Computable isomorphisms for certain classes of infinite graphs

We investigate (2,1):1 structures, which consist of a countable set [Formula: see text] together with a function [Formula: see text] such that for every element [Formula: see text] in [Formula: see text], [Formula: see text] maps either exactly one element or exactly two elements of [Formula: see text] to [Formula: see text]. These structures extend the notions of injection structures, 2:1 structures, and (2,0):1 structures studied by Cenzer, Harizanov, and Remmel, all of which can be thought of as infinite directed graphs. We look at various computability-theoretic properties of (2,1):1 structures, most notably that of computable categoricity. We say that a structure [Formula: see text] is computably categorical if there exists a computable isomorphism between any two computable copies of [Formula: see text]. We give a sufficient condition under which a (2,1):1 structure is computably categorical, and present some examples of (2,1):1 structures with different computability-theoretic properties.

Applications of Kolmogorov complexity to computable model theory

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-categorical saturated structure with a unique computable isomor-phism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ℵ1-categorical but not ℵ0-categorical saturated -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.

The settling-time reducibility ordering

To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >stA) if for every computable function f, for all but finitely many x, mB(x) > f(mA(x)). This settling-time ordering, which is a natural extension to an ordering of the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it. that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x) > f(mA(g(x))).

COMPUTABLY ENUMERABLE ALGEBRAS, THEIR EXPANSIONS, AND ISOMORPHISMS

Computably enumerable algebras are the ones whose positive atomic diagrams are computably enumerable. Computable algebras are the ones whose atomic diagrams are computable. In this paper we investigate computably enumerable algebras and provide several algebraic and computable theoretic distinctions of these algebras from the class of computable algebras. We give a characterization of computably enumerable but not computable algebras in terms of congruences and effective conjunctions of [Formula: see text]-sentences. Our characterization, for example, shows that computable conjunctions of negative atomic formulas true in a given c.e. algebra can be preserved in infinitely many of its homomorphic images. We also study questions on how expansions of algebras by finitely many new functions affect computable isomorphism types. In particular, we construct a c.e. algebra with unique computable isomorphism type but which has no finitely generated c.e. expansion.