Hostname: page-component-cb9f654ff-5jtmz Total loading time: 0 Render date: 2025-08-19T17:35:11.385Z Has data issue: false hasContentIssue false
  • English
  • Français

Undecidable relativizations of algebras of relations

Published online by Cambridge University Press:  12 March 2014

Szabolcs Mikulás  and
Maarten Marx
Szabolcs Mikulás
Affiliation:
Department of Computer Science, King's College London, E-mail: szabolcs@dcs.kcl.ac.uk
Maarten Marx
Affiliation:
Department of Wins, University of Amsterdam, E-mail: marx@wins.uva.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 747 - 760
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[AHN] Andréka, H., Hodkinson, I., and Németi, I., Finite algebras of relations are representable on finite sets, this Journal, to appear.Google Scholar
[GOR] Grädel, E., Otto, M., and Rosen, E., Two-variable logic with counting is decidable, Proceedings of 12th IEEE Symposium on Logic in Computer Science LICS'97 (Warsaw), 1997.Google Scholar
[HMT] Henkin, L., Monk, J.D., and Tarski, A., Cylindric Algebras I, II, North-Holland, Amsterdam, 1971, 1985.Google Scholar
[HMTAN] Henkin, L., Monk, J.D., Tarski, A., Andréka, H., and Németi, I., Cylindric Set Algebras, Lecture Notes in Mathematics, vol. 883, Springer Verlag, Berlin, 1981.Google Scholar
[HH97] Hirsch, R. and Hodkinson, I., Representability is not decidable for finite relation algebras, Transactions of the AMS, to appear.Google Scholar
[Ma82] Maddux, R.D., Some varieties containing relation algebras, Transactions of the AMS, vol. 272 (1982), pp. 501–526.Google Scholar
[Ma95] Marx, M., Algebraic Relativization and Arrow Logic, Ph.D. dissertation , ILLC dissertation series 1995–3, University of Amsterdam, 1995.Google Scholar
[Ma97] Marx, M., Complexity of modal logics of relations, ILLC research report and technical notes series ML-97-02, University of Amsterdam, 1997.Google Scholar
[MMN] Marx, M., Mikulás, Sz., and Németi, I., Taming logic, Journal of Logic, Language, and Information, vol. 4 (1995), pp. 207–226.Google Scholar
[MMP] Marx, M., Pólos, L., and Masuch, M. (editors), Arrow Logic and Multimodal Logics, Studies in Logic, Language and Information, FoLLI and CSLI Publications, 1996.Google Scholar
[Mi95] Mikulás, Sz., Taming Logics, Ph.D. dissertation , ILLC dissertation series 1995–12, University of Amsterdam, 1995.Google Scholar
[Mi97] Mikulás, Sz., A note on expressing infinity in cylindric-relativized set algebras, Proceedings of RelMiCS'97 (Tunisia), 1997.Google Scholar
[Mi98] Mikulás, Sz., Taming first-order logic, Journal of the IGPL, vol. 6 (1998), no. 2, pp. 305–316.Google Scholar
[Mo93] Monk, J.D., Lectures on cylindric set algebras, Algebraic Methods in Logic and in Computer Science (Rauszer, C., editor), Banach Center Publications, vol. 28, Polish Academy of Sciences, 1993, pp. 253–290.Google Scholar
[Né86] Németi, I., Free Algebras and Decidability in Algebraic Logic, Dissertation with the Hungarian Academy of Sciences, 1986.Google Scholar
[Né91] Németi, I., Algebraization of quantifier logics, an introductory overview, Studia Logica, vol. 50 (1991), pp. 485–569, updated version available from Mathematical Institute, Budapest.Google Scholar
[Né95] Németi, I., Decidable versions of first order logic and cylindric-relativized set algebras, Logic Colloquium '92 (Csirmaz, L., Gabbay, D.M., and de Rijke, M., editors), Studies in Logic, Language and Information, FoLLI and CSLI Publications, 1995, pp. 177–241.Google Scholar
[Ro71] Robinson, R.M., Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, vol. 12 (1971), pp. 177–209.Google Scholar
[TG87] Tarski, A. and Givant, S., A Formalization of Set Theory without Variables, AMS Colloquium Publications, vol. 41 (1987).Google Scholar
[vB96] van Benthem, J., Exploring Logic Dynamics, Studies in Logic, Language and Information, FoLLI and CSLI Publications, 1996.Google Scholar