The Reachability Problem for Vector Addition System with One Zero-Test

Abstract

Abstract. We consider here a variation of Vector Addition Systemswhere one counter can be tested for zero. We extend the reachability proof for Vector Addition System recently published by Leroux to this model. This provides an alternate, more conceptual proof of the reachability problem that was originally proved by Reinhardt. IntroductionContext Petri Nets, Vector Addition Systems (VAS) and Vector Addition System with control states (VASS) are equivalent well known classes of counter systems for which the reachability problem is decidable ([9], [5], [8]). If we add to VAS the ability to test at least two counters to zero, one obtains a model equivalent to Minsky machines, for which all nontrivial properties are undecidable. The study of VAS with a single zerotest transition (VAS0) began recently, and already a reasonable number of results are known for this model. Reinhardt [11] has shown that the reachability problem is decidable for VAS0 (as well as for hierarchical zero-tests). Abdulla and Mayr have shown that the coverability problem is decidable in [1] by using the backward procedure of Well Structured Transition Systems. The boundedness problem (whether the reachability set is nite), the termination and the reversal-boundedness problem (whether the counters can alternate innitely often between the increasing and the decreasing modes) are all decidable by using a forward procedure, a nite but non-complete Karp and Miller tree provided by Finkel and Sangnier in [4]. The decidability of the place-boundedness problem (whether one specic counter is unbounded), and more generally the possibility to compute a nite representation of the downward closure of the reachability set have been shown by Bonnet, Finkel, Leroux and Zeitoun in [3] using the notion of productive sequences.The reachability problem The decidability of reachability for VAS has been originally solved by Mayr (1981, [9]) and Kosaraju (1982, [5]). Lambert later simplied these proofs (1992, [6]) while still using the same proof techniques. Recently, Leroux gave another way to prove of Supported by the Agence Nationale de la Recherche, AVERISS (grant ANR-06-SETIN-001) this problem, by using Presburger invariants and productive sequences ([7], [8]). The history of the reachability problem for VAS0 is shorter. The only proofs are the dierent versions of Reinhardt proof (original unpublished manuscript in 1995 [10], then published in 2008, [11]), which is based on showing that any expression representing a reachability problem can be put under a "normal form" for which satisability is easy to solve. However, the denition of the normal form is complex, and the proof of termination of the algorithm reducing any expression to the normal form is dicult to understand. Since this publication, some new results were found by reduction to reachability in VAS0, for example decidability of minimal cost reachability in the Priced Timed Petri Nets of Abdulla and Mayr [1], and the decidability of reachability in a restricited class of pushdown counter automa...

Priced Timed Petri Nets

“…Definition 7.1. Petri nets with one inhibitor arc [Rei08,Bon11] are an extension of Petri nets. They contain a special inhibitor arc that prevents a certain transition from firing if a certain place is nonempty.…”

“…In Section 7, we instantiate this abstract method with AC-PTPN and prove the main result. This instantiation is nontrivial and requires several auxiliary lemmas, which ultimately use the decidability of the reachability problem for Petri nets with one inhibitor arc [Rei08,Bon11]. There exist close connections between timed Petri nets, Petri nets with one inhibitor arc, and transfer nets.…”

Undecidability of performance equivalence of Petri nets

“…Investigation of this question is a natural continuation of the work presented in this paper. We suspect that there may be a reduction from durational nets to nets with hierarchical inhibitor arcs, a model with decidable reachability problem according to [20] (see also [2] for a self-contained proof of reachabilty for nets with one zero test).…”

On Selective Unboundedness of VASS

“…It is shown in [21] that checking whether an initialized VASS with one zero-test is reversal-bounded is decidable, but with a nonprimitive recursive worst-case complexity, the existence of an ExpSpace upper bound being open; see also recent results on VASS with one zero-test [6,7].…”