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Symbolic Concurrent Semantics of Safe Petri nets

Symbolic Concurrent Semantics of Safe Petri nets. Application to Time Petri Nets. Claude Jard, ENS Cachan / IRISA, Rennes, France & Thomas Chatain, ENS Cachan / LSV, Cachan, France. Why are we interested in PN and unfoldings?.

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Symbolic Concurrent Semantics of Safe Petri nets

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  1. Symbolic Concurrent Semantics of Safe Petri nets Application to Time Petri Nets Claude Jard, ENS Cachan / IRISA, Rennes, France & Thomas Chatain, ENS Cachan / LSV, Cachan, France

  2. Why are we interested in PN and unfoldings? • Supervision and diagnosis: inferring causal dependencies from observations in a distributed system (guided unfolding) -> already in use on Alcatel platforms • Composition of QoS contracts in WS orchestrations (need a partial order view of the behaviours) -> concurrent semantics for ORC In such application domains, we do not need strong decidability results and thus consider extensions of PN with data, time, probas, …

  3. Focus of the talk • Generalize our symbolic approach about unfoldings of time Petri nets • Better understand time specificities in a concurrent setting • Safe colored PN with linear real constraints • Concurrent semantics for such nets • Translations of Time PN

  4. Background: PNs • Places P = {a,b,c}, Transitions T = {u,v,w} Consumed (pre(p,t)), read (cont(p,t)) or written (post(p,t)) by transitions • Marking: p M(p)  {0,1}, • initially: M0(a)= M0(b)=1, M0(c)=0 • t fireable iff ppre(t)cont(t), M(p)=1 • Sequential move by firing t: • p, M(p):=M(p)-pre(p,t)+post(p,t)

  5. Why do we need read arcs? • To be able to test the presence of tokens without serialisation

  6. Sequential behaviours: marking graph finite state automaton Exhaustive simulation

  7. Sequential behaviours: marking graph finite state automaton Exhaustive simulation

  8. Sequential behaviours: marking graph finite state automaton Exhaustive simulation

  9. Sequential behaviours: marking graph finite state automaton Exhaustive simulation

  10. Sequential behaviours: marking graph finite state automaton Exhaustive simulation

  11. Concurrent semantics: processes • v and w can be executed concurrently Processes (partially ordered executions):

  12. Unfolding: union of all the processes Prefix (*≤1):

  13. Notion of conflict • fg = (f ≤ g)  (cont(f)  pre(g)  ) • Conflict(F) = •  f,g  F, pre(f)  pre(g)   or •  (fi)i[1,n]  F, fn=f1  i[1,n-1] fi fi+1

  14. Unfolding: the puzzle game

  15. Unfolding: the puzzle game v

  16. Unfolding: the puzzle game w

  17. Unfolding: the puzzle game u

  18. Unfolding: the puzzle game u

  19. Unfolding: the puzzle game v

  20. Unfolding: the puzzle game Maximal co-sets of places correspond to markings -> notion of finite complete prefix -> bounded in space by the size of the marking graph (can be exponentially smaller) -> but the time complexity can be exponential (size of the prefix to the power of the degree of concurrency) w

  21. Representation as a set of events: event structure e=(e,e,Me)

  22. Our Safe Colored PNs • Places P: finite set of real variables • Transitions T: labeled (G(t)) with linear expressions over pre(t)+cont(t)+post(t)’ • Initial expression: ζ0

  23. Concurrent semantics • Set of events: • U={e=(e,e,Ce,Me)} • ⊥=(∅ζ0[x/x⊥]x∈M0, M0) ∈ U • pre(e)  cont(e)  f∈e Mf • Me=post(e) • Ce=G(e)[x/xe]x∈pre(e)cont(e) [x’/xe]x∈post(e) • e is conflict-free f∈e Cf  Ce satisfiable  e∈ U

  24. Unfolding / Process / Trace • Unfolding is the union of processes • Processes are the conflict-free and downward-causally-closed subsets of the unfolding • Linear extensions of processes are the sequential traces • No hope to obtain in general a complete finite prefix

  25. Safe Time PNs • Syntax: • TPN=(P,T,pre,post,efd,lfd,M0) • efd: T|R • lfd: T |R{} • Sequential semantics: • dob: P|R • (M,dob) -t,-> (M’,dob’) iff • - pre(t)M • - maxppre(t) dob(p) + efd(t) ≤  • - t’T, pre(t’)M   ≤ maxppre(t’) dob(p) + lfd(t’) • - maxpPdob(p) ≤  • M’=(M\pre(t))  post(t) • dob’(p)= if ppost(t), dob(p) otherwise

  26. PE(u) = {bc}, PE(v) = {a,ab}, PE(w)={b,ab,bc} Note: Conflict(abw,abv) Conflict(bcw,abv)

  27. TPN to CPN : read arcs are added to take into account the time dependencies -> duplication of transitions -> try to minimize the number of read arcs

  28. Short term perspectives • Experiments • Existence of finite complete prefixes ? OK • Coding of some TPN extensions ? Stopwatches, parametric PNs… • Study a similar approach for networks of Timed Automata. Experiment with different semantics for time.

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