Solutions of the Goncharov-Millar and Degree Spectra Problems in The Theory of Computable Models

The theory of computable models is an intensively developing area of mathematics that studies the interactions between the theory of models and computability theory. Analyzing the relationships between computable presentations of models, model-theoretic definability and the computable complexity of relations is one of the central problems in this area. A fundamental notion in the study of these problems is that of being computable isomorphic or autoequivalent as first introduced by A.I. Malcev [5]. We call a model B computable if its domain, basic predicates and operations are uniformly computable. If a model B is computable and is isomorphic to a model A, then B is called a computable presentation of A. Computable presentations B1 and B2 are computably isomorphic (autoequivalent) if there exists a computable isomorphism between B1 and B2. The maximal number of computable but not computably isomorphic presentations of a model B is called the computable (algorithmic) dimension of B and is denoted by dim(B). A model B is computably categorical (autostable) if dim(B) = 1. Atomless Boolean algebras and dense linearly ordered sets are typical examples of computably categorical models. The notion of computable dimension was introduced by Goncharov. He proved that for any natural number n ≥ 1 there exists a model whose computable dimension is n [2]. By an appropriate coding of these models of Goncharov, examples of groups, partially ordered sets, unary and other algebras of computable dimension n have been constructed in [2] [3] [6]. One of the important problems in this area is that of Goncharov-Millar about the relationship between the computable dimension of any given model B that of its expansion (B, c1, . . . , cm) by finitely many constants. We note that the following inequality always holds: dim(B) ≤ dim(B, c1, . . . , cm). The following theorem gives a full solution to the Goncharov-Millar problem.