ON THREE NOTIONS OF EFFECTIVE COMPUTATION OVER R

We compare three notions of effectiveness on uncountable struc- tures. The first notion is that of a R-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 �-definability over HF(R), defined by Ershov as a generalization of the observation that the computably enumerable sets are exactly those �1- definable in HF(N). We show that every R-computable structure has an F-parameterization, but that the expansion of the real field by the exponential function is F- parameterizable but not R-computable. We also show that the structures with R-computable copies are exactly the structures with copies �-definable over HF(R). One consequence of this equivalence is a method of approximating certain R-computable structures by Turing computable structures.