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The Complexity of Inductive Definability

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New Computational Paradigms (CiE 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3526))

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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.

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References

  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. Cenzer, D., Marek, W., Remmel, J.B.: Logic programming with snfinite sets. Annals of Mathematics and Artificial Intelligence (to appear)

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, FL, 32611, USA

    Douglas Cenzer

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    Jeffrey B. Remmel

Authors
  1. Douglas Cenzer
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  2. Jeffrey B. Remmel
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Editor information

Editors and Affiliations

  1. School of Mathematics, University of Leeds, LS2 9JT, Leeds, U.K.

    S. Barry Cooper

  2. Department Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Benedikt Löwe

  3. Universiteit van Amsterdam,  

    Leen Torenvliet

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© 2005 Springer-Verlag Berlin Heidelberg

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Cenzer, D., Remmel, J.B. (2005). The Complexity of Inductive Definability. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_11

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  • DOI: https://doi.org/10.1007/11494645_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26179-7

  • Online ISBN: 978-3-540-32266-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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