Undecidability of topological and arithmetical properties of infinitary rational relations

Finkel, Olivier

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 2, pp. 115-126.

We prove that for every countable ordinal $\alpha$ one cannot decide whether a given infinitary rational relation is in the Borel class ${\Sigma }_{\alpha }^{\mathbf{0}}$ (respectively ${\Pi }_{\alpha }^{\mathbf{0}}$). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a ${\Sigma }_{\mathbf{1}}^{\mathbf{1}}$-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.

Classification : 68Q45, 03D05, 03D55, 03E15
Mots clés : infinitary rational relations, topological properties, Borel and analytic sets, arithmetical properties, decision problems

[1] M.-P. Béal, O. Carton, C. Prieur and J. Sakarovitch, Squaring Transducers: An Efficient Procedure for Deciding Functionality and Sequentiality of Transducers, in Proc. of LATIN 2000, edited by G. Gonnet, D. Panario and A. Viola. Springer, Lect. Notes Comput. Sci. 1776 (2000) 397-406. | Zbl

[2] J. Berstel, Transductions and Context Free Languages. Teubner Verlag (1979). | MR | Zbl

[3] J.R. Büchi, On a Decision Method in Restricted Second Order Arithmetic, Logic Methodology and Philosophy of Science, in Proc. 1960 Int. Congr. Stanford University Press (1962) 1-11. | MR | Zbl

[4] C. Choffrut, Une Caractérisation des Fonctions Séquentielles et des Fonctions Sous-Séquentielles en tant que Relations Rationnelles. Theoret. Comput. Sci. 5 (1977) 325-338. | MR | Zbl

[5] C. Choffrut and S. Grigorieff, Uniformization of Rational Relations, Jewels are Forever, edited by J. Karhumäki, H. Maurer, G. Paun and G. Rozenberg. Springer (1999) 59-71. | MR | Zbl

[6] O. Finkel, On the Topological Complexity of Infinitary Rational Relations. Theoret. Informatics Appl. 37 (2003) 105-113. | Numdam | MR | Zbl

[7] O. Finkel, On Infinitary Rational Relations and Borel Sets, in Proc. of the Fourth International Conference on Discrete Mathematics and Theoretical Computer Science DMTCS'03, 7-12 July 2003, Dijon, France. Springer, Lect. Notes Comput. Sci. (to appear). | Zbl

[8] C. Frougny and J. Sakarovitch, Synchronized Relations of Finite and Infinite Words. Theoret. Comput. Sci. 108 (1993) 45-82. | MR | Zbl

[9] F. Gire, Relations Rationnelles Infinitaires, Thèse de troisième cycle. Université Paris-7, France (1981).

[10] F. Gire, Une Extension aux Mots Infinis de la Notion de Transduction Rationnelle, in 6th GI Conf. Springer, Lect. Notes Comput. Sci. 145 (1983) 123-139. | Zbl

[11] F. Gire and M. Nivat, Relations Rationnelles Infinitaires. Calcolo XXI (1984) 91-125. | MR | Zbl

[12] F. Gire, Contribution à l'Étude des Langages et Relations Infinitaires, Thèse d'État. Université Paris-7, France (1986).

[13] A.S. Kechris, Classical Descriptive Set Theory. Springer-Verlag (1995). | MR | Zbl

[14] L.H. Landweber, Decision Problems for $\omega$-Automata. Math. Syst. Theory 3 (1969) 376-384. | MR | Zbl

[15] H. Lescow and W. Thomas, Logical Specifications of Infinite Computations, in A Decade of Concurrency, edited by J.W. de Bakker et al. Springer, Lect. Notes Comput. Sci. 803 (1994) 583-621. | MR

[16] Y.N. Moschovakis, Descriptive Set Theory. North-Holland, Amsterdam (1980). | MR | Zbl

[17] D. Perrin and J.-E. Pin, Infinite Words. Book in preparation, available from http://www.liafa.jussieu.fr/jep/InfiniteWords.html

[18] J.-E. Pin, Logic, Semigroups and Automata on Words. Ann. Math. Artificial Intelligence 16 (1996) 343-384. | MR | Zbl

[19] C. Prieur, Fonctions Rationnelles de Mots Infinis et Continuité, Thèse de Doctorat. Université Paris-7, France (2000).

[20] C. Prieur, How to Decide Continuity of Rational Functions on Infinite Words. Theoret. Comput. Sci. 250 (2001) 71-82. | MR | Zbl

[21] P. Simonnet, Automates et Théorie Descriptive, Ph.D. Thesis. Université Paris-7, France (1992). | JFM

[22] L. Staiger, Hierarchies of Recursive $\omega$-Languages. J. Inform. Process. Cybernetics EIK 22 (1986) 219-241. | MR | Zbl

[23] L. Staiger, $\omega$-Languages, Handbook Formal Languages, Vol. 3, edited by G. Rozenberg and A. Salomaa. Springer-Verlag, Berlin (1997). | MR | Zbl

[24] W. Thomas, Automata on Infinite Objects, edited by J. Van Leeuwen. Elsevier, Amsterdam, Handb. Theoret. Comput. Sci. B (1990) 133-191. | MR | Zbl