Computable Structures: Presentations Matter
Richard A. Shore · 1999
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 on A or R 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 of R. For example, what conditions guarantee that A 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 of A 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 conne...