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Constructing Minimal Coverability Sets

Piipponen A., Valmari A.

Fundamenta Informaticae

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2016

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Vol. 143, nr 3/4

393--414

EN

This publication addresses two bottlenecks in the construction of minimal coverability sets of Petri nets: the detection of situations where the marking of a place can be converted to ω, and the manipulation of the set A of maximal ω-markings that have been found so far. For the former, a technique is presented that consumes very little time in addition to what maintaining A consumes. It is based on Tarjan’s algorithm for detecting maximal strongly connected components of a directed graph. For the latter, a data structure is introduced that resembles BDDs and Covering Sharing Trees, but has additional heuristics designed for the present use. Results from a few experiments are shown. They demonstrate significant savings in running time and varying savings in memory consumption compared to an earlier state-of-the-art technique.

2

Old and New Algorithms for Minimal Coverability Sets

Valmari A., Hansen H.

Fundamenta Informaticae

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2014

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Vol. 131, nr 1

1--25

EN

Many algorithms for computing minimal coverability sets for Petri nets prune futures. That is, if a newmarking strictly covers an old one, then not just the old marking but also some subset of its successor markings is discarded from search. In this publication, a simpler algorithm that lacks future pruning is presented and proven correct. Its performance is compared with future pruning. It is demonstrated, using examples, that neither approach is systematically better than the other. However, the simple algorithm has some attractive features. It never needs to re-construct pruned parts of the minimal coverability set. It automatically gives most of the advantage of future pruning, if the minimal coverability set is constructed in depth-first or most tokens first order, and if so-called history merging is applied. Some implementation aspects of minimal coverability set construction are also discussed. Some measurements are given to demonstrate the effect of construction order and other implementation aspects.

3

Simple Bisimilarity Minimization in O(m log n) Time

Valmari A.

Fundamenta Informaticae

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2011

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Vol. 105, nr 3

319-339

EN

A new algorithm for bisimilarity minimization of labelled directed graphs is presented. Its time consumption is O(mlog n), where n is the number of states and m is the number of transitions. Unlike earlier algorithms, it meets this bound even if the number of different labels of transitions is not fixed. It is based on refining a partition of states with respect to the labelled transitions. A splitter is a pair consisting of a label and a set in the partition. A transition belongs to a splitter, if and only if it has the same label and its end state is in the set. Earlier algorithms consume lots of time in scanning splitters that have no transitions. The new algorithm avoids this by maintaining also a partition of transitions. To facilitate this, a refinable partition data structure with amortized constant time operations is used. Detailed pseudocode of all nontrivial details is presented. Novel simplifications are applied that both reduce the practical memory consumption and shorten the program code.

4

Can Stubborn Sets Be Optimal?

Valmari A., Hansen H.

Fundamenta Informaticae

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2011

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Vol. 113, nr 3/4

377-397

EN

Literature on the stubborn set and similar state space reduction methods presents numerous seemingly ad-hoc conditions for selecting the transitions that are investigated in the current state. There are good reasons to believe that the choice between them has a significant effect on reduction results, but not much has been published on this topic. This article presents theoretical results and examples that aim at shedding light on the issue. Because the topic is extensive, we only consider the detection of deadlocks. We distinguish between different places where choices can be made and investigate their effects. It is usually impractical to aim at choices that are "best" in some sense. However, one non-trivial practical optimality result is proven.