Discrete Parameters in Petri Nets

Transcription

1 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

2 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

3 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

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

5 Classic Model? a marked Petri Net (PPN) p 1 t 1 3 t 2 p 4 2 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

6 Classic Model? a marked Petri Net (PPN) p 1 t 1 3 t 2 p 4 2 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

7 Classic Model? a marked Petri Net (PPN) p 1 t 1 3 t 2 p 4 2 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

8 Generalization toward Parametric marked Petri Net (PPN) p 1 t1 t p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

9 Generalization toward Parametric marked Petri Net (PPN) p 1 t 1 t p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

10 Generalization toward Parametric marked Petri Net (PPN) p 1 t 1 t p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

11 Generalization toward Parametric marked Petri Net (PPN) p 1 t1 t p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

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

13 Some concrete examples p 1 p 2 t Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 7 / 34

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

15 Some Concrete Examples p 1 p p t 2 1 t p Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 8 / 34

16 Instantiation p 1 p 1 ν 1 () = 2 ν 1 ( 1 ) = 3 ν 1 ( 2 ) = 1 ν 1 ( 3 ) = 2 t 1 3 p 2 t 2 2 t 1 t p 2 3 p 1 ν 2 () = 3 ν 2 ( 1 ) = 1 ν 2 ( 2 ) = 2 ν 2 ( 3 ) = 1 t 1 2 p 2 t 2 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

17 Instantiation p 1 p 1 ν 1 () = 2 ν 1 ( 1 ) = 3 ν 1 ( 2 ) = 1 ν 1 ( 3 ) = 2 t 1 3 p 2 t 2 2 t 1 t p 2 3 p 1 ν 2 () = 3 ν 2 ( 1 ) = 1 ν 2 ( 2 ) = 2 ν 2 ( 3 ) = 1 t 1 2 p 2 t 2 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

18 Instantiation p 1 p 1 ν 1 () = 2 ν 1 ( 1 ) = 3 ν 1 ( 2 ) = 1 ν 1 ( 3 ) = 2 t 1 3 p 2 t 2 2 t 1 t p 2 3 p 1 ν 2 () = 3 ν 2 ( 1 ) = 1 ν 2 ( 2 ) = 2 ν 2 ( 3 ) = 1 t 1 2 p 2 t 2 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

19 Reminders... Definition (Reachability) Let S = (N, m 0 ) = (P, T, Pre, Post, m 0 ) and m a marking of S, S reaches m iff m RS(S). Definition (Coverability) Let S = (N, m 0 ) = (P, T, Pre, Post, m 0 ) 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

20 Some Examples p 1 1 t p 2 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

21 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 N Par s.t. ν(sp) satisfies φ? Definition (P-Universality problem) (U -P): Does ν(sp) satisfies φ for each ν N Par? 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

22 Existence ν 1 (SP) ν 1 parametric model SP ν 2 ν 3... ν 2 (SP) ν 3 (SP) classic models ν n... ν n (SP) Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

23 Existence ν 1 (SP) ν 1 parametric model SP ν 2 ν 3... ν 2 (SP) ν 3 (SP) classic models ν 3 (SP) φ ν n... ν n (SP) Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

24 Existence ν 1 (SP) ν 1 parametric model SP ν 2 ν 3... ν 2 (SP) ν 3 (SP) classic models ν 3 (SP) φ ν n SP E φ... ν n (SP) Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

25 Universality ν 1 (SP) ν 1 parametric model SP ν 2 ν 3... ν 2 (SP) ν 3 (SP) classic models ν n... ν n (SP) Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

26 Universality ν 1 (SP) ν 1 (SP) φ ν 1 parametric model SP ν 2 ν 3... ν 2 (SP) ν 3 (SP) ν 2 (SP) φ classic models ν 3 (SP) φ ν n... ν n (SP) ν n (SP) φ Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

27 Universality ν 1 (SP) ν 1 (SP) φ ν 1 parametric model SP ν 2 ν 3... ν 2 (SP) ν 3 (SP) ν 2 (SP) φ classic models ν 3 (SP) φ ν n SP U φ... ν n (SP) ν n (SP) φ Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

28 Mixing Properties and Parameters... (U -cov) asks: Does each valuation of the parameters implies that the valuation of the PPN covers m? i.e. m is U -coverable in SP { ν N Par, 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

29 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

30 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

31 2-Counters Machine two counters c 1, c 2, states P = {p 0,...p m }, a terminal state labelled halt finite list of instructions l 1,..., l s 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

32 Example of 2-Counters Machine instructions sequence: (p 1, C 1 = 0, C 2 = 0) (p 2, C 1 = 1, C 2 = 0) (p 1, C 1 = 1, C 2 = 1) (p 2, C 1 = 2, C 2 = 1)... p 1. C 0 := C 0 + 1; goto p 2 ; p 2. C 1 := C 1 + 1; goto p 1 ; Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 19 / 34

33 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

34 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

35 Simulation of Instructions: m(c1) + m( C1) = π π π θ θ θ C1 C1 + + C1 C1 C1 C1 C1 0 0 C1 p i p j p i p j p i p j error p k incrementation of a counter decrementation of a counter zero test of a counter Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

36 Simulation of Instructions: m(c1) + m( C1) = π π π θ θ θ C1 C1 + + C1 C1 C1 C1 C1 0 0 C1 p i p j p i p j p i p j error p k incrementation of a counter decrementation of a counter zero test of a counter Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

37 Simulation of Instructions: m(c1) + m( C1) = π π π θ θ θ C1 C C1 C1 C1 C1 C1 0 0 C1 p i p j p i p j p i p j error p k incrementation of a counter decrementation of a counter zero test of a counter Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

38 Simulation of Instructions: m(c1) + m( C1) = π π π θ θ θ C1 C C1 C1 C1 C1 C1 0 0 C1 p i p j p i p j p i p j error p k incrementation of a counter decrementation of a counter zero test of a counter Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

39 Simulation of Instructions: m(c1) + m( C1) = π π π θ θ θ C1 C C1 C1 C1 C1 C1 0 0 C1 p i p j p i p j p i p j error p k incrementation of a counter decrementation of a counter zero test of a counter Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34

40 M halts iff there exists a valuation ν such that ν(sp M ) covers the corresponding p halt place. the counters are unbounded along the instructions sequence of M iff for each valuation ν, ν(sp M ) covers the error state. Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 23 / 34

41 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

42 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

43 From P-PPN to postt-ppn π p 1 p 2 replacement of the P parameters by postt parameters θ p 1 p 2 t t p 4 p 4 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 26 / 34

44 From P-PPN to postt-ppn π p 1 p 2 replacement of the P parameters by postt parameters θ p 1 p t t p 4 p 4 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 26 / 34

45 PPN T-PPN P-PPN distinctt-ppn pret-ppn postt-ppn Caption: : is a syntactical subclass of : is a weak-bisimulation subclass of PN Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 27 / 34

46 From postt-ppn to P-PPN p t replacement of the postt parameters by P parameters π t,1 t θ t π t,p,1 π t,2 θ t,p,1 θ t,p,2 p π t,p,2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

47 From postt-ppn to P-PPN p t replacement of the postt parameters by P parameters π t,1 t θ t π t,p,1 π t,2 θ t,p,1 θ t,p,2 p π t,p,2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

48 From postt-ppn to P-PPN p t replacement of the postt parameters by P parameters π t,1 t θ t π t,p,1 π t,2 θ t,p,1 θ t,p,2 p π t,p,2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

49 From postt-ppn to P-PPN p t replacement of the postt parameters by P parameters π t,1 t θ t π t,p,1 π t,2 θ t,p,1 θ t,p,2 p π t,p,2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

50 From postt-ppn to P-PPN p t replacement of the postt parameters by P parameters π t,1 t θ t π t,p,1 π t,2 θ t,p,1 θ t,p,2 p π t,p,2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 28 / 34

51 PPN T-PPN P-PPN distinctt-ppn pret-ppn postt-ppn Caption: : is a syntactical subclass of : is a weak-bisimulation subclass of : is a weak-cosimulation subclass of PN Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 29 / 34

52 p p replacement of the P parameters by token canons p θ p π p θ p π p Figure 2: From PPN to PN Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 30 / 34

53 PPN T-PPN distinctt-ppn U -Cov P-PPN pret-ppn postt-ppn E -Cov PN 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

54 Table 1: 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 Decidability results for parametric coverability and reachability Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 32 / 34

55 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

56 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

57 Richard M. Karp and Raymond E. Miller. Parallel program schemata. Journal of Computer and System Sciences, 3(2): , 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 , New York, NY, USA, ACM. Marvin L. Minsky. Computation: Finite and Infinite Machines. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 34 / 34