Computable Structures: Presentations Matter

The computability properties of a relation R not included in the language of a computable structure A can vary from one computable presentation to another. We describe some classic results giving conditions onA orR that restrict the possible variations in the computable dimension of A (i.e. the number of isomorphic copies of A up to computable isomorphism) and the computational complexity ofR. For example, what conditions guarantee thatA is computably categorical (i.e. of dimension 1) or that R is intrinsically computable (i.e. computable in every presentation). In the absence of such conditions, we discuss the possible computable dimensions ofA and variations (in terms of Turing degree) of R in different presentations (the degree spectrum of R). In particular, various classic theorems and more recent ones of the author, B. Khoussainov, D. Hirschfeldt and others about the possible degree spectra of computable relations on computable structures and the connections with computable dimension and categoricity will be discussed both in general model theoretic settings and in restricted classes of structures such as graphs, linear and partial orderings, lattices, Boolean algebras, Abelian and nilpotent groups, rings, integral domains, and real or algebraically closed fields.

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