Between Turing and Kleene

TL;DR

This paper proposes an extended computational framework that combines aspects of Turing's and Kleene's models, utilizing a fragment of the Axiom of Choice to handle higher-order objects and address limitations of existing theories.

Contribution

It introduces a novel extension of Turing's approach using continuous choice functions, bridging the gap between Turing and Kleene frameworks with higher-order computation.

Findings

A new relation 'is computationally stronger than' for third-order objects

Overcomes limitations of traditional Turing and Kleene models

Extends the scope of computability with real numbers and higher types

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 to computing with objects of any finite type. Both frameworks have their pros and cons and it is a natural question if there is an approach that marries the best of both the Turing and Kleene worlds. In answer to this question, we propose a considerable extension of the scope of Turing's approach. Central is a fragment of the Axiom of Choice involving continuous choice functions, going back to Kreisel-Troelstra and intuitionistic analysis. Put another way, we formulate a relation `is computationally stronger than' involving third-order objects that overcomes (many of) the pitfalls of the Turing and Kleene frameworks.