Skip to main content
Log in

Problems of Equivalence, Categoricity of Axioms and States Description in Databases

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

The paper is devoted to applications of algebraic logic to databases. In databases a query is represented by a formula of first order logic. The same query can be associated with different formulas. Thus, a query is a class of equivalent formulae: equivalence here being similar to that in the transition to the Lindenbaum-Tarski algebra. An algebra of queries is identified with the corresponding algebra of logic. An algebra of replies to the queries is also associated with algebraic logic. These relations lie at the core of the applications.

In this paper it is shown how the theory of Halmos (polyadic) algebras (a notion introduced by Halmos as a tool in the algebraization of the first order predicate calculus) is used to create the algebraic model of a relational data base. The model allows us, in particular, to solve the problem of databases equivalence as well as develop a formal algebraic definition of a database's state description. In this paper we use the term "state description" for the logical description of the model. This description is based on the notion of filters in Halmos algebras. When speaking of a state description, we mean the description of a function which realizes the symbols of relations as real relations in the given system of data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, M., and E. Simoudis, ‘Integrating distributed expertise’, 10th International Workshop on DAI, Texas, 1990.

  2. Beeri, C., A. Mendelzon, Y. Sagiv, J. Ullman, ‘Equivalence of relational database schemes’, Proc. of the Eleventh Annual (ACM) Symposium of Theory of Computing, 1979, 319–329.

  3. Beniaminov, E., ‘Algebraic invariants of relational databases’, Data Banks, 1980, 47–52.

  4. Biller, H., and E. Neuhold, ‘Semantics of data bases: the semantics of data models’ Information Systems 3, 1978, 11–30.

    Google Scholar 

  5. Biskup, J., ‘Null values in data base relations’, in Logic and Data Bases, eds. H. Gallaire and J. Minker, Plenum Press, 1978.

  6. Boose, J., and J. M. Bradshaw, ‘Expertie transfer and complex problems: using AQUINAS as a data-acquisition workbench for data-based systems’, International Journal of Man-Machine Studies 26(1), 1987, 3–28.

    Google Scholar 

  7. Borschev, V., and M. Homjakov, ‘On informational equivalence of databases’, NTI 7, 1979, 14–21.

    Google Scholar 

  8. Carnap, R., Meanimg and Necessity, 2nd edn., University of Chicago Press, 1956.

  9. Church, A., Introduction to Mathematical Logic, Princeton, NJ, Princeton Univ. Press, 1956.

    Google Scholar 

  10. Codd, E. F., ‘A relational model for large shared data banks’, Comm. ACM 13, 1970, 377–387.

    Google Scholar 

  11. Codd, E. F., ‘Extending the data base relational model to capture more meaning’, ACM Trans. Database Systems 4, 1979, 397–434.

    Google Scholar 

  12. Cohn, P. M., Universal Algebra, Harper & Row, New York, 1965.

    Google Scholar 

  13. Halmos, P. R., Algebraic Logic, New York, 1962.

  14. Henkin, L., I. Monk, and A. Tarski, Cylindric Algebras, Amsterdam-London, 1971.

  15. Imielinski, T., and W. Lipski, ‘The relational model of data and cylindric algebras’, J. Comp. Systems Sci. 28, 1984, 80–102.

    Google Scholar 

  16. Imielinski, T., and W. Lipski, ‘Incomplete information in relational databases’, Proceedings of the Conference on Very Large Databases, Cannes, France, 1981, 368–397.

  17. Israel, D. J., and R. J. Brachman, ‘Some remarks on the semantics of representation Languages. Towards a logical recognition of relational database theory’, in On Conceptual Modelling, eds. M. Brodie, J. Mylopoulos and J. Schmidt, Springer-Verlag, 1984, 119–146.

  18. Lawvere, F. W., ‘Some algebraic problems in the context of functorial semantics of algebraic theories’, Rep. Midwest Category Seminar II, Berlin, Springer-Verlag 61, 1968, 41–46.

    Google Scholar 

  19. Leveque, H., ‘Foundation of a functional approach to knowledge representation’, Artificial Intelligence 23, 1984, 155–212.

    Google Scholar 

  20. Malcev, A. I., Algebraic Systems, Nauka, 1970.

  21. Marcus, S., and J. McDermott, ‘SALT: a knowledge acquisition language* for propose-and-revise systems’, Artificial Intelligence 39, 1989, 1–37.

    Google Scholar 

  22. Plotkin, B. I., Universal Algebra, Algebraic Logic and Databases, Kluwer Academic Publishers, 1993.

  23. Plotkin, B. I., L. J. Greenglaz, and A. A. Gvaramija, ‘Algebraic structures in automata and databases theory’, World Scientific, 1992.

  24. Plotkin, B. I., ‘Halmos (polyadic) algebras in database theory’, Proc. of Int. Conf. on Algebraic Logic, Budapest, Colloqua Math. Janos Bolyai 54, North-Holland, Amsterdam, 1991, 503–518.

    Google Scholar 

  25. Plotkin, T. L., ‘Equivalence transformations of relational databases’, Latvian Math. Annual 29, 1985, 137–150.

    Google Scholar 

  26. Plotkin, T. L., ‘Algebraic logic in the problem of databases equivalence’, Logic Colloquium '94, 1994, 104.

  27. Reiter, R., ‘Towards a logical recognition of relational database theory’, in On Conceptual Modelling, eds. M. Brodie, J. Mylopoulos and J. Schmidt, Springer-Verlag, 1984, 191–233.

  28. Ullman, J. D. ‘Principles of database and knowledge-base systems’, Computer science press, 1988.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Plotkin, T.L., Kraus, S. & Plotkin, B.I. Problems of Equivalence, Categoricity of Axioms and States Description in Databases. Studia Logica 61, 347–366 (1998). https://doi.org/10.1023/A:1005066020883

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005066020883

Navigation