Olivier Finkel

Topological properties of omega context-free languages

Abstract This paper is a study of topological properties of omega context-free languages ( ω - CFL ). We first extend some decidability results for the deterministic ones ( ω - DCFL ), proving that one can decide whether an ω - DCFL is in a given Borel class, or in the Wadge class of a given ω -regular language. We prove that ω - CFL exhaust the hierarchy of Borel sets of finite rank, and that one cannot decide the borel class of an ω - CFL , giving an answer to a question of Lescow and Thomas (A Decade of Concurrency, Springer Lecture Notes in Computer Science, vol. 803, Springer, Berlin, 1994, pp. 583–621). We give also a (partial) answer to a question of Simmonet (Automates et theorie descriptive, Ph.D. Thesis, Universite Paris 7, March 1992) about omega powers of finitary languages. We show that Buchi–Landwebers Theorem cannot be extended to even closed ω - CFL : in a Gale–Stewart game with a (closed) ω - CFL winning set, one cannot decide which player has a winning strategy. From the proof of topological properties we derive some arithmetical properties of ω - CFL .

Undecidable problems about timed automata

We solve some decision problems for timed automata which were raised by S. Tripakis in [Tri04] and by E. Asarin in [Asa04]. In particular, we show that one cannot decide whether a given timed automaton is determinizable or whether the complement of a timed regular language is timed regular. We show that the problem of the minimization of the number of clocks of a timed automaton is undecidable. It is also undecidable whether the shuffle of two timed regular languages is timed regular. We show that in the case of timed Buchi automata accepting infinite timed words some of these problems are Π11-hard, hence highly undecidable (located beyond the arithmetical hierarchy).

Borel hierarchy and omega context free languages

We give in this paper additional answers to questions of Lescow and Thomas (A decade of Concurrency, Lecture Notes in Computer Science, Vol. 803, Springer, Berlin, 1994, pp. 583�621), proving topological properties of omega context free languages (�-CFL) which extend those of O. Finkel (Theoret. Comput. Sci. 262 (1�2) (2001) 669�697): there exist some �-CFL which are non Borel sets and one cannot decide whether an �-CFL is a Borel set. We give also an answer to a question of Niwinski (Problem on �-Powers Posed in the Proceedings of the Workshop “Logics and Recognizable Sets, 1990”) and of Simonnet (Automates et Th�orie Descriptive, Ph.D. Thesis, Universit� Paris 7, 1992) about �-powers of finitary languages, giving an example of a finitary context free language L such that L� is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an �-CFL is an arithmetical set. Finally we extend some results to context free sets of infinite trees.

Computer science and the fine structure of Borel sets

(I) Wadge defined a natural refinement of the Borel hierarchy, now called the Wadge hierarchy WH. The fundamental properties of WH follow from the results of Kuratowski, Martin, Wadge and Louveau. We give a transparent restatement and proof of Wadges main theorem. Our method is new for it yields a wide and unexpected extension: from Borel sets of reals to a class of natural but non Borel sets of infinite sequences. Wadges theorem is quite ineffective and our generalization clearly worsens in this respect. Yet paradoxically our method is appropriate to effectivize this whole theory in the context discussed below. (II) Wagner defined on Buchi automata (accepting words of length ω) a hierarchy and proved for it an effective analog of Wadges results. We extend Wagners results to more general kinds of automata: counters, push-down automata and Buchi automata reading transfinite words. The notions and methods developed in (I) are quite useful for this extension, and we start to use them in order to look for extensions of the fundamental effective determinacy results of Buchi–Landweber, Rabin; and of Courcelle–Walukiewicz.

Wadge hierarchy of omega context-free languages

Abstract The main result of this paper is that the length of the Wadge hierarchy of omega context-free languages is greater than the Cantor ordinal e 0 , and the same result holds for the conciliating Wadge hierarchy, defined by Duparc (J. Symbolic Logic, to appear), of infinitary context-free languages, studied by Beauquier (Ph.D. Thesis, Universite Paris 7, 1984). In the course of our proof, we get results on the Wadge hierarchy of iterated counter ω -languages, which we define as an extension of classical (finitary) iterated counter languages to ω -languages.

An Effective Extension of the Wagner Hierarchy to Blind Counter Automata

The extension of the Wagner hierarchy to blind counter automata accepting infinite words with a Muller acceptance condition is effective. We determine precisely this hierarchy.

Undecidability of topological and arithmetical properties of infinitary rational relations

We prove that for every countable ordinal a one cannot decide whether a given infinitary rational relation is in the Borel class Σ 0 α (respectively Π 0 α). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a Σ 1 1-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation.

ON THE TOPOLOGICAL COMPLEXITY OF INFINITARY RATIONAL RELATIONS

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].

Locally finite languages

We investigate properties of locally finite languages introduced by Ressayre (J. Symbolic Logic 53 (4) (1988) 1009–1026). These languages are defined by locally finite sentences and generalize languages recognized by automata or defined by monadic second-order sentences. We give many examples, showing that numerous context free languages are locally finite. Then we study closure properties of the family LOC of locally finite languages, and show that most undecidability results that hold for context free languages may be extended to locally finite languages. In a second part, we consider an extension of these languages to infinite and transfinite length words. We prove that each α-language which is recognized by a Buchi automaton (where α is an ordinal and ω⩽α<ωω) is defined by a locally finite sentence. This result, combined with a preceding one of (Finkel and Ressayre (J. Symbolic Logic 61 (2) (1996) 563–585), provides a generalization of Buchis result about decidability of monadic second-order theory of the structure (α,<).

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 (?-CFL). We proved in (Topological properties of omega context free languages, Theoretical Computer Science, 262 (1?2) (2001) 669?697) that the class of ?-CFL exhausts the finite ranks of the Borel hierarchy, and in (Borel hierarchy and omega context free languages, Theoretical Computer Science, to appear) that there exist some ?-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” (J.W. de Bakker et al. (Eds.), Springer LNCS 803 (1994) 583?621).