 [3, 4] t3 [3, 3]

• Strong semantics (T-TPN): t3 is never fired, t1 and t2 always are

Weak semantics (T-TPN): all t1, t2 and t3 can be fired, or not

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 10 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Priority) p1 p2 t1 [1, 2] t2 [3, 4] t3 [3, 3] Strong semantics (T-TPN): t3 is never fired, t1 and t2 always are Weak semantics (T-TPN): all t1, t2 and t3 can be fired, or not

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 10 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3]

• {p1, p3}

ν(t1) = 0 ν(t2) = 0

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3]

• {p1, p3}

ν(t1) = 0 ν(t2) = 0

θ1

− − − →

θ1≤3

{p1, p3} ν(t1) = θ1 ν(t2) = θ1

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3]

• {p1, p3}

ν(t1) = 0 ν(t2) = 0

θ1

− − − →

θ1≤3

{p1, p3} ν(t1) = θ1 ν(t2) = θ1

t1

→ {p2, p3} ν(t2) = θ1

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3]

• {p1, p3}

ν(t1) = 0 ν(t2) = 0

θ1

− − − →

θ1≤3

{p1, p3} ν(t1) = θ1 ν(t2) = θ1

t1

→ {p2, p3} ν(t2) = θ1

θ2

− − − − − →

θ1+θ2≤3

{p2, p3} ν(t2) = θ1 + θ2

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3]

• {p1, p3}

ν(t1) = 0 ν(t2) = 0

θ1

− − − →

θ1≤3

{p1, p3} ν(t1) = θ1 ν(t2) = θ1

t1

→ {p2, p3} ν(t2) = θ1

θ2

− − − − − →

θ1+θ2≤3

{p2, p3} ν(t2) = θ1 + θ2

t2

→ {p2, p4} ν(tsynch) = 0

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3]

• {p1, p3}

ν(t1) = 0 ν(t2) = 0

θ1

− − − →

θ1≤3

{p1, p3} ν(t1) = θ1 ν(t2) = θ1

t1

→ {p2, p3} ν(t2) = θ1

θ2

− − − − − →

θ1+θ2≤3

{p2, p3} ν(t2) = θ1 + θ2

t2

→ {p2, p4} ν(tsynch) = 0

1

→ {p2, p4} ν(tsynch) = 1

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Synchronisation) p1 p2 p3 p4 t1 [0, 4] tsynch [1, 1] t2 [0, 3] {p1, p3} ν(t1) = 0 ν(t2) = 0

θ1

− − − →

θ1≤3

{p1, p3} ν(t1) = θ1 ν(t2) = θ1

t1

→ {p2, p3} ν(t2) = θ1

θ2

− − − − − →

θ1+θ2≤3

{p2, p3} ν(t2) = θ1 + θ2

t2

→ {p2, p4} ν(tsynch) = 0

1

→ {p2, p4} ν(tsynch) = 1

tsynch

→ ∅

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 11 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Continuous enabling) p1 t1 [2, 2] t2 [3, 3]

• Transition t2 can never be fired, in strong semantics.
• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 12 / 39

Motivation Definition Expressiveness Conclusion

T-time Petri net (T-TPN): properties

Example (Continuous enabling) p1 t1 [2, 2] t2 [3, 3]

• Transition t2 can never be fired, in strong semantics.

{p1} ν(t1) = 0 ν(t2) = 0

2

→ {p1} ν(t1) = 2 ν(t2) = 2

t1

→ {p1} ν(t1) = 0 ν(t2) = 0

2

→ {p1} ν(t1) = 2 ν(t2) = 2

t1

→ · · ·

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 12 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 13 / 39

Motivation Definition Expressiveness Conclusion

Strong vs Weak Semantics

Strong and weak semantics description Weak semantics: a transition is never forced to be fired, no urgency Strong semantics: when the deadline is reached, the transition must be fired; time can not disable firing of a transition Strong semantics definition t ∈ firable(S + d) ⇒ ∀d′ ∈ [0, d] : t ∈ firable(S + d′)

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 14 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 15 / 39

Motivation Definition Expressiveness Conclusion

P-time Petri net

P-TPN: time interval are associated to places Example (A TPN ∈ P-TPN) P0 P1 P2 [1, 5] [1, 2] [2, 4] t0 t1

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 16 / 39

Motivation Definition Expressiveness Conclusion

P-time Petri net

P-TPN: time interval are associated to places Example (A TPN ∈ P-TPN) P0 P1 P2 [1, 5] [1, 2] [2, 4] t0 t1

• {p0, p2}

ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

1

→ {p1, p2} ν(p1) = 1 ν(p2) = 4

t1

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 16 / 39

Motivation Definition Expressiveness Conclusion

P-time Petri net

P-TPN: time interval are associated to places Example (A TPN ∈ P-TPN) P0 P1 P2 [1, 5] [1, 2] [2, 4] t0 t1

• {p0, p2}

ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

1

→ {p1, p2} ν(p1) = 1 ν(p2) = 4

t1

→

Weak semantics :

{p0, p2} ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

2

→ {p1, ˆ p2} ν(p1) = 2 ν(p2) = 5

···

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 16 / 39

Motivation Definition Expressiveness Conclusion

P-time Petri net

P-TPN: time interval are associated to places Example (A TPN ∈ P-TPN) P0 P1 P2 [1, 5] [1, 2] [2, 4] t0 t1

• {p0, p2}

ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

1

→ {p1, p2} ν(p1) = 1 ν(p2) = 4

t1

→

Strong semantics :

{p0, p2} ν(p0) = 0 ν(p2) = 0

5

→ {p0, ˆ p2} ν(p0) = 5 ν(p2) = 5

t0

→ {p1, ˆ p2} ν(p1) = 0 ν(p2) = 5

···

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 16 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 17 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

P-TPN: time interval are associated to arcs (p,t) Example (A TPN ∈ A-TPN without choice) P0 P1 P2 t0 t1 [1, 5] [1, 2] [2, 4]

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 18 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

P-TPN: time interval are associated to arcs (p,t) Example (A TPN ∈ A-TPN without choice) P0 P1 P2 t0 t1 [1, 5] [1, 2] [2, 4]

• {p0, p2}

ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

1

→ {p1, p2} ν(p1) = 1 ν(p2) = 4

t1

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 18 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

P-TPN: time interval are associated to arcs (p,t) Example (A TPN ∈ A-TPN without choice) P0 P1 P2 t0 t1 [1, 5] [1, 2] [2, 4]

• {p0, p2}

ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

1

→ {p1, p2} ν(p1) = 1 ν(p2) = 4

t1

→

Weak semantics :

{p0, p2} ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

2

→ {p1, ˆ p2} ν(p1) = 2 ν(p2) = 5

···

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 18 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

P-TPN: time interval are associated to arcs (p,t) Example (A TPN ∈ A-TPN without choice) P0 P1 P2 t0 t1 [1, 5] [1, 2] [2, 4]

• {p0, p2}

ν(p0) = 0 ν(p2) = 0

3

→ {p0, p2} ν(p0) = 3 ν(p2) = 3

t0

→ {p1, p2} ν(p1) = 0 ν(p2) = 3

1

→ {p1, p2} ν(p1) = 1 ν(p2) = 4

t1

→

Strong semantics :

{p0, p2} ν(p0) = 0 ν(p2) = 0

5

→ {p0, ˆ p2} ν(p0) = 5 ν(p2) = 5

t0

→ {p1, ˆ p2} ν(p1) = 0 ν(p2) = 5

···

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 18 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

Example (A TPN ∈ A-TPN, with choice) P1 P2 t1 t2 [1, 3] [4, 6[ [2, 7]

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 19 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

Example (A TPN ∈ A-TPN, with choice) P1 P2 t1 t2 [1, 3] [4, 6[ [2, 7]

• Strong semantics :

{p1, p2} ν(p1) = 0 ν(p2) = 0

3

→ {p1, p2} ν(p1) = 3 ν(p2) = 3

t2

→ {} . .

···

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 19 / 39

Motivation Definition Expressiveness Conclusion

A-time Petri net (A-TPN)

Example (A TPN ∈ A-TPN, with choice) P1 P2 t1 t2 [1, 3] [4, 6[ [2, 7]

• Strong semantics :

{p1, p2} ν(p1) = 0 ν(p2) = 0

3

→ {p1, p2} ν(p1) = 3 ν(p2) = 3

t2

→ {} . .

···

→

Weak semantics :

{p1, p2} ν(p1) = 0 ν(p2) = 0

5

→ {p1, p2} ν(p1) = 5 ν(p2) = 5

t1

→ {p2} ν(p2) = 5 .

3

→ { ˆ p2} ν(p2) = 5 .

···

→

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 19 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions

◮

Comparing model expressiveness

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 20 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 21 / 39

Motivation Definition Expressiveness Conclusion

Previous works

w.r.t. language acceptance From [CA99] : P-TPN =L T-TPN =L A-TPN P-TPN =L T-TPN =L A-TPN From [BV00] : T-TPN ⊂L A-TPN and P-TPN ⊂L T-TPN (with ≤ constraints only).

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 22 / 39

Motivation Definition Expressiveness Conclusion

Previous works

w.r.t. language acceptance From [CA99] : P-TPN =L T-TPN =L A-TPN P-TPN =L T-TPN =L A-TPN From [BV00] : T-TPN ⊂L A-TPN and P-TPN ⊂L T-TPN (with ≤ constraints only). w.r.t. bisimulation Dans [BV00] : T-TPN ⊂≈ A-TPN, P-TPN ⊆≈ A-TPN P-TPN ⊆≈ T-TPN (with ≤ constraints only). In [Kha92] : P-TPN and T-TPN are declared incomparable (without proof).

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 22 / 39

Motivation Definition Expressiveness Conclusion

Another result [BCH+05] l0 l1 a ; x < 1

Figure: The Timed Automaton A0

Theorem ([BCH+05]) There is no TPN ∈ T-TPN weakly timed bisimilar to A0 ∈ T A. Because: time can not disable an action (transition) in strong semantics, it can in automata.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 23 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 24 / 39

Motivation Definition Expressiveness Conclusion

T-TPN and T-TPN P1 a, [0, 1[

• Figure: The TPN NT0 ∈ T-TPN bisimilar to A0

Corollary There is no TPN ∈ T-TPN weakly timed bisimilar to NT0 ∈ T-TPN.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 25 / 39

Motivation Definition Expressiveness Conclusion

T-TPN and T-TPN P1 a, [0, 1[

• Figure: The TPN NT0 ∈ T-TPN bisimilar to A0

Corollary There is no TPN ∈ T-TPN weakly timed bisimilar to NT0 ∈ T-TPN. T-TPN ⊆≈ T-TPN

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 25 / 39

Motivation Definition Expressiveness Conclusion

T-TPN and T-TPN P1 a, [0, 1[

• Figure: The TPN NT0 ∈ T-TPN bisimilar to A0

Corollary There is no TPN ∈ T-TPN weakly timed bisimilar to NT0 ∈ T-TPN. T-TPN ⊆≈ T-TPN T-TPN and T-TPN are incomparable.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 25 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 26 / 39

Motivation Definition Expressiveness Conclusion

P-TPN and P-TPN p I(p) t

• Figure: The translation from P-TPN into P-TPN
• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 27 / 39

Motivation Definition Expressiveness Conclusion

P-TPN and P-TPN p pε1 pε2 [0, ∞[ [0, ∞[ I(p) ε ε t

• Figure: The translation from P-TPN into P-TPN
• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 27 / 39

Motivation Definition Expressiveness Conclusion

P-TPN and P-TPN p pε1 pε2 [0, ∞[ [0, ∞[ I(p) ε ε t

• Figure: The translation from P-TPN into P-TPN

N ∈ P-TPN and its translation N ∈ P-TPN are weakly timely bisimilar.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 27 / 39

Motivation Definition Expressiveness Conclusion

P-TPN and P-TPN p pε1 pε2 [0, ∞[ [0, ∞[ I(p) ε ε t

• Figure: The translation from P-TPN into P-TPN

N ∈ P-TPN and its translation N ∈ P-TPN are weakly timely bisimilar. P-TPN ⊂≈ P-TPN

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 27 / 39

Motivation Definition Expressiveness Conclusion

Corollaries and similar results P-TPN ⊆≈ T-TPN We are able to build a strong P-TPN equivalent to the weak P-TPN of the theorem P-TPN ⊆≈ T-TPN.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 28 / 39

Motivation Definition Expressiveness Conclusion

Corollaries and similar results P-TPN ⊆≈ T-TPN We are able to build a strong P-TPN equivalent to the weak P-TPN of the theorem P-TPN ⊆≈ T-TPN. The same way A-TPN ⊂≈ A-TPN

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 28 / 39

Motivation Definition Expressiveness Conclusion

Corollaries and similar results P-TPN ⊆≈ T-TPN We are able to build a strong P-TPN equivalent to the weak P-TPN of the theorem P-TPN ⊆≈ T-TPN. The same way A-TPN ⊂≈ A-TPN A-TPN ⊆≈ T-TPN

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 28 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 29 / 39

Motivation Definition Expressiveness Conclusion

T-TPN ⊆≈ P-TPN Lemma In P-TPN (P-TPN or P-TPN) a clock (of a token/place p) can be reset

• nly when the value is in the firing interval (ν(p) ∈ I(p)).

P1 P2 [3, 7[ [2, 4[ t1

• Figure: A TPN ∈ P-TPN
• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 30 / 39

Motivation Definition Expressiveness Conclusion

T-TPN ⊆≈ P-TPN P1 t1, [2, 2] t2, [0, ∞[

• Figure: The TPN NT1 ∈ T-TPN
• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 31 / 39

Motivation Definition Expressiveness Conclusion

T-TPN ⊆≈ P-TPN P1 t1, [2, 2] t2, [0, ∞[

• Figure: The TPN NT1 ∈ T-TPN

Theorem There is no TPN ∈ P-TPN weakly timed bisimilar to NT1 ∈ T-TPN.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 31 / 39

Motivation Definition Expressiveness Conclusion

T-TPN ⊆≈ P-TPN P1 t1, [2, 2] t2, [0, ∞[

• Figure: The TPN NT1 ∈ T-TPN

Theorem There is no TPN ∈ P-TPN weakly timed bisimilar to NT1 ∈ T-TPN. Corollary T-TPN ⊆≈ P-TPN

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 31 / 39

Motivation Definition Expressiveness Conclusion

Other corollaries Corollary T-TPN ⊆≈ P-TPN T-TPN ⊆≈ P-TPN

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 32 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions T-time Petri net Strong vs Weak Semantics P-time Petri net A-time Petri net

◮

Comparing model expressiveness Introduction, previous works Results from [BCH+05] Weak with strong for P-TPN and A-TPN Something that P-TPN can’t do, and T-TPN does Emulating T-TPN rule with A-TPN

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 33 / 39

Motivation Definition Expressiveness Conclusion

Theorem (Translation from T-TPN into A-TPN) T-TPN ⊆≈ A-TPN T-TPN ⊆≈ A-TPN Pb: how to emulate the rule “start counting time when everybody is there” with the rule “everyone has its own clock” Rep: count people, without time, start clock when everybody is there

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 34 / 39

Motivation Definition Expressiveness Conclusion

[0,2] [0,2] [1,1] u v t [1,1]

[0, 2] [0, 2] [0, 2] [0, 2]

t

• t0
• t1
• t2

u(t:0,1) u(t:1,2) v(t:0,1) v(t:1,2)

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 35 / 39

Motivation Definition Expressiveness Conclusion

Outline

◮

Motivation

◮

Model definitions

◮

Comparing model expressiveness

◮

Conclusion

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 36 / 39

Motivation Definition Expressiveness Conclusion

Overview [BR07] T-TPN P-TPN A-TPN T-TPN P-TPN A-TPN ⊆≈ ≈ ⊂≈ ⊂≈ ⊆≈ ≈ ⊂≈ ⊆≈ ⊂≈ ⊆≈ ≈ ⊂≈ ⊃≈

Figure: The classification explained

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 37 / 39

Motivation Definition Expressiveness Conclusion

In a few words

Several theorems, but, what to keep in mind ?

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

Expressiveness of Time Petri Nets june, 28th, 2007 - ICATPN’07 38 / 39

Motivation Definition Expressiveness Conclusion

In a few words

Several theorems, but, what to keep in mind ? A-TPN and P-TPN are very close from each other, A-TPN generalises T-TPN in theory, but not for human modelling, the strong semantics generalises the weak one for P-TPN and A-TPN, but not for T-TPN.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

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Motivation Definition Expressiveness Conclusion

Parosh Aziz Abdulla and Aletta Nyl´ en. Timed petri nets and bqos. In 22nd International Conference on Application and Theory of Petri Nets (ICATPN’01), volume 2075 of Lecture Notes in Computer Science, pages 53–70, Newcastle upon Tyne, United Kingdom, jun

• 2001. Springer-Verlag.

Beatrice B´ erard, Franck Cassez, Serge Haddad, Didier Lime, and Olivier (H.) Roux. Comparison of the expressiveness of timed automata and time Petri nets. In 3rd International Conference on Formal Modelling and Analysis of Timed Systems (FORMATS 05), volume 3829 of Lecture Notes in Computer Science, Uppsala, Sweden, September 2005. Springer. Bernard Berthomieu and Michel Diaz. Modeling and verification of time dependent systems using time petri nets.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

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Motivation Definition Expressiveness Conclusion

IEEE transactions on software engineering, 17(3):259–273, March 1991. Marc Boyer and Olivier (H.) Roux. Comparison of the expressiveness of arc, place and transition time Petri nets. In 28th International Conference on Application and Theory of Petri Nets and other models of concurrency (ICATPN’07), Lecture Notes in Computer Science, Siedlce, Poland, jun 2007. Springer-Verlag. to appear. M Boyer and F. Vernadat. Language and bisimulation relations between subclasses of timed petri nets with strong timing semantic. Technical report, LAAS, 2000.

• A. Cerone and Maggiolo-Schettini A.

Timed based expressivity of time petri nets for system specification. Theoretical Computer Science, 216:1–53, 1999.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

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Motivation Definition Expressiveness Conclusion

David de Frutos Escrig, Valent´ ın Valero Ruiz, and Olga Marroqu´ ın Alonso. Decidability of properties of timed-arc petri nets. In 21st International Conference on Application and Theory of Petri Nets (ICATPN’00), volume 1825 of Lecture Notes in Computer Science, pages 187–206, Aarhus, Denmark, jun 2000. Springer-Verlag. H.M. Hanisch. Analysis of place/transition nets with timed-arcs and its application to batch process control. In 14th International Conference on Application and Theory of Petri Nets (ICATPN’93), volume 691 of LNCS, page 282?299, 1993. Wael Khansa, J.-P Denat, and S. Collart-Dutilleul. P-time petri nets for manufacturing systems. In International Workshop on Discrete Event Systems, WODES’96, pages 94–102, Edinburgh (U.K.), august 1996. Wael Khanza.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

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Motivation Definition Expressiveness Conclusion

R´ eseau de Petri P-Temporels. Contribution ` a l’´ etude des syst` emes ` a ´ ev` enements discrets. PhD thesis, Universit´ e de Savoie, 1992. Philip M. Merlin. A study of the recoverability of computing systems. PhD thesis, Department of Information and Computer Science, University of California, Irvine, CA, 1974.

• M. Boyer / O.H. Roux (IRIT/IRCCyN)

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