MuPAD codes which implement limit-computable functions that cannot be bounded by any computable function

Abstract

For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system S \subseteq {x_k=1,x_i+x_j=x_k,x_i*x_j=x_k: i,j,k \in {1,...,n}} with a solution in non-negative integers x_1,...,x_n, there exists a solution of S in {0,...,b}^n. We prove that the function f is strictly increasing and dominates all computable functions. We present an infinite loop in MuPAD which takes as input a positive integer n and returns a non-negative integer on each iteration. Let g(n,m) denote the number returned on the m-th iteration, if n is taken as input. Then, g(n,m) \leq m-1, 0=g(n,1)<1=g(n,2) \leq g(n,3) \leq g(n,4) \leq ... and g(n,f(n))<f(n)=g(n,f(n)+1)=g(n,f(n)+2)=g(n,f(n)+3)=.... A MuPAD code constructed on another principle contains a repeat-until loop and implements a limit-computable function \xi: N-->N that cannot be bounded by any computable function. This code takes as input a non-negative integer n, immediately returns 0, and computes a system S of polynomial equations. If the loop terminates for S, then the next instruction is executed and returns \xi(n).