On the Undecidability of Equivalence Problems for Relational Expressions

  • Tomasz Imielinski
  • Witold Lipski

Abstract

We consider two versions of the relational algebra: (a) the attribute relational algebra, based on the natural join and relations with columns corresponding to attributes, and (b) the positional relational algebra, based on the cross product and relation with an order on the columns, and with any column identified by its position in that order. For the attribute relational algebra, we show that both the equivalence and the finite equivalence (i.e., equivalence over finite relations only) of expressions involving just one ternary relation and the operators of projection, selection, join and difference, are undecidable. For the positional relational algebra, we show that both the equivalence and finite equivalence of expressions involving just one binary relation and the operators of projection, selection, cross product, restriction and difference, are undecidable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aho, A. V., Sagiv, Y., and Ullman, J. D. [1979] “Equivalence among Relational Expressions”, SIAM Journal of Computing 8, 2 (1979) 218–246.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Church, A. [1936] “A Note on the Entscheidungs Problem”,J. Symbolic Logic 1 ,1 (1936) 40–41; correction, 1, 101–102.MATHCrossRefGoogle Scholar
  3. 3.
    Codd, E. F. [1972] “Relational Completeness of Data Base Sublanguages”. In: Data Base Systems (R. Rustin, Ed.), Prentice-Hall, Englewood Cliffs, New Jersey, 65–98.Google Scholar
  4. 4.
    Gurevich, Y. [1966] “The Word Problem for Certain Classes of Semigroups”, Algebra i Logika 5 (1966) 25–35 (in Russian).Google Scholar
  5. 5.
    Imielinski, T., and Lipski, W. [1982] “The Relational Model of Data and Cylindric Algebras,” ACM SIGACT-SIGMOD Symp. on Principles of Database Systems ,Los Angeles, March 1982. (Final version received too late to be included in the formal proceedings: see ICS PAS Report 446, Warsaw, Aug. 1981.)Google Scholar
  6. 6.
    Imielinski, T., and Lipski, W. [1982] “A Technique for Translating States between Database Schemata,” ACM SIGMOD Internat. Conf. on Management of Data ,Orlando, Florida, June 1982, 61–68.Google Scholar
  7. 7.
    Jaskowski, S. [1948] “Sur les Variables Propositionnelles Dependantes,” Studia Societatis Scientiarum Torunensis Sec. A 1 (1948) 17–21.MathSciNetGoogle Scholar
  8. 8.
    Kalmar, L. [1936] “Zurückführung des Entscheidungsproblems auf den Fall von Formeln mit einer einzigen, binären, Funktionsvariablen, Compositio Math. 4 ,1 (1936) 137–144.MathSciNetMATHGoogle Scholar
  9. 9.
    Lewis, H.R., and Papadimitriou, C. H. [1981] Elements of the Theory of Computation ,Prentice-Hall, Inc., Englewood Cliffs, New Jersey.MATHGoogle Scholar
  10. 10.
    Maddux, R. [1980] “The Equational Theory of CA3 is Undecidable,” J. Symbolic Logic 45 (1980) 311–316.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Monk, J. D. [1976] Mathematical Logic ,Springer-Verlag, N.Y.MATHGoogle Scholar
  12. 12.
    Mortimer, M. [1975] “On Languages with Two Variables”, Zeitschritt Math. Logik Grundlag. Math. 21 (1975) 135–140.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Pieczkowski, A. [1968] Undecidability of the Homogeneous Formulas of Degree 3 of the Predicate Calculus, Studia Logica 22 (1968) 7–16.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Post, E.L. [1947] “Recursive Unsolvability of a Problem of Thue”, Journal Symbolic Logic 12 ,1 (1947) 1–11.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Scott, D. [1962] “A Decision Method for Validity of Sentences in Two Variables”, Journal of Symbolic Logic 27 ,4 (1962) 477.Google Scholar
  16. 16.
    Solomon, M.K. [1979] Some Properties of Relational Expressions, ACM South-East Regional Conference (April 1979) 111–116.Google Scholar
  17. 17.
    Trakhtenbrot, B.A. [1950] “Impossibility of an Algorithm for the Decision Problem in Finite Classes”, Doklady Akademii Nauk SSSR 70 (1950) 569–572; translated in: Amer. Math. Soc. Transl. Ser. 2 ,23 (1963) 1–5.Google Scholar
  18. 18.
    Turing, A.M. [1937] “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proc. London Math. Soc. 42 ,1 (1937) 230–265; correction 43, 544–546.CrossRefGoogle Scholar
  19. 19.
    Ullman, J.D. [1980] Principles of Database Systems ,Computer Science Press, Potomac, MD.MATHGoogle Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • Tomasz Imielinski
    • 1
  • Witold Lipski
    • 2
  1. 1.McGill UniversityMontrealCanada
  2. 2.Polish Academy of SciencesWarsawPoland

Personalised recommendations