Three notions of effective computation on $\mathbb{R}$

TL;DR

This paper compares three notions of effectiveness for uncountable structures involving real computation, parameterizability, and definability, establishing their relationships and implications for approximating structures.

Contribution

It clarifies the relationships among three effectiveness notions on uncountable structures and connects real computability with definability over $HF( eal)$.

Findings

Every $ eal$-computable structure has an $F$-parameterization.

The expansion of the real field by exponential is $F$-parameterizable but not $ eal$-computable.

Structures with $ eal$-computable copies are exactly those $ ext{HF}( eal)$-definable.

Abstract

We compare three notions of effectiveness on uncountable structures. The first notion is that of a $\real$-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic. The second notion is that of an $F$-parameterizable structure, defined by Morozov and based on Mal'tsev's notion of a constructive structure. The third is $\Sigma$-definability over $HF(\real)$, defined by Ershov as a generalization of the observation that the computably enumerable sets are exactly those $\Sigma_1$-definable in $HF(\mathbb{N})$. We show that every $\real$-computable structure has an $F$-parameterization, but that the expansion of the real field by the exponential function is $F$-parameterizable but not $\real$-computable. We also show that the structures with $\real$-computable copies are exactly the structures with copies $\Sigma$-definable over $HF(\real)$. One consequence of this equivalence is a method of approximating certain $\real$-computable structures by Turing computable structures.