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 The only proofs are the dierent versions of Reinhardt proof (original unpublished manuscript in 1995 [10], then published in 2008, [11]), which is based on showing that any expression representing a reachability problem can be put under a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...

Abstract. We design a variation of the Karp-Miller algorithm to compute, in a forward manner, a finite representation of the cover (i.e., the downward closure of the reachability set) of a vector addition system with one zero-test. This algorithm yields decision procedures for several problems for these systems, open until now, such as place-boundedness or LTL model-checking. The proof techniques to handle the zero-test are based on two new notions of cover: the refined and the filtered cover. The refined cover is a hybrid between the reachability set and the classical cover. It inherits properties of the reachability set: equality of two refined covers is undecidable, even for usual Vector Addition Systems (with no zero-test), but the refined cover of a Vector Addition System is a recursive set. The second notion of cover, called the filtered cover, is the central tool of our algorithms. It inherits properties of the classical cover, and in particular, one can effectively compute a finite representation of this set, even for Vector Addition Systems with one zero-test.

Abstract. Basic Parallel Processes (BPPs) are a well-known subclass of Petri Nets. They are the simplest common model of concurrent programs that allows unbounded spawning of processes. In the probabilistic version of BPPs, every process generates other processes according to a probability distribution. We study the decidability and complexity of fundamental qualitative problems over probabilistic BPPs -in particular reachability with probability 1 of different classes of target sets (e.g. upward-closed sets). Our results concern both the Markov-chain model, where processes are scheduled randomly, and the MDP model, where processes are picked by a scheduler.