On the Degree Structure of Equivalence Relations Under Computable Reducibility

Abstract

We study the degree structure of the ω-c.e., n-c.e. and Π 0 1 equivalence relations under the computable many-one reducibility. In particular we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-c.e. equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e. and Π 0 1 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.

Comparing the isomorphism types of equivalence structures and preorders

“…Yet, computable reducibility has been used to analyze equivalence relations of various complexity, including some that are not even hyperarithmetical (such as the isomorphism relations for familiar classes of computable structures [15]). In recent times, following the work of Ng and Yu [16], we initiated a systematic study of ∆ 0 2 equivalence relations: we proved that theory of ceers, co-ceers, and ∆ 0 2 equivalence relations behave quite differently [17]. This motivates the following question (from which the present paper originated):…”

Punctual equivalence relations and their (punctual) complexity

“…Equivalence relations of the form R X are called 1-dimensional in [19], while cdegrees containing 1-dimensional equivalence relations are called set-induced in [30]. The interesting feature of set-induced degrees is that they offer algebraic and logical information about the overall structure of c-degrees: for example, in [4] it is proved that the first-order theory of ceers is undecidable, by showing that the interval rdeg c pIdq, deg c pR K qs of the c-degrees is isomorphic to the interval r0 1 , 0 1 1 s of the 1-degrees, where 0 1 is the 1-degree of an infinite and co-infinite computable set and 0 1 1 is the 1-degree of the halting set K. Yet, set-induced degrees are far from exhausting the collection of all c-degrees.…”

“…Less is known about larger structures of c-degrees; but recent studies considered the ∆ 0 2 case [30,9] and the global structure ER of all c-degrees [3]. Yet, despite its classificatory power, computable reducibility has an obvious shortcoming: it is simply too coarse for measuring the relative complexity of computable equivalence relations.…”

Classifying equivalence relations in the Ershov hierarchy

“…In this endeavour, we follow and extend the work of Andrews and Sorbi [ 3 ], that provides a very extensive analysis of the degree structure induced by computable reducibility on ceers. Ng and Yu [ 20 ] broadened the perspective by discussing some structural aspects of the c -degrees of n -c.e., -c.e., and -equivalence relations. We similarly rely on the Ershov hierarchy to pursue our analysis.…”

“… is the identity on , i.e., if and only if . The following definition is due to Gao and Gerdes [ 17 ] (but analogous ways of coding sets of numbers by equivalence relations occur frequently in the literature, see, for instance, the definition of a set-induced c -degree in [ 20 ]).…”