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

## Apoloniusz Tyszka

Citation: Proceedings of the 2014 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 2, pages 623629 ()

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Abstract. Let E_n={x_k=1, x_i+x_j=x_k, x_i · x_j=x_k: i,j,k ∊ {1,…,n}}. For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system S ⊆ E_n with a solution in non-negative integers x_1,…,x_n there exists a solution of S in non-negative integers not greater than b. We prove that if a function G:N\{0}-->N is computable, then f dominates G i.e. there exists a positive integer m such that G(n)<f(n) for any n ≥ m. For positive integers n,m, let g(n,m) denote the smallest non-negative integer b such that for each system S ⊆ E_n with a solution in {0,…,m-1}^n there exists a solution of S in {0,…,b}^n. Then, g(n,m) ≤ m-1, 0=g(n,1)<1=g(n,2) ≤ g(n,3) ≤ g(n,4) ≤ … and g(n,f(n))<f(n)=g(n,f(n)+1)=g(n,f(n)+2)=g(n,f(n)+3)=… We present an infinite loop in MuPAD which takes as input a positive integer n and returns g(n,m) on the m-th iteration.

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