## Access

You are not currently logged in.

Access JSTOR through your library or other institution:

# 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
Stable URL: http://www.jstor.org/stable/20015484
Page Count: 18
Preview not available

## 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.

• [133]
• 134
• 135
• 136
• 137
• 138
• 139
• 140
• 141
• 142
• 143
• 144
• 145
• 146
• 147
• 148
• 149
• [unnumbered]