Wadge Degrees of $\omega$-Languages of Petri Nets

TL;DR

This paper establishes that the topological complexity of omega-languages of Petri nets matches that of Turing machines, revealing their position within the Borel and Wadge hierarchies and identifying their maximal complexity.

Contribution

It proves the equivalence of topological hierarchies for Petri net and Turing machine omega-languages, extending understanding of their complexity and completeness levels.

Findings

Omega-languages of Petri nets reach the ordinal b3_2^1 in Borel hierarchy.

Existence of non-Borel a1_1^1-complete omega-languages of Petri nets.

The topological complexity of Petri net omega-languages matches that of Turing machines.

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 which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal $\alpha < \omega\_1^{{\rm CK}} $ there exist some ${\bf \Sigma}^0\_\alpha$-complete and some ${\bf \Pi}^0\_\alpha$-complete $\omega$-languages of Petri nets, and the supremum of the set of Borel ranks of $\omega$-languages of Petri nets is the ordinal $\gamma\_2^1$, which is strictly greater than the first non-recursive ordinal $\omega\_1^{{\rm CK}}$. We also prove that there are some ${\bf \Sigma}\_1^1$-complete, hence non-Borel, $\omega$-languages of Petri nets, and that it is consistent with ZFC that there exist some $\omega$-languages of Petri nets which are neither Borel nor ${\bf \Sigma}\_1^1$-complete. This answers the question of the topological complexity of $\omega$-languages of (non-deterministic) Petri nets which was left open in [DFR14,FS14].