Computably enumerable sets and related issues

Abstract

A topical direction in the theory of algorithms is studying reducibilities of arithmetic sets. Post defined concepts of m-, tt-, and T -reducibilities of arithmetic sets; subsequently, other kinds of reducibilities were introduced. A T -reducibility is being studied quite intensively today. Here a number of remarkable results were obtained. However, many questions concerning T -reducibility still await a solution. Progress in tt-reducibility is less noticeable. For an m-reducibility, exhaustive solutions were derived in several directions-especially if consideration was limited to computably enumerable sets. In the present review, we consider different aspects associated with computably enumerable sets and m-reducibility. Among these are an algebraic description of the structure of these formations both in upper and lower parts, definability, decidability problems, and so on. Many of the results cited in the paper are widely scattered over the literature. This makes it impossible to conceptualize the overall picture and spectrum of available investigations. It is also worth mentioning that a series of books and papers are not easily accessible to domestic specialists.Computably enumerable sets play a fundamental role in the theory of algorithms. Over a long period of time, these sets were referred to as recursively enumerable in Russian mathematical literature. A class of computably enumerable sets, along with computable sets (previously called recursive sets), occur at lower levels in the hierarchy of arithmetic sets. Given these sets, all other arithmetic sets, predicates, and functions are constructed by using logical tools. * Supported by the Program on Modern Problems in Pure Mathematics (Division of Mathematical Sciences of RAS) through the Project "Complexity Aspects of Computability Theory."