Betwixt Turing and Kleene

TL;DR

This paper compares Turing's and Kleene's models of computation for functions of bounded variation, revealing significant differences in their logical strength and the role of second-order arithmetic.

Contribution

It demonstrates that the Kleene approach requires full second-order arithmetic for key theorems, unlike the Turing approach, highlighting fundamental differences in their computational frameworks.

Findings

Kleene's approach involves higher logical complexity, exemplified by Kleene's quantifier ∃³.

The Jordan decomposition theorem's proof differs markedly between the two frameworks.

Full second-order arithmetic is essential in Kleene's approach for functions of bounded variation.

Abstract

Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach and provide a framework for computing with objects of any finite type. Various research programs have been proposed in which higher-order objects, like functions on the real numbers, are represented/coded as real numbers, so as to make them amenable to the Turing framework. It is then a natural question whether there is any significant difference between the Kleene approach or the Turing-approach-via-codes. Continuous functions being well-studied in this context, we study functions of bounded variation, which have at most countably many points of discontinuity. A central result is the Jordan decomposition theorem that a function of bounded variation on $[0, 1]$ equals the difference of two monotone functions. We show that for this theorem and related results, the difference between the Kleene approach and the Turing-approach-via-codes is huge, in that full second-order arithmetic readily comes to the fore in Kleenes approach, in the guise of Kleene's quantifier $\exists^3$.