Polynomial Generators of Recursively Enumerable Languages

Abstract

For each language L, let $\hat{\mathcal F}_\cap(L)$ be the smallest intersection-closed full AFL generated by the language L. Furthermore, for each natural number k≥ 2 let $P_k=\{a^{n^k}|n\in\mathbb N\}$. By applying certain classical and recent results on Diophantine equations we show that $\mathcal L_{RE}=\hat{\mathcal F}_\cap(P_k)$, i.e., the family of all recursively enumerable languages coincides with the smallest intersection-closed full AFL generated by the polynomial language P _k for all k≥ 2. This allows us to answer to an open problem of S. Ginsburg and J. Goldstine in [2].