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Victor Khomenko

Merged Processes of Petri nets. Victor Khomenko. Joint work with Alex Kondratyev, Maciej Koutny and Walter Vogler. Petri net unfoldings. An acyclic net obtained through unfolding the PN by successive firings of transition s:

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Victor Khomenko

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  1. Merged Processes of Petri nets Victor Khomenko Joint work with Alex Kondratyev, Maciej Koutny and Walter Vogler

  2. Petri net unfoldings • An acyclic net obtained through unfoldingthe PN by successive firings of transitions: • for each new firing a fresh transition (called an event) is generated • for eachnewly produced token a fresh place (called a condition) is generated • The full unfolding can beinfinite • If the PN has finitely manyreachable states then the unfolding eventually starts to repeat itself and can be truncated (byidentifying a set of cut-off events) without loss of essential information, yielding a finite prefix

  3. T7 P10 T2 P4 P12 P2 P7 P1 P14 P9 T10 T5 P6 T4 T6 T1 T9 P8 P3 T8 T3 P11 P13 P5 P2 T2 P4 P7 T1 T4 T5 P6 P1 P1 T3 P5 P8 P3 P7 T7 P12 P8 P10 P7 T9 P14 T10 P9 P9 T8 P13 T6 P8 P11 Example: Dining Philosophers

  4. Characteristics of unfoldings • Alleviate the state space explosion problem for highly concurrent systems • e.g. for Dining Philosophers the prefix size is linear in the number of philosophers even though the number of states is exponential • Efficient model checking algorithms • e.g. deadlock checking is PSPACE-complete for safe PNs but only NP-complete for prefixes • Do not cope well with other than concurrency sources of state space explosion, e.g. with sequence of choices • Do not cope well with non-safe PNs

  5. Example: sequence of choices No event is cut-off, the prefix is exponential

  6. Example: non-safe PN m m Tokens in the same place are distinguished in the unfolding, the prefix is exponential

  7. Wanted A data structure coping not only with concurrency but also with other sources of state space explosion

  8. Occurrence-depth 1 1 1 3 2 1 2 1 Merged Process: • Fuse conditions with the same label and occurrence-depth • Delete duplicate events

  9. Example: a Petri net 1 3 2 4

  10. Example: unfolding 3 1 4 3 2 4 Step 1: Fuse conditions of the nodes with the same label and occurrence-depth

  11. Example: (cont’d) 3 1 4 2 3 4 Step 2: Delete event replicas

  12. Examples m m MPs of these nets coincide with the original nets, even though unfoldings are exponential!

  13. Properties of MPs • Canonicity • Finiteness • Completeness • Theoretical upper bounds on size • Experimental results: size

  14. Canonicity • Easily follows from the canonicity of unfolding prefixes: Canonical MP = Merge(Canonical prefix)

  15. … Finiteness Proposition:Merge(Pref) is finite iff Pref is finite •  trivial, as Merge(Pref)is no larger than the prefix •  more difficult, as the Merge operation can collapse infinitely many nodes into one:

  16. Finiteness (cont’d) •  follows from the analog of Köning’s lemma for branching processes: • an infinite branching process contains an infinite causal chain • hence there are infinitely many instances of some place p along it • hence the occurrence-depth of instances of p is unbounded • hence there are infinitely many instances of p in the merged process

  17. Completeness • Preservation of firings is tricky – it’s hard to define cut-offs since an event can have multiple local configurations • Hence consider only marking-completeness (good enough for model checking as the firings can be retrieved from the original PN) Proposition: if Pref is marking-complete then Merge(Pref) is marking-complete

  18. Theoretical upper bounds on size • Trivial bound: Merge(Pref) is never larger than Pref, hence never larger than the reachability graph • too pessimistic in practice • MPs of acyclic PN coincide with the original PNs with the dead nodes removed • unfoldings can be exponential • MPs of live and safe free-choice PNs [with minor restrictions] are polynomial in the size of the original PNs • unfoldings can be exponential

  19. Experimental results: size

  20. Experimental results: PN/MP size

  21. Experimental results: summary • Corbett’s benchmarks were used • MPs are often by orders of magnitude smaller than unfolding prefixes • In many cases MPs are just slightly larger than the original PNs • In some cases MPs are smaller than the original PNs due to removal of dead nodes

  22. Model checking • MPs are small, but are they of any use in practice? • Can model checking algorithms developed for unfoldings be lifted to MPs? • In what follows, we consider safe PNs only

  23. Problem: cycles A Petri net

  24. Problem: cycles 1 1 2 Unfolding Criss-cross fusion results in a cycle! 2 1 1

  25. Problem: cycles MP with a cycle Still worse, the marking equation (ME) used for unfolding-based verification can have spurious solutions

  26. Problem: cycles Fire Borrow a token The borrowed token is returned Fire The current marking is unreachable

  27. Solution • Add to the marking equation another constraint, ACYCLIC, requiring the run to be acyclic: ME & ACYCLIC

  28. Example: an acyclic run

  29. Example: a run with a cycle

  30. SAT encoding • Associate a Boolean variable v to each node v of MP indicating whether it belongs to the run • View the run as a digraph induced in the MP by the variables whose value is true • Sort the nodes of the merged process so that the number of feedback vertices is (heuristically) minimised

  31. SAT encoding (cont’d) v • For each feedback vertex: • ignore the vertices on its left • generate the formula conveying that the sources of the feedback arcs are not reachable from this feedback vertex: • Formula size: O(|Vf|·|E|); can we do better?

  32. Another problem: spurious runs 2 Can visit this condition without first visiting the other one! not possible in the unfolding 1

  33. Solution • Add another constraint, NG (no-gap), conveying that • if a condition with occurrence-depth k>1 is visited then the condition with the same label and occurrence-depth k-1 is also visited • the conditions with the same label are visited in the order of increase of the occurrence depth (can be enforced by ACYCLIC by adding a few arcs)

  34. Solution (cont’d)

  35. Model checking ME & ACYCLIC & NG & VIOL • This is enough to lift unfolding-based model checking algorithms to merged processes! • Deadlock checking (and many other reachability-like problems) is NP-complete in the size of the MP – no worse than for unfoldings

  36. Experimental results: MC time

  37. Experimental results • Corbett’s benchmarks were used • Model checking is practical – running times are comparable with those of an unfolding-based algorithm • Still deteriorates on a couple of benchmarks – but it’s early days of this approach and we keep improving it

  38. Open problems / future work • Direct characterization of MPs (cf. the characterization of unfoldings by occurrence nets) • currently much is done via unfoldings • Improve the efficiency of model checking • the SAT encoding of ACYCLIC is the main problem • A direct algorithm for building MPs • currently built by fusing nodes in the unfolding prefix

  39. Algorithm for building MPs Idea: reduce the problem of finding a possible extension to the following problem: • Find a configuration C in the built part of the MP such that: • C can be extended by a new event and • C contains no cut-offs, i.e. for each event e in C there is no configuration C’ in the built part of MP such that Mark([e]C)=Mark(C’) and C’ [e]C • Reducible to QBF with 1(?) alternation • Reducible to SAT if the adequate order is

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