Sergei S. Goncharov

Computable Structure and Non-Structure Theorems

In a lecture in Kazan (1977), Goncharov dubbed a number of problems regarding the classification of computable members of various classes of structures. Some of the problems seemed likely to have nice answers, while others did not. At the end of the lecture, Shore asked what would be a convincing negative result. The goal of the present article is to consider some possible answers to Shores question. We consider structures Д of some computable language, whose universes are computable sets of constants. In measuring complexity, we identify Д with its atomic diagram D(Д), which, via the Gödel numbering, may be treated as a subset of ω. In particular, Д is computable if D(Д) is computable. If K is some class, then Kc denotes the set of computable members of K. A computable characterization for K should separate the computable members of K from other structures, that is, those that either are not in K or are not computable. A computable classification (structure theorem) should describe each member of Kc up to isomorphism, or other equivalence, in terms of relatively simple invariants. A computable non-structure theorem would assert that there is no computable structure theorem. We use three approaches. They all give the “correct” answer for vector spaces over Q, and for linear orderings. Under all of the approaches, both classes have a computable characterization, and there is a computable classification for vector spaces, but not for linear orderings. Finally, we formulate some open problems.

Computably categorical structures and expansions by constants

Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts. The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory, etc.) and divergent terminology. (We use “effective model theory” as the most general and descriptive designation. Harizanov [6] is an excellent introduction to the subject as is Millar [13].) The basic subjects of model theory include languages, structures, theories, models and various types of maps between these objects. There are many ways to introduce considerations of effectiveness into the area. The two most prominent derive from starting, on the one hand, with the notion of a theory and its models or, on the other, with just structures. If one begins with theories, then a natural version of effectiveness is to consider decidable theories (i.e., ones with a decidable (equivalently, computable or recursive) set of theorems). When one moves to models and wants them to be effective, one might start with the requirement that the model (of any theory) have a decidable theory (i.e., Th ( ), the set of sentences true in , is decidable). Typically, however, one wants to be able to talk about the elements of the model as well as its theory in the given language. Thus one naturally considers the model as a structure for the language expanded by adding a constant a i , for each element a i of . Of course, one requires that the mapping from the constants to the corresponding elements of be effective (computable). We are thus lead to the following basic definition: A structure or model is decidable if there is a computable enumeration a i of A , the domain of , such that Th( , a i ,) is decidable. (Of course, a i , is interpreted as a i , for each i Є ω .)

Degree Spectra of Relations on Boolean Algebras

AbstractWe show that every computable relation on a computable Boolean algebra

Decidability and Computability of Certain Torsion-Free Abelian Groups

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Categoricity of computable infinitary theories

is either definable by a quantifier-free formula with constants from

Introduction to the handbook of recursive mathematics

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