When Reachability Meets Grzegorczyk

Abstract

Vector addition systems with states, or equivalently vector addition systems, or Petri nets are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem is reachability: whether from a given initial configuration there exists a sequence of valid execution steps that reaches a given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of computation. In this paper, we survey results about the reachability problem focusing on the general problem. We also show how a recent paper about the reachability problem in fixed dimension combined with vector addition systems with states weakly computing Grzegorczyk hierarchy provides a logspace reduction of the general reachability problem to the bounded case. This result, not included in the original paper due to a lack of space shows that the reachability problem can obviously be decided by a deterministic brute-force exploration. We provide perspectives based on this observation.