The complexity of inductive definability

Authors:
Douglas Cenzer;Jeffrey B. Remmel
Affiliations:
Department of Mathematics, University of Florida, Gainesville, FL;Department of Mathematics, University of California at San Diego, La Jolla, CA
Venue:
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
Year:
2005

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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.