Generalization of Shapiro’s theorem to higher arities and noninjective notations

Abstract

In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination.

Degrees of relations on canonically ordered natural numbers and integers

“…In the end, degree spectra provide mathematical tools for investigating the following question: how difficult is it to compute the relation in notations in which all the basic relations are computable? For further discussion, we refer the reader to [1,24,25].…”