A Synergic Approach to the Minimal Uncompletable Words Problem
Sandrine Julia, Arnaud Malapert, Julien Provillard · HAL (Le Centre pour la Communication Scientifique Directe) · 2017
A finite language~$X$ over an alphabet~$\S$ is complete if any word in~$\S^*$ is a factor of a word in~$X^*$. A word which is not a factor of~$X^*$ is said uncompletable. Among them, some are \mi\ as all their proper factors belong to~$\fact(X^*)$. The problem is to find bounds on the length of the shortest \mi\ \un\ words depending on~$k$, the maximal length of words in~$X$. Though Restivo's conjecture stating an upper bound in~$2k^2$ was already contradicted twice, the problem of the existence of a quadratic upper bound is still open. Our approach is original and synergic. We start by characterizing minimal uncompletable words. An efficient in practice algorithm is given that speeds up the search of such words. Finally, a genetic algorithm using a SAT-solver allows us to obtain new results for the first values of~$k$.