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Reversing Unbounded Petri Nets

This paper discusses the concept of reversing unbounded Petri nets and their applications in solving reachability problems.

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Reversing Unbounded Petri Nets

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  1. ReversingUnboundedPetriNets Łukasz Mikulski1joint workwith Ivan Lanese2 1. Nicolaus CopernicusUniversityin Toruń (Poland) 2. Univeristy of Bologna/INRIA (Italy) 40th ATPN, Aachen 2019, Jun26

  2. Petrinets Tokens(marking) Transitions Places • 2 b a c Arcweigthfunctions 40th ATPN, Aachen 2019, Jun26

  3. Petrinets Step M[aM’ b b a a c c • 2 40th ATPN, Aachen 2019, Jun26

  4. Petrinets P/t-netis a tupleN = (P, T, W–, W+, M0), where • P and Tarefinitedisjointsets, of places and transitions, respectively • W–, W+ : TIN|P|arearcweigthfunctions • M0IN|P|istheinitialmarking Any multisetinIN|P| is a marking (global state). 40th ATPN, Aachen 2019, Jun26

  5. Petrinets A transitionbisenabledatMwhetherW–(b)M. We denoteit by M[b, whileEb(N) denotesthe set of allmarkingsenablingb. • Theefect of executionisM’=M– W–(b) + W+(b). We denoteit by M[bM’. • The set of allreachablemarkingsis[M0. • The set of allcoverablemarkingsis [M0. 40th ATPN, Aachen 2019, Jun26

  6. Cyclicity Mis a home stateifM[M’ for everyM’[M0. A net NiscyclicifM0is a home state. c c • M0 • [0,1,1,0] • p4 b • p3 • [0,0,1,1] • p1 c a b b a a • [1,1,0,0] • [1,0,0,1] b • p2 40th ATPN, Aachen 2019, Jun26

  7. Reversetransition – simpleapproach A newtransitionbis a reverse of transitionbTifW–(b) = W+(b) and W+(b) = W–(b) Notation for reverses: dashed lines • p1 b b • p3 • p2 40th ATPN, Aachen 2019, Jun26

  8. Reversing to fixcyclicity c • [0,1,1,0] • p4 b • b • p3 • [0,0,1,1] • [0,0,1,1] • p1 c a • b a b b • [1,1,0,0] • [1,0,0,1] • [1,0,0,1] b 40th ATPN, Aachen 2019, Jun26

  9. Negativeexample • p1 • [1,0,0] • [1,0,1] • [1,0,2] a a c c a c • b • b • b • c • c • c • [0,1,0] • [0,1,0] • [0,1,1] • [0,1,2] • p2 • b • b b b b b b • [0,0,1] • [0,0,1] • [0,0,2] • [0,0,3] • p3 40th ATPN, Aachen 2019, Jun26

  10. MESTR – no newstates Can we check, whetherany(new/unwanted) global system state can be reachedafteraddinga transitionreverse? Arethereachabilitysets of twogivenp/t-nets, wherethesecond one isobtainedfromthe first by adding a single transitionreverse, equal? MarkingEqualitywith a Single TransitionReverse We knowthataverysimilar problem (withadding an arbitrarytransition) isundecidable… 40th ATPN, Aachen 2019, Jun26

  11. A Letusconsider a net A: b1 b2 bk 40th ATPN, Aachen 2019, Jun26

  12. B a A b1 b1 b2 b2 bk bk 40th ATPN, Aachen 2019, Jun26

  13. C B a A b1 b1 e b2 b2 bk bk c 40th ATPN, Aachen 2019, Jun26

  14. D C B a A b1 b1 Reach. set (A) = Reach. set (B)if and onlyif Reach. set (C) = Reach. set (D) e b2 b2 bk bk c c

  15. MESTR – no newstates Can we check, whetheranyunwanted global system state can be reachedafteraddinga transitionreverse? No, this problemisundecidableeven for a single transitionreverse! 40th ATPN, Aachen 2019, Jun26

  16. FiniteLabelledTransition System TS=(S,→,T,s0) is a finitelabelledtransition systemif • Sis a finite set of states • Tis a finite set of transitions (disjoinedwithS) • →  STSis a set of labelledarcs • s0Sis an initial state Synthesis – construction of a net whichsolveslts 40th ATPN, Aachen 2019, Jun26

  17. Finitelts – splitting • p1 • [3,0,2] • [0,3,0] b b b b b b • 2 • [2,1,2] • [1,2,0] • p2 • 2 b' b b a b b a • 3 • 3 • [1,2,2] • [0,3,1] • 2 b b' b' b' b b b a • p3 • [0,3,2] • [1,2,1] • 2 40th ATPN, Aachen 2019, Jun26

  18. Theorem LetltsTS=(S,→,T,s0) be solvable and aT. Thenthereexists a set T of effectreverses of asuchthatTSwithTissolvable. We wouldsaythatsuchaisreversiblein a particular net solvingT. 40th ATPN, Aachen 2019, Jun26

  19. New question Do the constructionusingsplittingreversessolvethe general (possiblyunbounded) case? Can we find, for an arbitraryPetri net, a net withequalbehaviourthathas a set of feasibleeffectreverses? If not, istheexistence of suchfeasible set of reversesdecidable (twolevels)? 40th ATPN, Aachen 2019, Jun26

  20. Negativeanswer • [2,0] b • p1 • [1,1] • 2 b a • [0,2] • [2,1] • p2 b a b a • [1,2] • [3,1] • 2 b b a a • [2,2] • [0,3] b a • [1,3] b

  21. Negativeanswer • M0 b • p1 • M1 • 2 b a • M2 • M3 • p2 b a b a • M4 • M6 • 2 b b 1. M1[baM3 a a • M5 • M7 2. M0[b(ba) b a • M8 3. (M1M3) b

  22. Problematicpairs LetN=(P,T,F,M0) be a net, and bT a transition. A pair (M1,M2) of markingsreachableinNisb-problematicif: • M1<M2 • T* M0[bM1 • T*(M0[bM2) Pb(N) - the set of allb-problematicpairsin N.

  23. Properties of problematicpairs Lemma: LetN=(P,T,F,M0) be a net, and bT a transition. Decidingwhether a givenpair of markingsisb-problematicisequivalent to Reachability. Lemma: • LetN=(P,T,F,M0) be a net, and bT a transition, and (M1,M2) Pb(N). Then: • if M3<M1 and M3[M0 by bthen (M3,M2)  Pb(N), • if M4>M2 and M4[M0never by bthen(M1,M4)  Pb(N).

  24. Properties of problematicpairs Theorem: LetN=(P,T,F,M0) be a net, and bT a transition. A transitionbisreversibleinNif and onlyif Pb(N) = . Theorem: LetN=(P,T,F,M0) be a net, and bT a transition. • The set min(Pb(N)) isfinite, and • min(first(Pb(N)))  min(Eb(N)+{W+(b) – W–(b)})

  25. Evenmorenegativeanswer Theorem: The problem of theexistence of a b-problematicpairisundecidable.

  26. Somethingpositiveinstead of conclusions Supposethat M is a home state • M0 • M1 • M b  b • M2 Corollary:Eachcyclic net isreversible.

  27. Thankyou!

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