The study of effective choice problems has been initiated in [I, 2]. The very notion of effective choice for a class of constructive objects can be identified with effective construction of relevant objects.The problems in question admit distinct variations in classes of constructive and positive models, and in most cases, are amenable to transparent formalization. Thus, in studies dealing with the autostability of models and intrinsically recursive enumerability of relations on them, these notions can be specified naturally with regard for the possibility of constructing effectively a reducing function for ~ the two given constructivizations and, correspondingly, an enumeration procedure for the given constructivization and relation.Another important aspect in the study of effective choice problems arises from the fact (see [3]) that without effective choice, the notions of autostability and intrinsically recursive enumerability in classes of constructive models cannot be described naturally in terms of the algebraic properties of models.In this paper we adopt the notation and follow the definitions of [4][5][6][7].Let ,~![ be a countable model in a fixed finite signature _~, containing a predicate symbol for equality.By ~L~} we denote the set of all relations on L~ (subsets of an appropriate Cartesian power I~ v] ) that are stable under automorphisms of L~,~.
This paper is a continuation of the study of the branching properties of models [i]. We generalize the very concept of branching, and thus we are able to apply the basic theorems to a wider class of models.The study of branching models enables us to consider, from a unique point of view, many earlier results on the hyperimmunity of sets of atoms in Boolean algebras [2,3], on the complexity of ~ and ~)'in ~+~' , and on the relation of succession in linear orderings [5] (see [4]).Moreover, using Theorem i, we obtain a characterization of autostable dense linear orderings with a one-place predicate.
In [1], we described effectively enumerable relations on constructive and positive models. Similar problems arise if, instead of effective choice operators, we use effective operations --extensional partial recursive functions --that determine the indices of the corresponding recursively enumerable and recursive sets from the indices of constructivizations (positive enumerations).In this paper we analyze the potential for such a change and obtain relevant criteria for recursive and r.e. relations. Moreover, we demonstrate a number of counterexamples which elucidate the role of the domain and range in effective operations.Below we recall some definitions from the constructive model theory [2, 3] and give a list of the notation. Fix some finite language L containing predicate symbols P~°,...,P~ and functional symbols f~o°,...,fy '-A model fir of the language L is called constructivizable, provided there exists a surjection v : N --~ [NI such that the sets {< i, ml,...,,~, > I~ ~ e~(~ml,...,~-~,), i__ k}, are recursive. We call the pair (gR, v) a constructive model and v a constructivization of 9Yr.In what follows, Po is assumed to be always interpreted as the equality relation in a model, and we will concern ourselves with injective constructivizations only.Note that (93I, v) is a constructive model if and only if the diagram D~(99I) (i.e., the set of all atomic sentences of L U {co, ct,...} and their negations, true in fiR, under the interpretation c/-¢r~ v(i)) is recursive with respect to the GSdel numbering. By Dt(~Tt) we denote the subset of D~(gY~) containing only sentences with constants ci (i < t). Obviously, D~(gR) = [.J Dt~(g3~).t
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