Valentina S. Harizanov

Enumerations in computable structure theory

Abstract We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α , we transform a countable directed graph G into a structure G ∗ such that G has a Δ α 0 isomorphic copy if and only if G ∗ has a computable isomorphic copy. A computable structure A is Δ α 0 categorical (relatively Δ α 0 categorical, respectively) if for all computable (countable, respectively) isomorphic copies B of A , there is an isomorphism from A onto B , which is Δ α 0 ( Δ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is a computable, Δ α 0 categorical structure, which is not relatively Δ α 0 categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical. An additional relation R on the domain of a computable structure A is intrinsically Σ α 0 (relatively intrinsically Σ α 0 , respectively) on A if in all computable (countable, respectively) isomorphic copies B of A , the image of R is Σ α 0 ( Σ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is an intrinsically Σ α 0 relation on a computable structure, which not relatively intrinsically Σ α 0 . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable. The Δ α 0 dimension of a structure A is the number of computable isomorphic copies, up to Δ α 0 isomorphisms. We prove that for every computable successor ordinal α and every n ≥ 1 , there is a computable structure with Δ α 0 dimension n . This generalizes the result of Goncharov that there is a structure of computable dimension n for every n ≥ 1 . Finally, we prove that for every computable successor ordinal α , there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that Δ α 0 relative to X is not Δ α 0 . In particular, for every finite n , there is a structure with isomorphic copies in exactly the non- low n Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure A with isomorphic copies in exactly the nonzero Turing degrees.

Some effects of Ash–Nerode and other decidability conditions on degree spectra

Abstract With every new recursive relation R on a recursive model A , we consider the images of R under all isomorphisms from A to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on A , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on A , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum.

The possible turing degree of the nonzero member in a two element degree spectrum

Abstract We construct a recursive model A , a recursive subset R of its domain, and a (nonzero) Turing degree x ⩽ 00 satisfying the following condition. The nonrecursive images of R under all isomorphisms from A to other recursive models are of Turing degree x and cannot be recursively enumerable.

Uncountable degree spectra

Abstract We consider a recursive model A and an additional recursive relation R on its domain, such that there are uncountably many different images of R under isomorphisms from A to some (and therefore every) recursive model B isomorphic to A . We study properties of the set of Turing degrees of all these isomorphic images of R on the domain of B .

TURING DEGREES OF CERTAIN ISOMORPHIC IMAGES OF COMPUTABLE RELATIONS

Abstract A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A . That is, R is not named in the language of A . We define Dg A (R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models. We investigate conditions on A and R which are sufficient and necessary for Dg A (R) to contain every Turing degree. These conditions imply that if every Turing degree ⩽ 0″ can be realized in Dg A (R) via an isomorphism of the same Turing degree as its image of R, then Dg A (R) contains every Turing degree. We also discuss an example of A and R whose Dg A (R) coincides with the Turing degrees which are ⩽ 0′.

Trivial, strongly minimal theories are model complete after naming constants

We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(M M ) is a model complete £ M -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0-decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is Σ 0 5.

Chapter 1 Pure computable model theory

Publisher Summary This chapter discusses the foundations of computable model theory: the definitions and examples of decidable theories and computable and decidable models. Exploiting the fundamental concepts of computability theory, computable model theory introduces effective analogues of model-theoretic notions. The generalization of the definition of a particular computable algebraic structure to an arbitrary model yields one of the basic concepts of pure computable model theory, an area of logic developed in the past several years, that is, the notion of a computable model and a stronger notion of a decidable model. Lerman and Schmerl have given examples of theories with computable models. The chapter presents the Effective Completeness Theorem, the Effective Omitting Types Theorem, and the characterizations of decidable theories with decidable prime models and then with decidable saturated models. The chapter also provides results on decidable theories with only finitely many and decidable theories with only countably many nonisomorphic countable models and investigates the model-theoretic nature and the computability-theoretic complexity of models of such theories. The chapter discusses the isomorphisms of effective models and related subtopics, such as intrinsically relations, computably stable models, and computably categorical models.

Spectra of highn and non-lown degrees

We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of non-lown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order ℚ. New results include realizations of the set of non-lown Turing degrees as the spectrum of a relation on ℚ for all n≥1, and a realization of the set of all non-lown Turing degrees as the spectrum of a linear order whenever n≥2. The state of current knowledge is summarized in a table in the concluding section.

Describing free groups

We consider countable free groups of different ranks. For these groups, we investigate computability theoretic complexity of index sets within the class of free groups and within the class of all groups. For a computable free group of infinite rank, we consider the difficulty of finding a basis.

COMPACTNESS OF THE SPACE OF LEFT ORDERS

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.