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_1CK there exist some Σ0_α-complete and some Π0_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_21, which is strictly greater than the first non-recursive ordinal
ω_1CK. We also prove that there are some Σ_11-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_11-complete. This
answers the question of the topological complexity of ω-languages of
(non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326