On the expressive power of non-deterministic and unambiguous Petri nets over infinite words

TL;DR

This paper explores the topological complexity of omega-languages recognized by non-deterministic Petri nets and Turing machines, revealing high undecidability results and contrasting behaviors for unambiguous Petri nets.

Contribution

It establishes the equivalence in topological complexity between Petri nets and Turing machines, and introduces a determinisation procedure for unambiguous Petri nets.

Findings

Petri net omega-languages have the same topological complexity as Turing machine omega-languages.

Determining the topological complexity of Petri net omega-languages is highly undecidable.

Unambiguous Petri nets recognize omega-languages that are Delta^0_3 sets.

Abstract

We prove that $\omega$-languages of (non-deterministic) Petri nets and $\omega$-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of $\omega$-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of $\omega$-languages of (non-deterministic) Turing machines. We also show that it is highly undecidable to determine the topological complexity of a Petri net $\omega$-language. Moreover, we infer from the proofs of the above results that the equivalence and the inclusion problems for $\omega$-languages of Petri nets are $\Pi_2^1$-complete, hence also highly undecidable. Additionally, we show that the situation is quite the opposite when considering unambiguous Petri nets, which have the semantic property that at most one accepting run exists on every input. We provide a procedure of determinising them into deterministic Muller counter machines with counter copying. As a consequence, we entail that the $\omega$-languages recognisable by unambiguous Petri nets are $\Delta^0_3$ sets.