COMPUTABLE ISOMORPHISMS, DEGREE SPECTRA OF RELATIONS, AND SCOTT FAMILIES

The spectrum of a relation R on a computable structure is the set of T uring degrees of the image of R under all isomorphisms between A and a ny other computable structure B. The relation R is intrinsically compu tably 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 struct ure. Moreover, the isomorphism can be constructed in such a way that t he image of the minimum element (if it exists) of the partially ordere d 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 cate gorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computa bly categorical structures that remain computably categorical under ex pansions by constants but have no Scott family. (C) 1998 Published by Elsevier Science B.V. All rights reserved.