Computation over algebraic structures and a classification of undecidable problems

Abstract

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.