Computability of Fraïssé limits

Abstract

Fraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

Some new computable structures of high rank

“…We begin this section by proving the following: Our structure will be a Fraïssé limit, obtained from a class K of finite structures satisfying the hereditary, amalgamation, and joint embedding properties, abbreviated HP , AP , and JEP . For a discussion of Fraïssé limits from the point of view of computability, see [3]. We note that Henson [7] gave an example of a homogeneous triangle-free graph.…”

Computability and categoricity of weakly ultrahomogeneous structures

“…, a n of A such that any isomorphism between finitely generated substructures A becomes ultrahomogeneous when constants representing these elements are added to the language. Csima, Harizanov, R. Miller and A. Montalban [8] studied computable ages and the computability of the canonical ultrahomogeneous structures, called Fraïssé limits.…”

“…The notion of a computably homogeneous structure is defined in [8], as follows. Given a structure A and a tuple − → a = (a 1 , .…”

Hanf number for Scott sentences of computable structures

“…In [2], Csima et al give necessary and sufficient conditions for an age to give rise to a computable limit structure. They allow function symbols in the vocabulary, and the structures in the age are finitely generated, but not necessarily finite.…”

“…In particular, the embedding relation is not computable. The result in [2] was inspired by an old result of Goncharov [3] and Peretyat'kin [8], giving necessary and sufficient conditions for a countable homogeneous structure to have a decidable copy. The proof in [2], like those in [3] and [8], involves a priority construction, with guesses at the extension relation, and injury resulting from guesses that are not correct.…”