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Olivier Finkel

Publications and source records attributed to Olivier Finkel.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2011

A Hierarchy of Tree-Automatic Structures

We consider $ω^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $ω^n$ for some integer $n\geq 1$. We show that all these structures are $ω$-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $ω^2$-automatic (resp. $ω^n$-automatic for $n>2$) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for $ω^n$-automatic boolean algebras, $n > 1$, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a $Σ_2^1$-set nor a $Π_2^1$-set. We obtain that there exist infinitely many $ω^n$-automatic, hence also $ω$-tree-automatic, atomless boolean algebras $B_n$, $n\geq 1$, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].

math.LO↗
arXiv2011

Some Problems in Automata Theory Which Depend on the Models of Set Theory

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a Büchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter Büchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A)^-$ has cardinal $2^{\aleph_0}$. (3). There is a model $V_3$ of ZFC in which $L(A)^-$ has cardinal $\aleph_1$ with $\aleph_0<\aleph_1<2^{\aleph_0}$. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton $B$. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter omega-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter omega-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter omega-language (respectively, infinitary rational relation) is countable is in $Σ_3^1 \setminus (Π_2^1 \cup Σ_2^1)$. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

cs.FL↗
arXiv2011

Decision Problems for Recognizable Languages of Infinite Pictures

Altenbernd, Thomas and Wöhrle have considered in [ATW02] acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We first review in this paper some recent results of [Fin09b] where we gave the exact degree of numerous undecidable problems for Büchi-recognizable languages of infinite pictures, which are actually located at the first or at the second level of the analytical hierarchy, and "highly undecidable". Then we prove here some more (high) undecidability results. We first show that it is $Π_2^1$-complete to determine whether a given Büchi-recognizable languages of infinite pictures is unambiguous. Then we investigate cardinality problems. Using recent results of [FL09], we prove that it is $D_2(Σ_1^1)$-complete to determine whether a given Büchi-recognizable language of infinite pictures is countably infinite, and that it is $Σ_1^1$-complete to determine whether a given Büchi-recognizable language of infinite pictures is uncountable. Next we consider complements of recognizable languages of infinite pictures. Using some results of Set Theory, we show that the cardinality of the complement of a Büchi-recognizable language of infinite pictures may depend on the model of the axiomatic system ZFC. We prove that the problem to determine whether the complement of a given Büchi-recognizable language of infinite pictures is countable (respectively, uncountable) is in the class $Σ_3^1 \setminus (Π_2^1 \cup Σ_2^1)$ (respectively, in the class $Π_3^1 \setminus (Π_2^1 \cup Σ_2^1)$).

math.LO↗
arXiv2011

Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language

It was noticed by Harel in [Har86] that "one can define $Σ_1^1$-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular $ω$-language is $Σ_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $Π_1^1$-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about $ω$-rational functions realized by finite state Büchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given $ω$-rational function is continuous, while we show here that it is $Σ_1^1$-complete to determine whether a given $ω$-rational function has at least one point of continuity. Next we prove that it is $Π_1^1$-complete to determine whether the continuity set of a given $ω$-rational function is $ω$-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].

cs.LO↗
arXiv2011

Borel Hierarchy and Omega Context Free Languages

We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621], proving new topological properties of omega context free languages : there exist some omega-CFL which are non Borel sets. And one cannot decide whether an omega-CFL is a Borel set. We give also an answer to questions of Niwinski and Simonnet about omega powers of finitary languages, giving an example of a finitary context free language L such that L^omega is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an omega-CFL is an arithmetical set.

cs.LO↗
arXiv2010

The Isomorphism Relation Between Tree-Automatic Structures

An $ω$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $ω$-tree-automatic structures. We prove first that the isomorphism relation for $ω$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $ω$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a $Σ_2^1$-set nor a $Π_2^1$-set.

math.LO↗
arXiv2010

On Infinitary Rational Relations and Borel Sets

We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are Delta^0_4-sets but not (Sigma^0_3U Pi^0_3)-sets. These results give additional answers to questions of Simonnet and of Lescow and Thomas.

cs.LO↗
arXiv2010

On Omega Context Free Languages which are Borel Sets of Infinite Rank

This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621].

cs.LO↗
arXiv2009

The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter Büchi automaton and a 2-tape Büchi automaton such that the ω-language of the first and the infinitary rational relation of the second in one model of ZFC are π_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an ω-language accepted by a 1-counter Büchi automaton or of an infinitary rational relation accepted by a 2-tape Büchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by Büchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author.

cs.LO↗
arXiv2009

On Some Sets of Dictionaries Whose omega-Powers Have a Given Complexity

A dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({\bf\Si}^0_{k})$ (respectively, $W({\bfΠ}^0_{k})$, $W({\bfΔ}_1^1)$) of dictionaries $V \subseteq 2^\star$ whose omega-powers are ${\bf\Si}^0_{k}$-sets (respectively, ${\bfΠ}^0_{k}$-sets, Borel sets). In this paper we first establish a new relation between the sets $W({\bfΣ}^0_{2})$ and $W({\bfΔ}_1^1)$, showing that the set $W({\bfΔ}_1^1)$ is "more complex" than the set $W({\bfΣ}^0_{2})$. As an application we improve the lower bound on the complexity of $W({\bfΔ}_1^1)$ given by Lecomte. Then we prove that, for every integer $k\geq 2$, (respectively, $k\geq 3$) the set of dictionaries $W({\bfΠ}^0_{k+1})$ (respectively, $W({\bf\Si}^0_{k+1})$) is "more complex" than the set of dictionaries $W({\bfΠ}^0_{k})$ (respectively, $W({\bf\Si}^0_{k})$) .

math.LO↗
arXiv2009

Decision Problems For Turing Machines

We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(Σ_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2(Σ_1^1)$ is the class of 2-differences of $Σ_1^1$-sets. Secondly, it is $Σ_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.

cs.LO↗
arXiv2009

On Recognizable Tree Languages Beyond the Borel Hierarchy

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer $n \geq 1$, there is a $D_{ω^n}({\bf Σ}^1_1)$-complete tree language L_n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Büchi tree automaton must be Borel. Then we consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i, k)$. We prove that the $D_{ω^n}({\bf Σ}^1_1)$-complete tree languages L_n are Wadge reducible to the game tree language $W_{(i, k)}$ for $k-i \geq 2$. In particular these languages $W_{(i, k)}$ are not in any class $D_α({\bf Σ}^1_1)$ for $α< ω^ω$.

math.LO↗
arXiv2009

Classical and Effective Descriptive Complexities of omega-Powers

We prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232].

math.LO↗
arXiv2009

On Decidability Properties of One-Dimensional Cellular Automata

In a recent paper Sutner proved that the first-order theory of the phase-space $\mathcal{S}_\mathcal{A}=(Q^\mathbb{Z}, \longrightarrow)$ of a one-dimensional cellular automaton $\mathcal{A}$ whose configurations are elements of $Q^\mathbb{Z}$, for a finite set of states $Q$, and where $\longrightarrow$ is the "next configuration relation", is decidable. He asked whether this result could be extended to a more expressive logic. We prove in this paper that this is actuallly the case. We first show that, for each one-dimensional cellular automaton $\mathcal{A}$, the phase-space $\mathcal{S}_\mathcal{A}$ is an omega-automatic structure. Then, applying recent results of Kuske and Lohrey on omega-automatic structures, it follows that the first-order theory, extended with some counting and cardinality quantifiers, of the structure $\mathcal{S}_\mathcal{A}$, is decidable. We give some examples of new decidable properties for one-dimensional cellular automata. In the case of surjective cellular automata, some more efficient algorithms can be deduced from results of Kuske and Lohrey on structures of bounded degree. On the other hand we show that the case of cellular automata give new results on automatic graphs.

cs.LO↗
arXiv2009

On Recognizable Languages of Infinite Pictures

In a recent paper, Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. The authors asked for comparing the tiling system acceptance with an acceptance of pictures row by row using an automaton model over ordinal words of length $ω^2$. We give in this paper a solution to this problem, showing that all languages of infinite pictures which are accepted row by row by Büchi or Choueka automata reading words of length $ω^2$ are Büchi recognized by a finite tiling system, but the converse is not true. We give also the answer to two other questions which were raised by Altenbernd, Thomas and Wöhrle, showing that it is undecidable whether a Büchi recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable).

cs.LO↗
arXiv2009

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $Π_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $Π_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.

cs.LO↗