1. Victor Khomenko and Andrey Mokhov
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

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
until slice = ∅ ∧
￢∃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 μ contains a cut-off
MaybeCutOff(μ, e, sz) ≡ // 2QBF
∀ local conf C of e in μ such 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
until slice = ∅ ∧
￢∃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?