On the topological complexity of ω-languages of non-deterministic Petri nets

Finkel, Olivier; Skrzypczak, Michał

doi:10.1016/j.ipl.2013.12.007

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…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].…”

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confidence: 59%

“…They are ∆ 0 3 -sets and their Wadge hierarchy has been determined by Duparc, Finkel and Ressayre in [9]; its length is the ordinal ω ω 2 . On the other side, Finkel and Skrzypczak proved in [19] that there exist Σ 0 3 -complete, hence non ∆ 0 3 , ω-languages accepted by non-deterministic one-partially-blind-counter Büchi automata. But, up to our knowledge, this was the only known result about the topological complexity of ω-languages of non-deterministic Petri nets.…”

Section: Introductionmentioning

confidence: 99%

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Wadge Degrees of $ω$-Languages of Petri Nets

Finkel¹

2017

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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].

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confidence: 59%

Section: Introductionmentioning

confidence: 99%

Wadge Degrees of $ω$-Languages of Petri Nets

Finkel¹

2017

Preprint

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“…They are ∆ 0 3 -sets and their Wadge hierarchy has been determined by Duparc, Finkel, and Ressayre in [11]; its length is the ordinal ω ω 2 . On the other side, Finkel and Skrzypczak proved in [27] that there exist Σ 0 3 -complete, hence non ∆ 0 3 , ω-languages accepted by non-deterministic one-partially-blind-counter Büchi automata.…”

Section: Introductionmentioning

confidence: 99%

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

Finkel,

Skrzypczak

2021

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We prove that ω-languages of (non-deterministic) Petri nets and ωlanguages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ωlanguages 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.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 ω-languages recognisable by unambiguous Petri nets are ∆ 0 3 sets.

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“…3 -sets and their Wadge hierarchy has been determined by Duparc, Finkel and Ressayre in [7]; its length is the ordinal ω ω 2 . On the other side, Finkel and Skrzypczak proved in [18] that there exist Σ 0 3 -complete, hence non Δ 0 3 , ω-languages accepted by non-deterministic onepartially-blind-counter Büchi automata. The existence of a Σ 1 1 -complete, hence non Borel, ω-language accepted by a Petri net was independently proved by Finkel and Skrzypczak in [17,34].…”

Section: Introductionmentioning

confidence: 99%

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

Finkel

2020

Application and Theory of Petri Nets and Concurrency

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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

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On the topological complexity of ω-languages of non-deterministic Petri nets

Cited by 6 publications

References 18 publications

Wadge Degrees of $ω$-Languages of Petri Nets

Wadge Degrees of $ω$-Languages of Petri Nets

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

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

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