On the Upward/Downward Closures of Petri Nets

TL;DR

This paper investigates the computational complexity of automata recognizing upward and downward closures of Petri net languages, providing bounds, algorithms, and decidability results for general Petri nets and BPP subclasses.

Contribution

It introduces complexity bounds and algorithms for constructing finite automata for upward/downward closures of Petri nets, including special cases like BPP nets, and establishes decidability of closure properties.

Findings

Upward closure automata can be constructed in doubly-exponential time.

Downward closure automata may have non-primitive recursive size.

Deciding language inclusion in closures is EXPSPACE-complete for Petri nets and NP-complete for BPP nets.

Abstract

We study the size and the complexity of computing finite state automata (FSA) representing and approximating the downward and the upward closure of Petri net languages with coverability as the acceptance condition. We show how to construct an FSA recognizing the upward closure of a Petri net language in doubly-exponential time, and therefore the size is at most doubly exponential. For downward closures, we prove that the size of the minimal automata can be non-primitive recursive. In the case of BPP nets, a well-known subclass of Petri nets, we show that an FSA accepting the downward/upward closure can be constructed in exponential time. Furthermore, we consider the problem of checking whether a simple regular language is included in the downward/upward closure of a Petri net/BPP net language. We show that this problem is EXPSPACE-complete (resp. NP-complete) in the case of Petri nets (resp. BPP nets). Finally, we show that it is decidable whether a Petri net language is upward/downward closed. To this end, we prove that one can decide whether a given regular language is a subset of a Petri net coverability language.