Estimation of the algorithmic complexity of classes of computable models

Abstract

We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ 0 2 -complete), computable models with ω-categorical theories (Δ 0 ω -complex Π 0 ω+2 -set), prime models (Δ 0 ω -complex Π 0 ω+2 -set), models with ω 1 -categorical theories (Δ 0 ω -complex Σ 0 ω+1 -set). We obtain a universal lower bound for the model-theoretic properties preserved by Marker's extensions (Δ 0 ω ).While studying the structural properties of theories and the algebraic properties of models we are especially interested in the complexity of the definitions of these properties from an algorithmic viewpoint. The characterization of classes, and hence, the abstract properties of models in this case can be obtained through bounds for index sets in the universal numbering of computable models [1, § 4]. Fix some computable signature σ with no function symbols. A model M of this signature is called computable if the universe of M and underlying relations are uniformly computable. Without loss of generality we will consider the models based on nonnegative integers.It is shown in [2] that given some finite signature with no function symbols there exists a universal computable numbering of computable models of this signature; i.e., some computable numbering of all computable models of the fixed signature such that every other computable numbering of computable models reduces to it. In the case of an infinite signature σ there exists a universal computable numbering of all computable models of σ together with all finite computable models of the finite parts of this signature [3, § 1.4; 4, § 1.3]. Both of these numberings are principal, i.e., every computable sequence of computable models of σ reduces to these numberings.The existence of this type of universal numberings of models allows us to obtain a characterization of classes of computable models by relating the index sets of concrete classes of models with a level in a suitable algorithmic hierarchy. The classes of algebraic systems in a signature with function symbols can be characterized using this approach via the representation of functions by their graphs which are predicates.Denote by ν σ = {M 0 , M 1 , . . . , M n , . . . } the universal computable numbering of models of the corresponding signature σ. Take some class K of models of σ closed under isomorphism.Definition 1. We call the index set I(K) of the class K of models the setMany articles deal with index sets [5][6][7][8][9][10][11][12], as well as some estimate of the index sets of classes of models [13].In this article we construct bounds in the corresponding complexity hierarchies for the following index sets of the available classes of computable models considered with various signatures:Fin σ = {i | M i is finite}, Cat σ = {i | M i has an ℵ 0 -categorical theory}, UCat σ = {i | M i has an ℵ 1 -categorical theory}, Prim σ = {i | M i is a prime model}.

Index Sets of Constructive Models of Bounded Signature that are Autostable Relative to Strong Constructivizations

“…In [6][7][8], it was shown that the given constructions allow of preserving the property of being prime and, as it is not hard to see, of being almost prime. The properties of being d-decidable for constructed models and of being d-autostable relative to d-strong constructivizations are also inherited.…”

“…PROPOSITION 1.1 [6][7][8]. For any finite relational signature σ, there exists a signature σ 1 that consists of a single predicate P and is such that for any model M of the signature σ, there exists a model M of the signature σ 1 such that: [6][7][8].…”