Commodification of accelerations for the Karp and Miller Construction

• Published in: Discrete event dynamic systems Vol. 31; no. 2; pp. 251 - 270
• Main Authors: Finkel, Alain, Haddad, Serge, Khmelnitsky, Igor
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Computer Science; Control; Convex and Discrete Geometry; Electrical Engineering; Machines; Manufacturing; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Petri nets; Processes; Systems Theory; Topical Collection on Recent Trends in Reactive Systems; Petri nets; Verification tool; Coverability; Minimal coverability set; Minimal coverability tree; Karp-Miller tree algorithm; Clover
• ISSN: 0924-6703; 1573-7594
• DOI: 10.1007/s10626-020-00331-z

Karp and Miller’s algorithm is based on an exploration of the reachability tree of a Petri net where, the sequences of transitions with positive incidence are accelerated. The tree nodes of Karp and Miller are labeled with ω -markings representing (potentially infinite) coverability sets. This set of ω -markings allows us to decide several properties of the Petri net, such as whether a marking is coverable or whether the reachability set is finite. The edges of the Karp and Miller tree are labeled by transitions but the associated semantic is unclear which yields to a complex proof of the algorithm correctness. In this work we introduce three concepts: abstraction, acceleration and exploration sequence. In particular, we generalize the definition of transitions to ω -transitions in order to represent accelerations by such transitions. The notion of abstraction makes it possible to greatly simplify the proof of the correctness. On the other hand, for an additional cost in memory, which we theoretically evaluated, we propose an “accelerated” variant of the Karp and Miller algorithm with an expected gain in execution time. Based on a similar idea we have accelerated (and made complete) the minimal coverability graph construction, implemented it in a tool and performed numerous promising benchmarks issued from realistic case studies and from a random generator of Petri nets.