Computing with infinitary logic

  • Serge Abiteboul
  • Moshe Vardi
  • Victor Vianu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 646)

Abstract

Most recursive extensions of relational calculus converge around two central classes of queries: fixpoint and while. Infinitary logic (with finitely many variables) is a very powerful extension of these languages which provides an elegant unifying formalism for a wide variety of query languages. However, neither the syntax nor the semantics of infinitary logic are effective, and its connection to practical query languages has been largely unexplored. We relate infinitary logic to another powerful extension of fixpoint and while, called relational machine, which highlights the computational style of these languages. Relational machines capture the kind of computation occurring when a query language is embedded in a host programming language, as in C+SQL. The main result of this paper is that relational machines correspond to the natural effective fragment of infinitary logic. Other well-known query languages are related to infinitary logic using syntactic restrictions formulated in language-theoretic terms. For example, it is shown that while corresponds to infinitary logic formulas which can be described by a regular language. As a side effect to these results, we obtain interesting normal forms for infinitary logic formulas.

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References

  1. [ACY91]
    Afrati, F., S.S. Cosmadakis, M. Yannakakis, On Datalog vs. polynomial time, Proc. 10th ACM Symp. on Principles of Database Systems (1991), 13–25.Google Scholar
  2. [AV88]
    Abiteboul, S., V. Vianu, Datalog extensions for database updates and queries, J. Computer and System Sciences 43:1 (1991), 62–124.Google Scholar
  3. [AV91]
    Abiteboul, S., V. Vianu, Generic computation and its complexity, Proc. 23rd ACM Symposium on Theory of Computing (1991), 209–219.Google Scholar
  4. [AV92]
    Abiteboul, S., V. Vianu, Computing with first-order logic, to appear in J. Computer and System Sciences.Google Scholar
  5. [AVV92]
    Abiteboul, S., M.Y. Vardi, V. Vianu, Fixpoint logics, relational machines, and computational complexity, to appear in Proc. 7th IEEE Conf. on Structure in Complexity Theory (1992).Google Scholar
  6. [Ba75]
    Barwise, J., Admissible Sets and Structures, Springer-Verlag, 1975.Google Scholar
  7. [Ba77]
    Barwise, J., On Moschovakis closure ordinals, J. Symbolic Logic, 42 (1977), 292–296.Google Scholar
  8. [BF85]
    Barwise, J., S. Feferman (eds.), Model-Theoretic Logics, Springer-Verlag, 1985.Google Scholar
  9. [BIS90]
    D. A. M. Barrington, N. Immerman, and H. Straubing, On uniformity within NC1, J. Computer and System Sciences 41 (1990), 274–306.Google Scholar
  10. [CH80]
    Chandra, A.K., D. Harel, Computable Queries for Relational Databases, J. Computer and System Sciences 21:2 (1980), 156–178.Google Scholar
  11. [CH82]
    Chandra, A.K., D. Harel, Structure and Complexity of Relational Queries, J. Computer and System Sciences 25:1 (1982), 99–128.Google Scholar
  12. [DLW91]
    Dawar, A., S. Lindell and S. Weinstein, Infinitary logic and inductive definability over finite structures, Technical Report, Univ. of Pennsylvania, 1991.Google Scholar
  13. [F90]
    Fagin R., Finite-Model Theory—a Personal Perspective, Proc. 3rd Int'l. Conf. on Database Theory, Springer-Verlag, Lecture Notes in Computer Science 470 (1990), 3–24, to appear in Theoretical Computer Science.Google Scholar
  14. [HU79]
    Hopcroft J.E., J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, 1979.Google Scholar
  15. [186]
    Immerman N., Relational queries computable in polynomial time, Information and Control 68 (1986), 86–104.CrossRefGoogle Scholar
  16. [K90]
    Kanellakis, P.C., Elements of Relational Database Theory, Handbook of Theoretical Computer Science (J. van Leeuwen, A.R. Meyer, N. Nivat, M.S. Paterson, and D. Perrin, eds.), Vol. B, Chapter 17, North-Holland, 1990.Google Scholar
  17. [KV87]
    Kolaitis, P., M.Y. Vardi, The decision problem for the probabilities of higher-order properties, Proc. 19th ACM Symp. on Theory of Computing (1987), 425–435.Google Scholar
  18. [KV90a]
    Kolaitis, P., M.Y. Vardi, On the expressive power of Datalog: tools and a case study, Proc. 9th ACM Symp. on Principles of Database Systems (1990), 61–71. To appear in J. Computer and System Sciences.Google Scholar
  19. [KV90b]
    Kolaitis, P., M.Y. Vardi, 0–1 laws for infinitary logic, Proc. 5th IEEE Symp. on Logic in Computer Science (1990), 156–167. To appear in Information and Computation.Google Scholar
  20. [KV92]
    Kolaitis, P., M.Y. Vardi, Fixpoint vs. infinitary logic in finite-model theory, to appear in Proc. 7th IEEE Symp. on Logic in Computer Science (1992).Google Scholar
  21. [SF88]
    Shemesh Y., N. Francez, Finite-state Datalog automata and relational languages, unpublished manuscript, 1988.Google Scholar
  22. [U88]
    Ullman, J.D., Principles of Database and Knowledge Base Systems, Computer Science Press, 1988.Google Scholar
  23. [V82]
    Vardi, M.Y., The complexity of relational query languages, Proc. 14th ACM Symp. on Theory of Computing (1982), 137–146.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Serge Abiteboul
    • 1
  • Moshe Vardi
    • 2
  • Victor Vianu
    • 3
  1. 1.INRIALe Chesnay CedexFrance
  2. 2.IBM Almaden Research CenterSan JoseUSA
  3. 3.CSE C-0114UC San DiegoLa JollaUSA

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