On the High Complexity of Petri Nets $$\omega $$-Languages

Abstract

We prove that ω-languages of (non-deterministic) Petri nets and ω-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Turing machines. We also show that it is highly undecidable to determine the topological complexity of a Petri net ω-language. Moreover, we infer from the proofs of the above results that the equivalence and the inclusion problems for ω-languages of Petri nets are Π 1 2-complete, hence also highly undecidable. Keywords: Automata and formal languages • Petri nets • Infinite words • Logic in computer science • Cantor topology • Borel hierarchy • Wadge hierarchy • Wadge degrees • Highly undecidable properties

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

“…Moreover, it is inferred from the proofs of the above results that also the equivalence and the inclusion problems for ω-languages of Petri nets are Π 1 2 -complete, hence also highly undecidable. A particular instance of the above construction of simulation is for the Wadge degrees of analytic sets Σ 1 1 -it follows that there exists a real time 4-blindcounter Büchi automaton recognising a Σ 1 1 -complete ω-language, but also that it is consistent with the axiomatic system ZFC of set theory that there exists some non-Borel non-Σ 1 1 -complete ω-language of 4-blind-counter Büchi automata, [25,28].…”

“…The first author proved in [25,28] that ω-languages of non-deterministic Petri nets and effective analytic sets have the same topological complexity. More precisely the Borel and Wadge hierarchies of the class of ω-languages of Petri nets are equal to the Borel and Wadge hierarchies of the class of effective analytic sets.…”

“…In particular, for each non-null recursive ordinal α ă ω CK 1 there exist some Σ 0 α -complete and some Π 0 α -complete ω-languages of Petri nets, and the supremum of the set of Borel ranks of ω-languages of Petri nets is the ordinal γ 1 2 , which is strictly greater than the first non-recursive ordinal ω CK 1 . Moreover it is proved in [25,28] that it is highly undecidable to determine the topological complexity of a Petri net ω-language. Moreover, it is inferred from the proofs of the above results that also the equivalence and the inclusion problems for ω-languages of Petri nets are Π 1 2 -complete, hence also highly undecidable.…”

“…In the present journal paper we gather results of both authors from [25,28] and [59] with some complements which did not appear in these papers:…”

“…The topological complexity of ω-languages of unambiguous Petri nets may also been compared with the complexity of ω-languages of unambiguous Turing machines: they form the class of effective ∆ 1 1 -sets which contains Σ 0 α -complete sets and Π 0 α -complete sets for each recursive ordinal α ă ω CK 1 [24]. This paper is an extended journal version of both the paper [28] of the first author which appeared in the Proceedings of the 41st International Conference on Application and Theory of Petri Nets and Concurrency, Petri Nets 2020, which took place virtually in Paris on June 2020, and of the paper [59] of the second author which appeared in the Proceedings of the 12th International Conference on Reachability Problems, which took place in Marseille on September 2018.…”