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On a Complexity-Based Way of Constructivizing the Recursive Functions

F. W. Kroon and W. A. Burkhard
Studia Logica: An International Journal for Symbolic Logic
Vol. 49, No. 1 (Mar., 1990), pp. 133-149
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015484
Page Count: 18
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On a Complexity-Based Way of Constructivizing the Recursive Functions
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Abstract

Let $g_{E}(m,n)$ = o mean that n is the Gödel-number of the shortest derivation from E of an equation of the form φ(m) = k. Hao Wang suggests that the condition for general recursiveness $\forall m\exists n(g_{E}(m,n)=o)$ can be proved constructively if one can find a speedfunction $\phi _{S}$, with $\phi _{S}(m)$ bounding the number of steps for getting a value of φ(m), such that $\forall m\exists n\leq \phi _{S}(m)$ s.t. $g_{E}(m,n)$ = o. This idea, he thinks, yields a constructivist notion of an effectively computable function, one that doesn't get us into a vicious circle since we intuitively know, to begin with, that certain proofs are constructive and certain functions effectively computable. This paper gives a broad 'possibility' proof for the existence of such classes of effectively computable functions, with Wang's idea of effective computability generalized along a number of dimensions.

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