On a complexity-based way of constructivizing the recursive functions

Studia Logica 49 (1):133 - 149 (1990)
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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 (m)=k. Hao Wang suggests that the condition for general recursiveness mn(g E(m, n)=o) can be proved constructively if one can find a speedfunction s s, with s(m) bounding the number of steps for getting a value of (m), such that mn 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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Frederick Kroon
University of Auckland

Citations of this work

The Intrinsic Difficulty of Recursive Functions.F. W. Kroon - 1996 - Studia Logica 56 (3):427 - 454.

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References found in this work

From Mathematics to Philosophy.Hao Wang - 1974 - London and Boston: London.
A Survey of Mathematical Logic.Georg Kreisel - 1966 - Philosophical Review 75 (2):240-244.
A Hierarchy of Primitive Recursive Functions.J. P. Cleave - 1963 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9 (22):331-346.
A Hierarchy of Primitive Recursive Functions.J. P. Cleave - 1963 - Mathematical Logic Quarterly 9 (22):331-346.

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