The Computable Dimension of I-Trees of Infinite Height

Abstract

shown that their computable dimension may assume only the value 1 or ω, and a complete characterization of computable categoricity was given.In the present paper, we study the question about spectrum of possible computable dimensions of trees enriched by an initial subtree (briefly, I-trees). It is proved that the computable dimension of any computable I-tree of infinite height is ω. Moreover, this dimension is effectively infinite, in the sense that, given any uniformly presented list of computable copies of the same I-tree, we can construct another computable copy of that tree, which is not computably isomorphic to any of the copies on the list. Notice that the results obtained can be naturally generalized to the case of several distinguished initial subtrees.

Categoricity properties for computable algebraic fields

“…For over fifty years since that first result in [9], computable categoricity for fields has remained largely a mystery. For many other classes of structures, mathematicians have found structural definitions equivalent to computable categoricity: see for instance [13], [14], [20], [24], [26], [35], and [36]. As an example, Goncharov and Dzgoev, and independently Remmel, showed that a linear order is computably categorical if and only if it has only finitely many pairs of adjacent elements.…”

Computable categoricity for algebraic fields with splitting algorithms

“…In both cases, the structural criterion identifies the obstacle to computing isomorphisms, in the class of structures under consideration. On the other hand, the criteria equivalent to computable categoricity for trees (viewed either as partial orders, or under the meet relation, and also with distinguished subtrees), established in [16] and [18] by Lempp, McCoy, Miller, and Solomon and in [13] by Kogabaev, Kudinov, and Miller, are not easy to describe in any known way, even though they are "structural," in any reasonable sense of the word. In terms of computational complexity, they are Σ 0 3 , as defined in Section 2, just like the conditions for linear orders and Boolean algebras.…”