Abstract
We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let \(\mathcal{B}\) be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving profinite equations characterizing \(\mathcal{B} \cap \operatorname{Reg}\) via ultrafilter equations satisfied by \(\mathcal{B}\). This suffices to establish the decidability of the membership problem for \(\mathcal{B} \cap \operatorname{Reg}\).
Work supported by the project ANR 2010 BLAN 0202 02 FREC.
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
Almeida, J.: Residually finite congruences and quasi-regular subsets in uniform algebras. Portugaliæ Mathematica 46, 313–328 (1989)
Almeida, J.: Finite semigroups and universal algebra. World Scientific Publishing Co. Inc., River Edge (1994), Translated from the 1992 Portuguese original and revised by the author
Barrington, D.A.M., Straubing, H., Thérien, D.: Non-uniform automata over groups. Information and Computation 89, 109–132 (1990)
Chaubard, L., Pin, J.-É., Straubing, H.: First order formulas with modular predicates. In: 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006), pp. 211–220. IEEE (2006)
Gehrke, M., Grigorieff, S., Pin, J.-É.: Duality and equational theory of regular languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008)
Gehrke, M., Grigorieff, S., Pin, J.-É.: A topological approach to recognition. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. Part II. LNCS, vol. 6199, pp. 151–162. Springer, Heidelberg (2010)
McKenzie, P., Thomas, M., Vollmer, H.: Extensional uniformity for Boolean circuits. SIAM J. Comput. 39(7), 3186–3206 (2010)
Pin, J.-É.: Profinite methods in automata theory. In: Albers, S., Marion, J.-Y. (eds.) 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), pp. 31–50. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl (2009)
Pin, J.-É.: Equational descriptions of languages. Int. J. Found. Comput. S. 23, 1227–1240 (2012)
Straubing, H.: Constant-depth periodic circuits. Internat. J. Algebra Comput. 1(1), 49–87 (1991)
Straubing, H.: Finite automata, formal logic, and circuit complexity. Birkhäuser Boston Inc., Boston (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Gehrke, M., Krebs, A., Pin, JÉ. (2014). From Ultrafilters on Words to the Expressive Power of a Fragment of Logic. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2014. Lecture Notes in Computer Science, vol 8614. Springer, Cham. https://doi.org/10.1007/978-3-319-09704-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-09704-6_13
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09703-9
Online ISBN: 978-3-319-09704-6
eBook Packages: Computer ScienceComputer Science (R0)