On the expressive power of nondeterministic and unambiguous Petri nets over infinite words
Abstract
We prove that $\omega$languages of (nondeterministic) Petri nets and $\omega$languages of (nondeterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of $\omega$languages of (nondeterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of $\omega$languages of (nondeterministic) 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 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.04025
 Bibcode:
 2021arXiv210704025F
 Keywords:

 Computer Science  Formal Languages and Automata Theory
 EPrint:
 arXiv admin note: text overlap with arXiv:1712.07945