Discrete Parameters in Petri Nets SYNCOP2015 Based on a Paper - - PowerPoint PPT Presentation

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Discrete Parameters in Petri Nets SYNCOP2015 Based on a Paper - - PowerPoint PPT Presentation

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Discrete Parameters in Petri Nets SYNCOP2015 Based on a Paper accepted in PN2015 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux April 22, 2015 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets

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Discrete Parameters in Petri Nets

SYNCOP2015 Based on a Paper accepted in PN2015

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux April 22, 2015

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 1 / 34

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1 Introducing Parameters 2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 2 / 34

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1 Introducing Parameters

Preliminaries On the Use of Parameters Parametric Properties

2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 3 / 34

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SLIDE 4

Why Introducing Parameters ?

modeling arbitrary large amount of processes (markings) modeling unspecified aspect of the environement ...

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 4 / 34

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Classic Model ? a marked Petri Net (PPN)

p1 p2 p3 p4 t2 2 t1 3

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

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Classic Model ? a marked Petri Net (PPN)

p1 p2 p3 p4 t2 2 t1 3

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

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SLIDE 7

Classic Model ? a marked Petri Net (PPN)

p1 p2 p3 p4 t2 2 t1 3

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

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Generalization toward Parametric marked Petri Net (PPN)

p1 p2 p3 p3 t2

λ2

t1

λ1 λ2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

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Generalization toward Parametric marked Petri Net (PPN)

p1 p2 p3 p3 t2

λ3 2

t1

3 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

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SLIDE 10

Generalization toward Parametric marked Petri Net (PPN)

λ

p1 p2 p3 p3 t2

2

t1

3 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

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SLIDE 11

Generalization toward Parametric marked Petri Net (PPN)

λ

p1 p2 p3 p3 t2

λ3 λ2

t1

λ1 λ2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

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Some concrete examples

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 7 / 34

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Some concrete examples

λ

p1 p2 t

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 7 / 34

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Some Concrete Examples

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 8 / 34

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Some Concrete Examples

p1 p2 t λ1 p1 p2 t λ2

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 8 / 34

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Instantiation

λ

p1 p2 p3 p3 t2

λ3 λ2

t1

λ1 λ2

p1 p2 p3 p3 t2

2

t1

3

p1 p2 p3 p3 t2

2

t1

2

ν1(λ) = 2 ν1(λ1) = 3 ν1(λ2) = 1 ν1(λ3) = 2 ν2(λ) = 3 ν2(λ1) = 1 ν2(λ2) = 2 ν2(λ3) = 1

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

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Instantiation

λ

p1 p2 p3 p3 t2

λ3 λ2

t1

λ1 λ2

p1 p2 p3 p3 t2

2

t1

3

p1 p2 p3 p3 t2

2

t1

2

ν1(λ) = 2 ν1(λ1) = 3 ν1(λ2) = 1 ν1(λ3) = 2 ν2(λ) = 3 ν2(λ1) = 1 ν2(λ2) = 2 ν2(λ3) = 1

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

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Instantiation

λ

p1 p2 p3 p3 t2

λ3 λ2

t1

λ1 λ2

p1 p2 p3 p3 t2

2

t1

3

p1 p2 p3 p3 t2

2

t1

2

ν1(λ) = 2 ν1(λ1) = 3 ν1(λ2) = 1 ν1(λ3) = 2 ν2(λ) = 3 ν2(λ1) = 1 ν2(λ2) = 2 ν2(λ3) = 1

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

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Reminders...

Definition (Reachability) Let S = (N, m0) = (P, T, Pre, Post, m0) and m a marking of S, S reaches m iff m ∈ RS(S). Definition (Coverability) Let S = (N, m0) = (P, T, Pre, Post, m0) and m a marking of S, S covers m if there exists a reachable marking m′ of S such that m′ is greater or equal to m i.e. ∃m′ ∈ RS(S)s.t. ∀p ∈ P, m′(p) ≥ m(p) (1) Decidability studied in [2] and [1]

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 10 / 34

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Some Examples

p1 p2 p3 t 1 1 RS = {(2, 1, 0), (1, 0, 1)} CS = {m|m ≤ (2, 1, 0) ∨ m ≤ (1, 0, 1)}

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 11 / 34

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Parametric Properties

Given a class of problem P (coverability, reachability,...), SP a PPN and φ is an instance of P Definition (P-Existence problem) (E -P): Is there a valuation ν ∈ NPar s.t. ν(SP) satisfies φ ? Definition (P-Universality problem) (U -P): Does ν(SP) satisfies φ for each ν ∈ NPar ? Definition (P-Synthesis problem) (S -P): Give all the valuation ν, s.t. ν(SP) satisfies φ.

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 12 / 34

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Existence

SP ... ... ν1 ν2 ν3 νn ν1(SP) ν2(SP) ν3(SP) νn(SP) parametric model classic models

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

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Existence

SP ... ... ν1 ν2 ν3 νn ν1(SP) ν2(SP) ν3(SP) ν3(SP) φ νn(SP) parametric model classic models

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

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Existence

SP SP E φ ... ... ν1 ν2 ν3 νn ν1(SP) ν2(SP) ν3(SP) ν3(SP) φ νn(SP) parametric model classic models

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

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Universality

SP ... ... ν1 ν2 ν3 νn ν1(SP) ν2(SP) ν3(SP) νn(SP) parametric model classic models

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

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Universality

SP ... ... ν1 ν2 ν3 νn ν1(SP) ν1(SP) φ ν2(SP) ν2(SP) φ ν3(SP) ν3(SP) φ νn(SP) νn(SP) φ parametric model classic models

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

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Universality

SP SP U φ ... ... ν1 ν2 ν3 νn ν1(SP) ν1(SP) φ ν2(SP) ν2(SP) φ ν3(SP) ν3(SP) φ νn(SP) νn(SP) φ parametric model classic models

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

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Mixing Properties and Parameters...

(U -cov) asks: ”Does each valuation of the parameters implies that the valuation

  • f the PPN covers m ?”

i.e. m is U -coverable in SP ⇔ ∀ν ∈ NPar, ∃m′ ∈ RS(ν(SP)) s.t. m′ ≥ m (2)

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 15 / 34

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1 Introducing Parameters 2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 16 / 34

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Results

Theorem (Undecidability of E -cov on PPN) The E -coverability problem for PPN is undecidable. Theorem (Undecidability of U -cov on PPN) The U -coverability problem for PPN is undecidable.

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 17 / 34

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2-Counters Machine

two counters c1, c2, states P = {p0, ...pm}, a terminal state labelled halt finite list of instructions l1, ..., ls among the following list:

increment a counter decrement a counter check if a counter equals zero

Counters are assumed positive.

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 18 / 34

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Example of 2-Counters Machine

  • p1. C0 := C0 + 1; goto p2;
  • p2. C1 := C1 + 1; goto p1;

instructions sequence: (p1, C1 = 0, C2 = 0) → (p2, C1 = 1, C2 = 0) → (p1, C1 = 1, C2 = 1) → (p2, C1 = 2, C2 = 1) → ...

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 19 / 34

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Undecidability

halting problem (whether state halt is reachable) is undecidable counters boundedness problem (whether the counters values stay in a finite set) is undecidable proved by Minksy [3]

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 20 / 34

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Why ?

simulation of a counter machine E -cov can be reduced to halting problem U -cov can be reduced to counter boundedness

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 21 / 34

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Simulation of Instructions: m(C1) + m(¬C1) = λ

C1 ¬C1 π pi pj error λ C1 + + θ λ C1 ¬C1 π pi pj C1 − − θ λ C1 ¬C1 π pi pj pk λ λ ¬0 θ λ incrementation

  • f a counter

decrementation

  • f a counter

zero test of a counter

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

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Simulation of Instructions: m(C1) + m(¬C1) = λ

C1 λ ¬C1 π pi pj error λ C1 + + θ λ C1 ¬C1 π pi pj C1 − − θ λ C1 ¬C1 π pi pj pk λ λ ¬0 θ λ incrementation

  • f a counter

decrementation

  • f a counter

zero test of a counter

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

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Simulation of Instructions: m(C1) + m(¬C1) = λ

C1

λ − 1

¬C1 π pi pj error λ C1 + + θ λ C1 ¬C1 π pi pj C1 − − θ λ C1 ¬C1 π pi pj pk λ λ ¬0 θ λ incrementation

  • f a counter

decrementation

  • f a counter

zero test of a counter

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

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SLIDE 38

Simulation of Instructions: m(C1) + m(¬C1) = λ

C1

λ − 1

¬C1 π pi pj error λ C1 + + θ λ C1 ¬C1 π pi pj C1 − − θ λ C1 ¬C1 π pi pj pk λ λ ¬0 θ λ incrementation

  • f a counter

decrementation

  • f a counter

zero test of a counter

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

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SLIDE 39

Simulation of Instructions: m(C1) + m(¬C1) = λ

C1

λ − 1

¬C1 π pi pj error λ C1 + + θ λ C1 ¬C1 π pi pj C1 − − θ λ C1 ¬C1 π pi pj pk λ λ ¬0 θ λ incrementation

  • f a counter

decrementation

  • f a counter

zero test of a counter

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

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M halts iff there exists a valuation ν such that ν(SPM) covers the corresponding phalt place. the counters are unbounded along the instructions sequence of M iff for each valuation ν, ν(SPM) covers the error state.

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 23 / 34

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1 Introducing Parameters 2 Undecidability of the General Case 3 Toward Decidable Subclasses

Restrain the Use of Parameters Some Translations Results

4 Conclusion

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 24 / 34

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PPN T-PPN P-PPN distinctT-PPN preT-PPN postT-PPN PN

Figure 1: Syntaxical subclasses of PPN and inclusions between them

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 25 / 34

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From P-PPN to postT-PPN

p1

λ1

p2

λ2

p3 p4 t p1 p2 π p3 p4 t θ 1 λ2 λ1 replacement of the P parameters by postT parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 26 / 34

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From P-PPN to postT-PPN

p1

λ1

p2

λ2

p3 p4 t p1

λ1

p2 π

λ2

p3 p4 t θ 1 λ2 λ1 replacement of the P parameters by postT parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 26 / 34

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PPN T-PPN P-PPN distinctT-PPN preT-PPN postT-PPN PN

Caption: : is a syntactical subclass of : is a weak-bisimulation subclass of Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 27 / 34

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From postT-PPN to P-PPN

p t λ πt,1 p πt,p,2

λ

πt,p,1 πt,2 t θt θt,p,2 θt,p,1 replacement

  • f the postT

parameters by P parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

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From postT-PPN to P-PPN

p t λ πt,1 p πt,p,2

λ

πt,p,1 πt,2 t θt θt,p,2 θt,p,1 replacement

  • f the postT

parameters by P parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

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SLIDE 48

From postT-PPN to P-PPN

p t λ πt,1

λ

p

λ

πt,p,2 πt,p,1 πt,2 t θt θt,p,2 θt,p,1 replacement

  • f the postT

parameters by P parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

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From postT-PPN to P-PPN

p t λ πt,1

λ

p

λ

πt,p,2 πt,p,1 πt,2 t θt θt,p,2 θt,p,1 replacement

  • f the postT

parameters by P parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

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From postT-PPN to P-PPN

p t λ πt,1

λ

p πt,p,2

λ

πt,p,1 πt,2 t θt θt,p,2 θt,p,1 replacement

  • f the postT

parameters by P parameters

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

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PPN T-PPN P-PPN distinctT-PPN preT-PPN postT-PPN PN

Caption: : is a syntactical subclass of : is a weak-bisimulation subclass of : is a weak-cosimulation subclass of Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 29 / 34

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λp p p πp π′

p

θp θ′

p

replacement

  • f the P

parameters by token canons

Figure 2: From PPN to PN

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 30 / 34

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PPN T-PPN P-PPN distinctT-PPN preT-PPN postT-PPN PN E -Cov U -Cov

Figure 3: What is decidable among the subclasses ? (for coverability)

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 31 / 34

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U -problem E -problem Reach. Cov. Reach. Cov. preT-PPN ? ? ? D postT-PPN ? D ? D PPN U U U U distinctT-PPN ? ? ? D P-PPN ? D D D

Table 1: Decidability results for parametric coverability and reachability

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 32 / 34

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1 Introducing Parameters 2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 33 / 34

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Future Work

Fill the Table (especially U -cov on preT-PPN...) Synthesis Problem

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 34 / 34

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Richard M. Karp and Raymond E. Miller. Parallel program schemata. Journal of Computer and System Sciences, 3(2):147 – 195, 1969. Ernst W. Mayr. An algorithm for the general petri net reachability problem. In Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, STOC ’81, pages 238–246, New York, NY, USA, 1981. ACM. Marvin L. Minsky. Computation: Finite and Infinite Machines. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1967.

Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 34 / 34