Abstract
We show that the family of tree languages recognized by weak alternating automata is closed by three set theoretic operations that correspond to sum, multiplication by ordinals < ω ω, and pseudo-exponentiation with the base ω 1 of the Wadge degrees. In consequence, the Wadge hierarchy of weakly recognizable tree languages has the height of at least ε 0, that is the least fixed point of the exponentiation with the base ω.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnold, A., Niwiński, D.: Continuous separation of game languages. Manuscript, 2006 (submitted)
Duparc, J.: Wadge hierarchy and Veblen hierarchy. Part I: Borel sets of finite rank. The Journal of Symbolic Logic 66 (2001)
Duparc, J.: A hierarchy of deterministic context-free ω-languages. Theoret. Comput. Sci. 290, 1253–1300 (2003)
Finkel, O.: Wadge Hierarchy of Omega Context Free Languages. Theoret. Comput. Sci. 269, 283–315 (2001)
Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics 156 (1995)
Muller, D.E., Saoudi, A., Schupp, P.E.: Alternating automata. The weak monadic theory of the tree, and its complexity. In: Kott, L. (ed.) Automata, Languages and Programming. LNCS, vol. 226, pp. 275–283. Springer, Heidelberg (1986)
Murlak, F.: On deciding topological classes of deterministic tree languages. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 428–441. Springer, Heidelberg (2005)
Murlak, F.: The Wadge hierarchy of deterministic tree languages. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 408–419. Springer, Heidelberg (2006)
Niwiński, D., Walukiewicz, I.: A gap property of deterministic tree languages. Theoret. Comput. Sci. 303, 215–231 (2003)
Niwiński, D., Walukiewicz, I.: Deciding nondeterministic hierarchy of deterministic tree automata. In: Proc. WoLLiC 2004. Electronic Notes in Theoret. Comp. Sci., pp. 195–208 (2005)
Perrin, D., Pin, J.-E.: Infinite Words. Automata, Semigroups, Logic and Games. In: Pure and Applied Mathematics, vol. 141, Elsevier, Amsterdam (2004)
Rabin, M.O.: Weakly definable relations and special automata. In: Mathematical Logic and Foundations of Set Theory, North-Holland, pp. 1–70 (1970)
Selivanov, V.: Wadge Degrees of ω-languages of deterministic Turing machines. Theoret. Informatics Appl. 37, 67–83 (2003)
Skurczyński, J.: The Borel hierarchy is infinite in the class of regular sets of trees. Theoret. Comput. Sci. 112, 413–418 (1993)
Urbański, T.F.: On deciding if deterministic Rabin language is in Büchi class. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 663–674. Springer, Heidelberg (2000)
Wagner, K.: Eine topologische Charakterisierung einiger Klassen regulärer Folgenmengen. J. Inf. Process. Cybern. EIK 13, 473–487 (1977)
Wagner, K.: On ω-regular sets. Inform. and Control 43, 123–177 (1979)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duparc, J., Murlak, F. (2007). On the Topological Complexity of Weakly Recognizable Tree Languages. In: Csuhaj-Varjú, E., Ésik, Z. (eds) Fundamentals of Computation Theory. FCT 2007. Lecture Notes in Computer Science, vol 4639. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74240-1_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-74240-1_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74239-5
Online ISBN: 978-3-540-74240-1
eBook Packages: Computer ScienceComputer Science (R0)