Moschovakis Extension of Represented Spaces

TL;DR

This paper constructs appropriate representations for the Moschovakis extension of a represented space and explores TTE computability results, including for single-valued and multi-valued functions, using iterative combinatory spaces.

Contribution

It introduces a new representation for the Moschovakis extension and establishes computability results for functions within this framework, expanding TTE theory.

Findings

Constructed representations for Moschovakis extensions in effective topological and metric spaces.

Proved computability of absolutely prime computable functions in the extended space.

Extended computability results to multi-valued functions using iterative combinatory spaces.

Abstract

Given a represented space (in the sense of TTE theory), an appropriate representation is constructed for the Moschovakis extension of its carrier (with paying attention to the cases of effective topological spaces and effective metric spaces). Some results are presented about TTE computability in the represented space obtained in this way. For single-valued functions, we prove, roughly speaking, the computability of any function which is absolutely prime computable in some computable functions. A similar result holds for multi-valued functions, but with an analog of absolute prime computability. The formulation of this result makes use of the notion of computability in iterative combinatory spaces - a notion studied by the author in other publications.