📝 SummarXiv

Abstract

It is known that there exists a limit-computable function $f:{\mathbb N} \to {\mathbb N}$ which is not computable. Every known proof of this fact does not lead to the existence of a short computer program that computes $f$ in the limit. For $n \in {\mathbb N}$, let $E_n=\{1=x_k,~x_i+x_j=x_k,~x_i \cdot x_j=x_k:~i,j,k \in \{0,\ldots,n\}\}$. For $n \in {\mathbb N}$, $f(n)$ denotes the smallest $b \in {\mathbb N}$ such that if a system of equations $S \subseteq E_n$ has a solution in ${\mathbb N}^{n+1}$, then $S$ has a solution in $\{0,\ldots,b\}^{n+1}$. The author proved earlier that the function $f:{\mathbb N} \to {\mathbb N}$ is computable in the limit and eventually dominates every computable function $g:{\mathbb N} \to {\mathbb N}$. We present a short program in MuPAD which for $n \in {\mathbb N}$ prints the sequence $\{f_i(n)\}_{i=0}^\infty$ of non-negative integers converging to $f(n)$. For $n \in {\mathbb N}$, $\beta(n)$ denotes the smallest $b \in {\mathbb N}$ such that if a system of equations $S \subseteq E_n$ has a unique solution in ${\mathbb N}^{n+1}$, then this solution belongs to $\{0,\ldots,b\}^{n+1}$. The author proved earlier that the function $\beta:{\mathbb N} \to {\mathbb N}$ is computable in the limit and eventually dominates every function $\delta:{\mathbb N } \to {\mathbb N}$ with a single-fold Diophantine representation. The computability of $\beta$ is unknown. We present a short program in MuPAD which for $n \in {\mathbb N}$ prints the sequence $\{\beta_i(n)\}_{i=0}^\infty$ of non-negative integers converging to $\beta(n)$.