Fine Covers of a VAS Language

In recent years much attention has been paid to parallelism and concurrent systems. Petri nets are a formalism which is commonly used for these studies [8]. The theory of Petri nets can be described in the mathematical frame of vector addition systems (VAS). Although these two approaches are equivalent, we formulate our results here in the VAS formalism. Karp and Miller provided in [5] a tool which is adopted by everyone who wants to study Petri nets or VAS: the coverability tree. It is usual to study the language associated to Petri nets or VAS. The coverability tree of Karp and Miller allows us to give a rational approximation of this language: from the coverability tree it is easy to derive a finite automaton, called the coverability automaton, accepting a language that contains the VAS language. Unfortunately this inclusion is not always a strict one. The aim of this note is to provide a constructible refinement of the Karp and Miller automaton: the covering automaton. This automaton recognizes a language included in the one recognized by the coverability automaton, and still containing the language associated to the VAS. This last approximation is the best possible in the sense that if one substitutes to a cycle any finite set of elementary paths then some words of the language are no more accepted by the new automaton.