Abstract
We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associative-commutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative.
Highlights
The purpose of this paper is to study Branching VASS (BVASS), a natural extension of both vector addition systems with states (VASS) and Parikh images of context-free grammars, and to show that emptiness, coverability and boundedness are decidable for this common extension, by extending the usual KarpMiller tree construction
We have studied a natural generalization of both Parikh images of context-free languages and of vector addition systems with states (VASS, special case: Petri nets), where derivations are both two-way and branching
For these so-called branching VASS, we have constructed an analogue of the Karp-Miller coverability tree construction for Petri nets
Summary
The purpose of this paper is to study Branching VASS (BVASS), a natural extension of both vector addition systems with states (VASS) and Parikh images of context-free grammars, and to show that emptiness, coverability and boundedness are decidable for this common extension, by extending the usual KarpMiller tree construction. The fact that BVASS are a natural common generalization of two already well-established tools in computer science – Parikh images of context free languages, and VASS – and that they are useful in domains as diverse as equational tree automata and linear logic, confirms that BVASS are interesting objects to study. This connection between MELL and VATA ( BVASS) generalizes the already known connection between ordinary VASS and the !-Horn fragment of MELL, which was used to obtain decidability result for the latter [Kan95]. Our study of BVASS was initially prompted by certain classes of these automata
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