On polynomial-modular recursive sequences

Abstract

Abstract We consider recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function |x|; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [x]. $\begin{array}{} \displaystyle [\sqrt{x}]. \end{array}$ For the set of such recursive sequences, an algorithmically unsolvable problem is indicated.