Bakhadyr Khoussainov

Computable isomorphisms, degree spectra of relations, and Scott families

Abstract The spectrum of a relation R on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between A and any other computable structure B . The relation R is intrinsically computably enumerable (c.e.) if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of the minimum element (if it exists) of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family.

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 .)

Models and Computability: Effective Model Theory: The Number of Models and Their Complexity

Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

Finite nondeterministic automata: simulation and minimality

Abstract Motivated by recent applications of finite automata to theoretical physics, we study the minimization problem for nondeterministic automata (with outputs, but no initial states). We use Ehrenfeucht–Fraisse-like games to model automata responses and simulations. The minimal automaton is constructed and, in contrast with the classical case, proved to be unique up to an isomorphism. Finally, we investigate the partial ordering induced by automata simulations. For example, we prove that, with respect to this ordering, the class of deterministic automata forms an ideal in the class of all automata.

Randomness, computability and algebraic specifications

Abstract This paper shows how the notion of randomness defines, in a natural way, an algebra. It turns out that the algebra is computably enumerable and finitely generated. The paper investigates algebraic and effective properties of this algebra.

Decidable Kripke models of intuitionistic theories

Abstract In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T .

Recursive unary algebras and trees

Abstract A unary algebra is an algebraic system A = (A, ƒ 0 ,…,ƒ n ) , where ƒ 0 ,…,ƒ n are unary operations on A and n ∈ ω. In the paper we develop the theory of effective unary algebras. We investigate well-known questions of constructive (recursive) model theory with respect to the class of unary algebras. In the paper we construct unary algebras with a finite number of recursive isomorphism types. We give the notions of program, uniform, and algebraic dimensions of models, and then we investigate these notions on unary algebras. We find connections between algebraic and effective properties of r.e. representable unary algebras. We also deal with finitely generated r.e. (positive) unary algebras. We show the connections between trees and unary algebras. Our interests also concern recursive automorphisms groups, r.e. subalgebra and congruence lattices of effective unary algebras.

Computable Kripke models and intermediate logics

We introduce effectiveness considerations into model theory of intuitionistic logic. We investigate effectiveness of completeness (by Kripke) results for intermediate logics such as intuitionistic logic, classical logic, constant domain logic, directed frames logic, and Dummetts logic.

Effective Properties of Finitely Generated R.E. Algebras

In this paper we are interested in the properties of finitely generated r.e. structures. Well-known examples of such structures are finitely presented groups, rings, etc. The class of r.e. structures is a very natural class which contains the class of recursive structures. Though the theory of recursive structures has been intensively investigated and some study of r.e. structures has been done, we think that a well-developed theory of r.e. structures is still missing. Our interest in finitely generated r.e. structures also comes from the theoretical computer science, mostly from the theory of algebraic specifications and abstract data types. We would like to briefly explain this phenomenon.

Games with Unknown Past

We define a new type of two player game occurring on a tree. The tree may have no root and may have arbitrary degrees of nodes. These games extend the class of games considered by Gurevich-Harrington in [5]. We prove that in the game one of the players has a winning strategy which depends on finite bounded information about the past part of a play and on future of each play that is isomorphism types of tree nodes. This result extends further the Gurevich-Harrington determinacy theorem from [5].