Structural liveness of Petri nets is ExpSpace-hard and decidable
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Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is verification of safety and liveness properties in this model; despite an extensive research effort, some basic problems remain open, which is exemplified by the complexity status of the reachability problem that is still not fully clarified. The liveness problems are known to be closely related to the reachability problem, and various structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem of determining whether a net is structurally live, i.e. whether there is an initial marking for which it is live, remained open for some time; e.g. Best and Esparza (Inf Process Lett 116(6):423–427, 2016. https://doi.org/10.1016/j.ipl.2016.01.011) emphasize this open question. Here we show that the structural liveness problem for Petri nets is ExpSpace-hard and decidable. In particular, given a net N and a semilinear set S, it is decidable whether there is an initial marking of N for which the reachability set is included in S; this is based on results by Leroux (28th annual ACM/IEEE symposium on logic in computer science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, IEEE Computer Society, pp 23–32, 2013. https://doi.org/10.1109/LICS.2013.7).
Both authors thank the anonymous reviewers for their helpful comments, and Dmitry Chistikov for discussions initiating their cooperation when meeting at the University of Warwick. P. Jančar thanks Eike Best for a discussion about the open decidability status of the structural liveness problem, and acknowledges funding by the Grant Agency of the Czech Republic (18-11193S). D. Purser acknowledges funding by the UK Engineering and Physical Sciences Research Council (EP/L016400/1), the EPSRC Centre for Doctoral Training in Urban Science.
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