Decidability in Analysis

Abstract

In this paper, we introduce the notion of almost decidable predicates of real variables. The notion comes from the concept of decidability of number theoretical predicates together with the idea of effective convergence in computable analysis. The weakness of traditional definitions of decidability is discussed. The definition of almost decidable predicates of real variables is given. It is proved that some commonly used predicates on such as between computable reals and recursive open/closed sets are almost decidable, which justifies our definition of decidability in Euclidean spaces. Additionally, the relation between almost decidability and computability of reals and recursiveness of subsets of is considered, which provides a bridge to include our works here to the earlier literature on computability in analysis.