Effective choice operations on constructive and positive models

Abstract

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<w Fix a computable family S of all recursively enumerable diagrams of the language L U C (C = {co, ct,...}) and its certain computable enumeration 7. By a recursively enumerable index of constructivization g we mean 3' --an r.e. index of D~(ffR).