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Olivier Finkel

Publications and source records attributed to Olivier Finkel.

At least 19 recordsLinked to original sources
arXiv2026

On the Complexity of the Bi-infinite Post Correspondence Problem

In the bi-infinite Post Correspondence Problem ($\Z$PCP), it is asked whether the same bi-infinite word can be constructed correspondingly from a given finite set of pairs of words. In this article, we study its complexity with respect to the arithmetical hierarchy and prove that it is in $\Si^0_2 \setminus (Π^0_1 \cup \Si^0_1)$ and, therefore, at the level 2 of the arithmetical hierarchy. For the proof, we present a sequence of reductions starting from the nonhalting of the Turing machine all the way to $\Z$PCP via infinite PCP, an $s$-shift infinite PCP and $s$-shift $\Z$PCP for all natural numbers $s$. In the process, we prove that the infinite PCP is undecidable for injective morphisms, and that the infinite injective PCP, $s$-shift infinite PCP, $s$-shift $\Z$PCP and the non-termination problem for (deterministic and reversible) semi-Thue systems are all $Π^0_1$-complete.

cs.FL↗
arXiv2025

Toward higher-order infinite time Turing machines: simulational $Γ$-machines

Infinite time Turing machines (ITTMs) have been introduced by Hamkins and Lewis in their seminal article arXiv:math/9808093. The strength of the model comes from a limit rule which allows the ITTM to compute through ordinal stages. This rule is simple to describe and another rule would lead to a different model of ordinal computation. The aim of this article is to define a collection of limit rules for which the models of infinite Turing machine they induce have nice properties, akin to that of the ITTM. Through the analysis of the universal ITTM and of its preponderant role in the study of the ITTM, we devise a set of four constraints. A limit rule satisfying those constraints yields a model of infinite machine for which we can define a universal machine. Adding a fifth constraint, we show that any limit rule definable in set theory which meets those constraints produces a model of infinite machine with the desired properties. Among those, the fact that the supremum of the writable ordinal matches that of the clockable ordinal. That is, with the usual notations, the equality $λ= γ$ holds for any of those limit rules. Eventually, we provide a counter-example to show that the four first constraints alone are not sufficient.

math.LO↗
arXiv2024

Wadge degrees of $Δ^0_2$ omega-powers

We provide, for each natural number $n$ and each class among $D_n(Σ^0_1)$, $\bar D_n(Σ^0_1)$ and $D_{2n+1}(Σ^0_1)\oplus\bar D_{2n+1}(Σ^0_1)$, a regular language whose associated omega-power is complete for this class.

cs.LO↗
arXiv2022

On Bi-infinite and Conjugate Post Correspondence Problems

We study two modifications of the Post Correspondence Problem (PCP), namely 1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and 2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the bi-infinite PCP we show that it is in the class $Σ_2^0$ of the arithmetical hierarchy and for the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems.

cs.DM↗
arXiv2021

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

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 $Π_2^1$-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 accepting 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 $Δ^0_3$ sets.

cs.FL↗
arXiv2020

Descriptive Set Theory and $ω$-Powers of Finitary Languages

The $ω$-power of a finitary language L over a finite alphabet $Σ$ is the language of infinite words over $Σ$ defined by L $\infty$ := {w 0 w 1. .. $\in$ $Σ$ $ω$ | $\forall$i $\in$ $ω$ w i $\in$ L}. The $ω$-powers appear very naturally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{ü}chi automata or B{ü}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $ω$-powers.

cs.LO↗
arXiv2019

Polishness of some topologies related to word or tree automata

We prove that the Büchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the Büchi and the Muller topologies are not Polish in this case.

math.LO↗
arXiv2018

An Effective Property of $ω$-Rational Functions

We prove that $ω$-regular languages accepted by Büchi or Muller automata satisfy an effective automata-theoretic version of the Baire property. Then we use this result to obtain a new effective property of rational functions over infinite words which are realized by finite state Büchi transducers: for each such function $F: Σ^ω\rightarrow Γ^ω$, one can construct a deterministic Büchi automaton $\mathcal{A}$ accepting a dense ${\bf Π}^0_2$-subset of $Σ^ω$ such that the restriction of $F$ to $L(\mathcal{A})$ is continuous.

math.LO↗
arXiv2018

Wadge Degrees of $ω$-Languages of Petri Nets

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\_1^{\rm CK} $ there exist some ${\bf Σ}^0\_α$-complete and some ${\bf Π}^0\_α$-complete $ω$-languages of Petri nets, and the supremum of the set of Borel ranks of $ω$-languages of Petri nets is the ordinal $γ\_2^1$, which is strictly greater than the first non-recursive ordinal $ω\_1^{\rm CK}$. We also prove that there are some ${\bf Σ}\_1^1$-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 ${\bf Σ}\_1^1$-complete. This answers the question of the topological complexity of $ω$-languages of (non-deterministic) Petri nets which was left open in [DFR14,FS14].

cs.LO↗
arXiv2015

An Upper Bound on the Complexity of Recognizable Tree Languages

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bfΣ}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^ω$ into the class ${\bfΔ}^1\_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bfΔ}^1\_2$.

cs.FL↗
arXiv2014

Ambiguity of ω-Languages of Turing Machines

An ω-language is a set of infinite words over a finite alphabet X. We consider the class of recursive ω-languages, i.e. the class of ω-languages accepted by Turing machines with a Büchi acceptance condition, which is also the class Σ11 of (effective) analytic subsets of Xω for some finite alphabet X. We investigate here the notion of ambiguity for recursive ω-languages with regard to acceptance by Büchi Turing machines. We first present in detail essentials on the literature on ω-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambiguity of Büchi Turing machines and of the ω-languages they accept. To obtain our new results, we make use of results and methods of effective descriptive set theory.

cs.LO↗
arXiv2014

On the Topological Complexity of omega-Languages of Non-Deterministic Petri Nets

We show that there are $Σ_3^0$-complete languages of infinite words accepted by non-deterministic Petri nets with Büchi acceptance condition, or equivalently by Büchi blind counter automata. This shows that omega-languages accepted by non-deterministic Petri nets are topologically more complex than those accepted by deterministic Petri nets.

cs.LO↗
arXiv2013

Infinite Games Specified by 2-Tape Automata

We prove that the determinacy of Gale-Stewart games whose winning sets are infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. Then we prove that winning strategies, when they exist, can be very complex, i.e. highly non-effective, in these games. We prove the same results for Gale-Stewart games with winning sets accepted by real-time 1-counter Büchi automata, then extending previous results obtained about these games. Then we consider the strenghs of determinacy for these games, and we prove that there is a transfinite sequence of 2-tape Büchi automata (respectively, of real-time 1-counter Büchi automata) $A_α$, indexed by recursive ordinals, such that the games $G(L(A_α))$ have strictly increasing strenghs of determinacy. Moreover there is a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) B such that the determinacy of G(L(B)) is equivalent to the (effective) analytic determinacy and thus has the maximal strength of determinacy. We show also that the determinacy of Wadge games between two players in charge of infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the (effective) analytic determinacy, and thus not provable in ZFC.

cs.LO↗
arXiv2013

The Determinacy of Context-Free Games

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(A), L(B)).

cs.LO↗
arXiv2013

Topological Complexity of Context-Free omega-Languages: A Survey

We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of ambiguity, and the special case of omega-powers.

cs.LO↗
arXiv2012

Automatic Ordinals

We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than $ω^{ω^ω}$. Then we show that the injectively $ω^n$-automatic ordinals, where $n>0$ is an integer, are the ordinals smaller than $ω^{ω^n}$. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word $ω^n$-automatic ordinals. As a by-product we obtain that the hierarchy of injectively $ω^n$-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.

math.LO↗
arXiv2011

The Determinacy of Context-Free Games

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(A), L(B)).

cs.GT↗