Abstract
We study tree languages that can be defined in Δ 2. These are tree languages definable by a first-order formula whose quantifier prefix is \(\exists^*\forall^*\), and simultaneously by a first-order formula whose quantifier prefix is \(\forall^*\exists^*\), both formulas over the signature with the descendant relation. We provide an effective characterization of tree languages definable in Δ 2. This characterization is in terms of algebraic equations. Over words, the class of word languages definable in Δ 2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil [11].
Work partially funded by the AutoMathA programme of the ESF, the PHC programme Polonium, and by the Polish government grant no. N206 008 32/0810.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Arfi, M.: Opérations polynomiales et hiérarchies de concaténation. Theor. Comput. Sci. 91(1), 71–84 (1991)
Benedikt, M., Segoufin, L.: Regular tree languages definable in FO and in FO+mod (preliminary version in STACS 2005) (manuscript, 2008)
Bojańczyk, M.: Forest expressions. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 146–160. Springer, Heidelberg (2007)
Bojańczyk, M.: Two-way unary temporal logic over trees. In: Logic in Computer Science, pp. 121–130 (2007)
Bojańczyk, M., Walukiewicz, I.: Characterizing EF and EX tree logics. Theoretical Computer Science 358(2-3), 255–273 (2006)
Bojańczyk, M., Walukiewicz, I.: Forest algebras. In: Automata and Logic: History and Perspectives, pp. 107–132. Amsterdam University Press (2007)
Etessami, K., Vardi, M.Y., Wilke, T.: First-order logic with two variables and unary temporal logic. Inf. Comput. 179(2), 279–295 (2002)
Straubing, H., Bojańczyk, M., Segoufin, L.: Piecewise testable tree languages. In: Logic in Computer Science (2008)
McNaughton, R., Papert, S.: Counter-Free Automata. MIT Press, Cambridge (1971)
Pin, J.-É.: Logic, semigroups and automata on words. Annals of Mathematics and Artificial Intelligence 16, 343–384 (1996)
Pin, J.-É., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Systems 30, 1–30 (1997)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Information and Control 8, 190–194 (1965)
Schwentick, T., Thérien, D., Vollmer, H.: Partially-ordered two-way automata: A new characterization of DA. In: Devel. in Language Theory, pp. 239–250 (2001)
Simon, I.: Piecewise testable events. In: Automata Theory and Formal Languages, pp. 214–222 (1975)
Thérien, D., Wilke, T.: Over words, two variables are as powerful as one quantifier alternation. In: STOC, pp. 256–263 (1998)
Wilke, T.: Classifying discrete temporal properties. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 32–46. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bojańczyk, M., Segoufin, L. (2008). Tree Languages Defined in First-Order Logic with One Quantifier Alternation. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70583-3_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-70583-3_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70582-6
Online ISBN: 978-3-540-70583-3
eBook Packages: Computer ScienceComputer Science (R0)