Advertisement

The Complexity of Inductive Definability

  • Douglas Cenzer
  • Jeffrey B. Remmel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)

Abstract

We study the complexity of computable and Σ\(_{\rm 1}^{\rm 0}\) inductive definitions of sets of natural numbers. For we example, we show how to assign natural indices to monotone Σ\(_{\rm 1}^{\rm 0}\)-definitions and we use these to calculate the complexity of the set of all indices of monotone Σ\(_{\rm 1}^{\rm 0}\)-definitions which are computable. We also examine the complexity of new type of inductive definition which we call weakly finitary monotone inductive definitions. Applications are given in proof theory and in logic programming.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cenzer, D., Marek, W., Remmel, J.B.: Using logic programs to reason about infinite sets. In: Proceedings of the Symposium on Mathematics and Artificial Intelligence (AIM 2004) (2004), http://rutcor.rutgers.edu/~amai/aimath04
  2. 2.
    Cenzer, D., Marek, W., Remmel, J.B.: Logic programming with snfinite sets. Annals of Mathematics and Artificial Intelligence (to appear)Google Scholar
  3. 3.
    Cenzer, D., Remmel, J.B.: Index sets in computable analysis. Theoretical Computer Science 219, 111–150 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gelfond, M., Lifschitz, V.: The stable semantics for logic programs. In: Kowalski, R., Bowen (eds.) ICLP 1988, pp. 1070–1080 (1988)Google Scholar
  5. 5.
    Hinman, P.G.: Recursion-Theoretic Hierarchies. Springer, Heidelberg (1978)zbMATHGoogle Scholar
  6. 6.
    Soare, R.E.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Douglas Cenzer
    • 1
  • Jeffrey B. Remmel
    • 2
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

Personalised recommendations