Towards a theory of recursive structures

  • David Harel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)

Abstract

In computer science, one is interested mainly in finite objects. Insofar as infinite objects are of interest, they must be computable, i.e., recursive, thus admitting an effective finite representation. This leads to the notion of a recursive graph, or, more generally, a recursive structure or data base. In this paper we summarize our recent work on recursive structures and data bases, including (i) the high undecidability of many problems on recursive graphs, (ii) somewhat surprising ways of deducing results on the classification of NP optimization problems from results on the degree of undecidability of their infinitary analogues, and (iii) completeness results for query languages on recursive data bases.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AV]
    S. Abiteboul and V. Vianu, “Generic Computation and Its Complexity”, Proc. 23rd ACM Symp. on Theory of Computing, pp. 209–219, ACM Press, New York, 1991.Google Scholar
  2. [AMS]
    R. Aharoni, M. Magidor and R. A. Shore, “On the Strength of König's Duality Theorem”, J. of Combinatorial Theory (Series B) 54:2 (1992), 257–290.Google Scholar
  3. [B1]
    D.R. Bean, “Effective Coloration”, J. Sym. Logic 41 (1976), 469–480.Google Scholar
  4. [B2]
    D.R. Bean, “Recursive Euler and Hamiltonian Paths”, Proc. Amer. Math. Soc. 55 (1976), 385–394.Google Scholar
  5. [BG1]
    R. Beigel and W. I. Gasarch, unpublished results, 1986–1990.Google Scholar
  6. [BG2]
    R. Beigel and W. I. Gasarch, “On the Complexity of Finding the Chromatic Number of a Recursive Graph”, Parts I & II, Ann. Pure and Appl. Logic 45 (1989), 1–38, 227–247.Google Scholar
  7. [Bu]
    S. A. Burr, “Some Undecidable Problems Involving the Edge-Coloring and Vertex Coloring of Graphs”, Disc. Math. 50 (1984), 171–177.Google Scholar
  8. [CH]
    A. K. Chandra and D. Harel, “Computable Queries for Relational Data Bases”, J. Comp. Syst. Sci. 21, (1980), 156–178.Google Scholar
  9. [F]
    R. Fagin, “Generalized First-Order Spectra and Polynomial-Time Recognizable Sets”, In Complexity of Computations (R. Karp, ed.), SIAM-AMS Proceedings, Vol. 7, 1974, pp. 43–73.Google Scholar
  10. [GL]
    W. I. Gasarch and M. Lockwood, “The Existence of Matchings for Recursive and Highly Recursive Bipartite Graphs”, Technical Report 2029, Univ. of Maryland, May 1988.Google Scholar
  11. [H]
    D. Harel, “Hamiltonian Paths in Infinite Graphs”, Israel J. Math. 76:3 (1991), 317–336. (Also, Proc. 23rd ACM Symp. on Theory of Computing, New Orleans, pp. 220–229, 1991.)Google Scholar
  12. [HH1]
    T. Hirst and D. Harel, “Taking it to the Limit: On Infinite Variants of NP-Complete Problems”, Proc. 8th IEEE Conf. on Structure in Complexity Theory, IEEE Press, New York, 1993.Google Scholar
  13. [HH2]
    T. Hirst and D. Harel, “Completeness Results for Recursive Data bases”, 12th ACM Symp. on Principles of Database Systems, ACM Press, New York, 1993, 244–252.Google Scholar
  14. [KT]
    P. G. Kolaitis and M. N. Thakur, “Logical definability of NP optimization problems”, 6th IEEE Conf. on Structure in Complexity Theory, pp. 353–366, 1991.Google Scholar
  15. [MR]
    A. Manaster and J. Rosenstein, “Effective Matchmaking (Recursion Theoretic Aspects of a Theorem of Philip Hall)”, Proc. London Math. Soc. 3 (1972), 615–654.Google Scholar
  16. [NR]
    A. Nerode and J. Remmel, “A Survey of Lattices of R. E. Substructures”, In Recursion Theory, Proc. Symp. in Pure Math. Vol. 42 (A. Nerode and R. A. Shore, eds.), Amer. Math. Soc., Providence, R. I., 1985, pp. 323–375.Google Scholar
  17. [PR]
    A. Panconesi and D. Ranjan, “Quantifiers and Approximation”, Theor. Comp. Sci. 107 (1993), 145–163.Google Scholar
  18. [PY]
    C. H. Papadimitriou and M. Yannakakis, “Optimization, Approximation, and Complexity Classes”, J. Comp. Syst. Sci. 43, (1991), 425–440.Google Scholar
  19. [R]
    H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • David Harel
    • 1
  1. 1.Dept. of Applied Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations