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 $\Re$-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 $HF(\Re )$, 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 $\Re$-computable structure has an $F$-parameterization, but that the expansion of the real field by the exponential function is $F$-parameterizable but not $\Re$-computable. We also show that the structures with $\Re$-computable copies are exactly the structures with copies $\Sigma$-definable over $HF(\Re )$. One consequence of this equivalence is a method of approximating certain $\Re$-computable structures by Turing computable structures.

Added a major section comparing real computation to $\Sigma$-definability, plus some additional references. Formerly entitled "$\mathbb{R}$-computation and $F$-parameterizability."