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

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 $Σ$-definability over $HF(\real)$, defined by Ershov as a generalization of the observation that the computably enumerable sets are exactly those $Σ_1$-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 $Σ$-definable over $HF(\real)$. One consequence of this equivalence is a method of approximating certain $\real$-computable structures by Turing computable structures.