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Victor Khomenko and Andrey Mokhov

An Algorithm for Direct Construction of Complete Merged Processes. Victor Khomenko and Andrey Mokhov. Rationale. Merged processes (MPs) – a condense representation of the set of reachable states very compact – good to cope with the state space explosion in model checking

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Victor Khomenko and Andrey Mokhov

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  1. An Algorithm for Direct Construction of Complete Merged Processes Victor Khomenko and Andrey Mokhov

  2. Rationale • Merged processes (MPs) – a condense representation of the set of reachable states • very compact – good to cope with the state space explosion in model checking • amenable to efficient model checking • similar to unfoldings, but much smaller (copes not only with concurrency, but also with sequences of choices) • The only known algorithm for constructing MPs was based on merging nodes in the unfoldings • hence cancels all the advantages of MPs • Contribution: an algorithm that avoids the intermediate construction of the unfolding

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

  4. Example: a Petri net 1 3 2 4

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

  6. Example: MP 3 1 4 2 3 4 Step 2: Delete event replicas

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

  8. Properties of MPs • Canonicity, Finiteness, Marking-Completeness – follow from the corresponding properties of unfoldings • Theoretical upper bounds on size • Experimental results: MPs are usually much smaller than unfoldings

  9. 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

  10. Experimental results: PN/Unf/MP size

  11. Experimental results: PN/MP size

  12. 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

  13. Model checking • Model checking algorithms developed for unfoldings can be lifted to MPs • Reduces to SAT: ME & ACYCLIC & NG & VIOL • Still need efficient encoding of ACYCLIC

  14. Unravelling algorithm μ:=the MP comprised of the initial conditions sz := 0 // current configuration size repeat sz++ pe := possible extensions of μ//SAT cand := {e∈pe | e has a local conf of size sz in μ}//SAT // filter out potential cut-offs slice := {e∈cand | ￢MaybeCutOff(μ⊕cand, e, sz)} //2QBF μ:= μ⊕slice untilslice = ∅ ∧ ￢∃e∈pe: e has a local conf of size >sz in μ⊕pe// SAT

  15. Computing the possible extensions • Reduces to model checking (and so to SAT): Find a configuration C enabling a new instance of t

  16. Cut-off criterion // Check if each local conf of e of size sz in μcontains a cut-off MaybeCutOff(μ, e, sz) ≡ // 2QBF ∀ local conf C of e in μsuch that |C|=sz: ∃ f∈C: ∃ conf C’ in μ: Mark([f]C)=Mark(C’) ∧ [f]CC’ • Problem: cannot definitely declare e a cut-off, as it can acquire new configurations as the MP grows • Solution: if configurations are checked in the size order then can detect events that are definitely not cut-offs • All configurations (not only the local ones) are allowed as cut-off correspondents • The adequate order  must refine the size order

  17. Termination criterion • Not trivial! • Check that no possible extension e has a local configuration of size >sz • Reduces to model checking (and so to SAT): Find a configuration C enabling e such that |C|>sz

  18. Age of reductions μ:=the MP comprised of the initial mp-conditions sz := 0 // current configuration size repeat sz++ pe := possible extensions of μ//SAT cand := {e∈pe | e has a local conf of size sz in μ}//SAT // filter out potential cut-offs slice := {e∈cand | ￢MaybeCutOff(μ⊕cand, e, sz)} //2QBF μ:= μ⊕slice untilslice = ∅ ∧ ￢∃e∈pe: e has a local conf of size >sz in μ⊕pe// SAT

  19. Experimental results • A prototype tool was developed • Showed the feasibility of the approach • Loses to unfoldings • Much headroom for improving the tool Back to the future – improvements since the paper: • Significant speedups in the tool • Total adequate order • Comparable with unfoldings • Still much headroom for improving the tool

  20. Future work Potential improvements: • Improving the SAT encoding of the ACYCLIC constraint • Home-brewed 2QBF solver – definitely needs improving • Using incremental SAT wherever possible • Improving the top-level structure of the unravelling algorithm?

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