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Step by step – Building representations in algebraic logic

Published online by Cambridge University Press:  12 March 2014

Robin Hirsch  and
Ian Hodkinson
Robin Hirsch
Affiliation:
Department of Computer Science, University College, Gower Street, London WC1, UK
Ian Hodkinson
Affiliation:
Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK
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Abstract

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.

An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.

Other instances of this approach are looked at, and include the step by step method.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997 , pp. 225 - 279
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1] Allen, J. F., Maintaining knowledge about temporal intervals, Communications of the ACM, vol. 26 (1983), no. 11, pp. 832–843.CrossRefGoogle Scholar
[2] Andréka, H., Weakly representable but not representable relation algebras, Algebra Universalis, vol. 32 (1994), pp. 31–43.CrossRefGoogle Scholar
[3] Andréka, H., Düntsch, I., and Németi, I., A nonpermutational integral relation algebra, Michigan Mathematics Journal, vol. 39 (1992), pp. 371–384.CrossRefGoogle Scholar
[4] Andréka, H., Monk, J. D., and Németi, I., Algebraic logic, North-Holland, 1991.Google Scholar
[5] Andréka, H. and Thompson, R. J., A Stone type representation theorem for algebras of relations of higher rank, Transactions of the American Mathematical Society, vol. 309 (1988), no. 2, pp. 671–682.Google Scholar
[6] Barwise, J. and Robinson, A., Completing theories by forcing, Annals of Mathematical Logic, vol. 2 (1970), pp. 119–142.CrossRefGoogle Scholar
[7] Blackburn, P., Reasoning in and about time, Ph.D. thesis, Centre for Cognitive Science, University of Edinburgh, 1989.Google Scholar
[8] Burgess, J. P., Axioms for tense logic I: “since” and “until”, Notre Dame Journal of Formal Logic, vol. 23 (1982), no. 2, pp. 367–374.CrossRefGoogle Scholar
[9] Chang, C. C. and Keisler, H. J., Model theory, 3rd ed., North-Holland, Amsterdam, 1990.Google Scholar
[10] Dechter, R., Meiri, I., and Pearl, J., Temporal constraint networks, Artificial Intelligence, vol. 49 (1991), pp. 61–95.CrossRefGoogle Scholar
[11] Fraïssé, R., Sur l'extension aux relations de quelques propriétés des ordres, Annales Scientifiques de l'École Normale Supérieure, vol. 71 (1954), pp. 363–388.CrossRefGoogle Scholar
[12] Gale, D. and Stewart, F., Infinite games with perfect information, Contributions to the theory of games II, Annals of Mathematical Studies, vol. 28 (1953), pp. 291–296.Google Scholar
[13] Givant, S., The structure of relation algebras generated by relativizations, Contemporary mathematics, vol. 156, American Mathematics Society, 1994.Google Scholar
[14] Henkin, L., Some interconnections between modern algebra and mathematical logic, Transactions of the American Mathematical Society, vol. 74 (1953), pp. 410–427.CrossRefGoogle Scholar
[15] Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras part I, North-Holland, 1971.Google Scholar
[16] Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras part II, North-Holland, 1985.Google Scholar
[17] Hirsch, R., Relation algebras of intervals, Ph,D. thesis, Imperial College, 1994.Google Scholar
[18] Hirsch, R. and Hodkinson, I., Complete representations in algebraic logic, accepted for publication in Journal of Symbolic Logic, 1996.Google Scholar
[19] Hodges, W., Building models by games, London Mathematical Society Student Texts, no. 2, Cambridge University Press, 1985.Google Scholar
[20] Hughes, D. and Piper, F., Projective planes, Graduate Texts in Mathematics, Springer-Verlag, 1973.Google Scholar
[21] Jónsson, B. and Tarski, A., Representation problems for relation algebras, Bulletin of the American Mathematical Society, vol. 54 (1948).Google Scholar
[22] Jónsson, B. and Tarski, A., Boolean algebras with operators, II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.CrossRefGoogle Scholar
[23] Ladkin, P. and Maddux, R., The algebra of convex time intervals, Technical report, The Kestrel Institute, Palo Alto, California, 1987.Google Scholar
[24] Lyndon, R., The representation of relational algebras, Annals of Mathematics, vol. 51 (1950), no. 3, pp. 707–729.CrossRefGoogle Scholar
[25] Lyndon, R., The representation of relation algebras, II, Annals of Mathematics, vol. 63 (1956), no. 2, pp. 294–307.CrossRefGoogle Scholar
[26] Lyndon, R., Relation algebras and projective geometries, Michigan Mathematics Journal, vol. 8 (1961), pp. 207–210.CrossRefGoogle Scholar
[27] Maddux, R., Topics in relation algebra, Ph.D. thesis, University of California, Berkeley, 1969.Google Scholar
[28] Maddux, R., Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), no. 2, pp. 501–526.CrossRefGoogle Scholar
[29] Maddux, R., Finite integral relation algebras, Universal algebra and lattice theory (Comer, S. D., editor), Lecture Notes in Mathematics, no. 1149, Springer-Verlag, 1985, pp. 175–197.CrossRefGoogle Scholar
[30] Maddux, R., Non-finite axiomatizability results for cylindric and relation algebras, this Journal, vol. 54 (1989), no. 3, pp. 951–974.Google Scholar
[31] Maddux, R., The origin of relation algebras in the development and axiomatization of the calculus of relations, Studia Logica, vol. 3/4 (1991).Google Scholar
[32] Marx, M., Algebraic relativization and arrow logic, Ph.D. thesis, University of Amsterdam, 1995, ILLC Dissertation Series 95-3.Google Scholar
[33] McKenzie, R., The representation of relation algebras, Ph.D. thesis, University of Colorado at Boulder, 1966.Google Scholar
[34] McKenzie, R., Representation of integral relation algebras, Michigan Mathematics Journal, vol. 17 (1970), pp. 279–287.CrossRefGoogle Scholar
[35] Monk, J. D., On representable relation algebras, Michigan Mathematics Journal, vol. 11 (1964), pp. 207–210.CrossRefGoogle Scholar
[36] Monk, J. D., Lectures on cylindric set algebras, Algebraic methods in logic and in computer science, Institute of Mathematics, Polish Academy of Sciences, vol. 28, Banach Center publications, 1993.Google Scholar
[37] Németi, I., Algebraisations of quantifier logics, an introductory overview, Studia Logica, vol. 50 (1991), no. 3/4, pp. 485–570.CrossRefGoogle Scholar
[38] Németi, I., Algebraisations of quantifier logics, an introductory overview, Technical report, Mathematics Institute, Hungarian Academy of Science, Budapest; also to appear in Algebraic logic and the methodology of applying it, Proceedings of Summer School Budapest 1995, CSLI Publications, 1995.Google Scholar
[39] Reynolds, M., Axiomatising first-order temporal logic: Until and since over linear time, to appear in Studia Logica.Google Scholar
[40] Robinson, A., Selected papers: Model theory and algebra, North-Holland, 1979.Google Scholar
[41] Tarski, A., On the calculus of relations, this Journal, vol. 6 (1941), pp. 73–89.Google Scholar
[42] Tarski, A. and Givant, S., A formalization of set theory without variables, American Mathematical Society Colloquium Publications, vol. 41 (1987).CrossRefGoogle Scholar
[43] van Beek, P. and Cohen, R., Exact and approximate reasoning about temporal relations, Computational intelligence, vol. 6 (1990), pp. 132–144.CrossRefGoogle Scholar
[44] Venema, Y., Cylindric modal logic, this Journal, vol. 60 (1995), pp. 591–623.Google Scholar
[45] Villain, M. and Kautz, H., Constraint propagation algorithmsfor temporal reasoning, Proceedings of the fifth AAAI, 1986, pp. 377–382.Google Scholar