Advertisement

Two applications of equational theories to database theory

  • Stavros S. Cosmadakis
  • Paris C. Kanellakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 202)

Abstract

Databases and equational theorem proving are well developed and seemingly unrelated areas of Computer Science Research. We provide two natural links between these fields and demonstrate how equational theorem proving can provide useful and tools for a variety of database tasks.

Our first application is a novel way of formulating functional and inclusion dependencies (the most common database constraints) using equations. The central computational problem of dependency implication is directly reduced to equational reasoning. Mathematical techniques from universal algebra provide new proof procedures and better lower bounds for dependency implication. The use of REVE, a general purpose transformer of equations into term rewriting systems, is illustrated on nontrivial sets of functional and inclusion dependencies.

Our second application demonstrates that the uniform word problem for lattices is equivalent to implication of dependencies expressing transitive closure, together with functional dependencies. This natural generalization of functional dependencies, which is not expressible using conventional database theory formulations, has a natural inference system and an efficient decision procedure.

Keywords

Word Problem Equational Theory Transitive Closure Horn Clause Proof Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beeri, C. and Bernstein, P.A. "Computational Problems Related to the Design of Normal Form Relational Schemas". ACM Transactions on Database Systems 4, 1 (March 1979), 30–59..Google Scholar
  2. 2.
    Beeri, C. and Vardi, M.Y. "Formal Systems for Tuple and Equality Generating Dependencies". SIAM Journal of Computing 13, 1 (February 1984), 76–98..Google Scholar
  3. 3.
    Beeri, C. and Vardi, M.Y. "A Proof Procedure for Data Dependencies". Journal of the Association for Computing Machinery 31, 4 (October 1984), 718–741..Google Scholar
  4. 4.
    Birkhoff, G. "On the Structure of Abstract Algebras". Proceedings of the Cambridge Philosophical Society 31, (1935).Google Scholar
  5. 5.
    Bloniarz, P.A., Hunt, H.B. III and Rosenkrantz, D.J. "Algebraic Structures with Hard Equivalence and Minimization Problems". Journal Of The ACM 31, 4 (October 1984), 879–904..Google Scholar
  6. 6.
    Casanova, M.A., Fagin, R. and Papadimitriou, C.H. "Inclusion Dependencies and Their Interaction with Functional Dependencies". Journal of Computer and System Sciences 28, 1 (February 1984), 29–59..Google Scholar
  7. 7.
    Chandra, A.K. and Vardi, M.Y. The Implication Problem for Functional and Inclusion Dependencies is Undecidable. IBM Tech. Rep. RC 9980,,, 1983.Google Scholar
  8. 8.
    Cosmadakis, S.S. Equational Theories and Database Constraints. Ph.D.Th., Massachusetts Institute of Technology, 1985.Google Scholar
  9. 9.
    Cosmadakis, S.S. and Kanellakis, P.C. "Functional and Inclusion Dependencies: A Graph Theoretic Approach". Proceedings of the 3rd ACM Symposium on Principles of Database Systems (April 1984), 24–37.Google Scholar
  10. 10.
    Cosmadakis, S.S. and Kanellakis, P.C. "Equational Theories and Database Constraints". Proceedings of the 17th Annual ACM Symposium on Theory of Computing (May 1984).Google Scholar
  11. 11.
    Cosmadakis, S.S., Kanellakis, P.C. and Spyratos, N. "Partition Semantics for Relations". Proceedings of the 4th ACM Symposium on Principles of Database Systems (March 1985),.Google Scholar
  12. 12.
    Downey, P.J., Sethi, R. and Tarjan, R.E. "Variations on the Common Subexpression Problem". Journal of the Association for Computing Machinery 27, 4 (October 1980), 758–771..Google Scholar
  13. 13.
    Fagin, R. "Horn Clauses and Database Dependencies". Journal of the ACM 29, 4 (October 1982), 952–985..Google Scholar
  14. 14.
    Forgaard, R. and Guttag, J.V. "REVE: A Term Rewriting System Generator with Failure Resistant Knuth-Bendix". Proceedings of an NSF Workshop on the Rewrite Rule Laboratory (April 1984), 5–31.Google Scholar
  15. 15.
    Huet, G. and Oppen, D. Equations and Rewrite Rules: a Survey. In Formal Languages: Perspectives and Open Problems, Eds., Academic Press,, 1980.Google Scholar
  16. 16.
    Kanellakis, P.C., Cosmadakis, S.S. and Vardi, M.Y. "Unary Inclusion Dependencies Have Polynomial Time Inference Problems". Proceedings of the 15th Annual ACM Symposium on Theory of Computing (1983).Google Scholar
  17. 17.
    Kozen, D. "Complexity of Finitely Presented Algebras". Proceedings of the Ninth Annual ACM Symposium on Theory of Computing. ACM SIGACT (May 1977),.Google Scholar
  18. 18.
    Lescanne, P. "Computer Experiments with the REVE Term Rewriting System Generator". Proceedings of the 10th ACM Symposium on Principles of Programming Languages (January 1983), 99–108.Google Scholar
  19. 19.
    Mitchell, J.C. "The Implication Problem for Functional and Inclusion Dependencies". Information and Control 56, 3 (March 1983), 154–173..Google Scholar
  20. 20.
    Ullman, J.D.. Principles of Database Systems. Computer Science Press, Inc.,, 1983.Google Scholar
  21. 21.
    Vardi, M.Y. "Personal Communication". ().Google Scholar
  22. 22.
    Whitman, P.M. "Free Lattices". Annals of Mathematics 42, (1941).Google Scholar
  23. 23.
    Yannakakis, M. and Papadimitriou C.H. "Algebraic Dependencies". J. Comput. Systems Sci. 21, 1 (August 1982), 2–41..Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Stavros S. Cosmadakis
    • 1
  • Paris C. Kanellakis
    • 1
  1. 1.MITUSA

Personalised recommendations