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FIRST-ORDER AXIOMATISATIONS OF REPRESENTABLE RELATION ALGEBRAS NEED FORMULAS OF UNBOUNDED QUANTIFIER DEPTH

Part of: Algebraic logic Graph theory

Published online by Cambridge University Press:  29 October 2021

ROB EGROT Open the ORCID record for ROB EGROT [Opens in a new window]  and
ROBIN HIRSCH
ROB EGROT
Affiliation:
FACULTY OF INFORMATION AND COMMUNICATION TECHNOLOGY MAHIDOL UNIVERSITY, SALAYA 73170, THAILAND E-mail: robert.egr@mahidol.ac.th
ROBIN HIRSCH
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY COLLEGE LONDON LONDON WC1E 6BT, UK E-mail: r.hirsch@ucl.ac.uk
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Abstract

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

Keywords

RRArepresentable relation algebrasfinite variable axiomatisationbounded quantifier depth axiomatisation

MSC classification

Primary: 03G15: Cylindric and polyadic algebras; relation algebras
Secondary: 05C90: Applications

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1283 - 1300
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Berghammer, R. and Winter, M., Solving computational tasks on finite topologies by means of relation algebra and the RelView tool . Journal of Logical and Algebraic Methods in Programming, vol. 88 (2017), pp. 1–25.CrossRefGoogle Scholar
Chin, L. H. and Tarski, A., Distributive and modular laws in the arithmetic of relation algebras . University of California Publication in Mathematics (New Series), vol. 1 (1951), pp. 341–384.Google Scholar
Egrot, R. and Hirsch, R., A corrected strategy for proving no finite variable axiomatisation exists for RRA, preprint, 2021, https://arxiv.org/abs/2109.01357.Google Scholar
Egrot, R. and Hirsch, R., Seurat games on Stockmeyer graphs . Journal of Graph Theory, in press.Google Scholar
Fletcher, G. H. L., Gyssens, M., Paredaens, J., Van Gucht, D., and Wu, Y., Structural characterizations of the navigational expressiveness of relation algebras on a tree . Journal of Computer and System Sciences, vol. 82 (2016), no. 2, pp. 229–259.CrossRefGoogle Scholar
Givant, S., Tarski’s development of logic and mathematics based on the calculus of relations , Algebraic Logic (H. Andréka, J. Monk, and I. Németi, editors), Colloq. Math. Soc. J. Bolyai., vol. 54, North-Holland, Amsterdam, 1991, pp. 189–215.Google Scholar
Givant, S., The calculus of relations as a foundation for mathematics . Journal of Automated Reasoning, vol. 37 (2006), no. 4, pp. 277–322.CrossRefGoogle Scholar
Guttmann, W., Verifying minimum spanning tree algorithms with Stone relation algebras . Journal of Logical and Algebraic Methods in Programming, vol. 101 (2018), pp. 132–150.CrossRefGoogle Scholar
Hirsch, R., Completely representable relation algebras . Bulletin of the interest group in propositional and predicate logics, vol. 3 (1995), no. 1, pp. 77–92.Google Scholar
Hirsch, R. and Hodkinson, I., Complete representations in algebraic logic , this Journal, vol. 62 (1997), no. 3, pp. 816–847.Google Scholar
Hirsch, R. and Hodkinson, I., Relation algebras from cylindric algebras, II . Annals of Pure and Applied Logic, vol. 112 (2001), pp. 267–297.CrossRefGoogle Scholar
Hirsch, R. and Hodkinson, I., Relation Algebras by Games, North-Holland, Elsevier Science, Amsterdam, 2002.Google Scholar
Hirsch, R., Hodkinson, I., and Maddux, R., Relation algebra reducts of cylindric algebras and an application to proof theory , this Journal, vol. 67 (2002), no. 1, pp. 197–213.Google Scholar
Hodkinson, I. and Venema, Y., Canonical varieties with no canonical axiomatisation. Transactions of the American Mathematical Society, vol. 357 (2005), pp. 4579–4605.CrossRefGoogle Scholar
Immerman, N., Descriptive Complexity, Graduate Texts in Computer Science, Springer-Verlag, New York, 1999.CrossRefGoogle Scholar
Jónsson, B., The theory of binary relations . In Algebraic Logic (H. Andréka, J. Monk, and I. Németi, editors), Colloq. Math. Soc. J. Bolyai., vol. 54, North-Holland, Amsterdam, 1991, pp. 245–292.Google Scholar
Jónsson, B. and Tarski, A., Representation problems for relation algebras . Bulletin of the American Mathematical Society, vol. 54 (1948), no. 80, p. 1192.Google Scholar
Lyndon, R., The representation of relational algebras . Annals of Mathematics, vol. 51 (1950), no. 3, pp. 707–729.CrossRefGoogle Scholar
Maddux, R., Non-finite axiomatizability results for cylindric and relation algebras , this Journal, vol. 54 (1989), no. 3, pp. 951–974.Google Scholar
Maddux, R., The origin of relation algebras in the development and axiomatization of the calculus of relations . Studia Logica, vol. 50 (1991), no. 3–4, pp. 421–455.CrossRefGoogle Scholar
Monk, J., On representable relation algebras . The Michigan Mathematical Journal, vol. 11 (1964), pp. 207–210.CrossRefGoogle Scholar
Tarski, A., On the calculus of relations , this Journal, vol. 6 (1941), pp. 73–89.Google Scholar
Tarski, A., Contributions to the theory of models, I, II . Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, vol. 57 (1954), pp. 572–581 and 582–588 resp.Google Scholar
Tarski, A. and Givant, S., A Formalization of Set Theory Without Variables. Colloquium Publications, vol. 41, American Mathematical Society, Providence, 1987.Google Scholar
Venema, Y.. Atom structures. In M. Kracht, M. D. Rijke, H. Wansing, and M. Zakharyaschev, editors, Advances in Modal Logic ’96, pages 291–305. CSLI Publications, Stanford, 1997.Google Scholar