Ordered Subrings of the Reals in which Output Sets are Recursively Enumerable

Abstract

In On a theory of computation and complexity over the real numbers ..., Bull. Amer. Math. Soc. 21 (1989), 1-46, Blum, Shub, and Smale investigated computability over the reals and over ordered rings in general. They showed that over the reals, output sets of machines are recursively enumerable (i.e., halting sets of machines). It is asked in the aforementioned paper which ordered rings have this property (which we abbreviate $O = R.E.$). In Ordered rings over which output sets are recursively enumerable, Proc. Amer. Math. Soc. 112 (1991), 569-575, Michaux characterized the members of a certain class of ordered rings of infinite transcendence degree over $\mathbb {Q}$ satisfying $O = R.E.$ In this paper we characterize the subrings of $\mathbb {R}$ of finite transcendence degree over $\mathbb {Q}$ satisfying $O = R.E.$ as those rings recursive in the Dedekind cuts of members of a transcendence base. With Michaux’s result, this answers the question for subrings of $\mathbb {R}$ (i.e., archimedean rings).