Wadge Degrees of $ω$-Languages of Petri Nets

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 which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal α < 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 . We also prove that there are some Σ 1 1 -complete, hence non-Borel, ω-languages of Petri nets, and that it is consistent with ZFC that there exist some ω-languages of Petri nets which are neither Borel nor Σ 1 1 -complete. This answers the question of the topological complexity of ω-languages of (non-deterministic) Petri nets which was left open in [9,19].

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.…”

“…The first author also proved in [25] that the determinacy of Wadge games between two players in charge of ω-languages of Petri nets is equivalent to the effective analytic (Wadge) determinacy, and thus is not provable in the axiomatic system ZFC.…”

“…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:…”