On the length of uncompletable words in unambiguous automata

This paper deals with uncomplete unambiguous automata. In this setting, we investigate the minimal length of uncompletable words. This problem is connected with a well-known conjecture formulated by A. Restivo. We introduce the notion of relatively maximal row for a suitable monoid of matrices. We show that, if M is a monoid of {0, 1}-matrices of dimension n generated by a set S, then there is a matrix of M containing a relatively maximal row which can be expressed as a product of O(n(3)) matrices of S. As an application, we derive some upper bound to the minimal length of an uncompletable word of an uncomplete unambiguous automaton, in the case that its transformation monoid contains a relatively maximal row which is not maximal. Finally we introduce the maximal row automaton associated with an unambiguous automaton AA. It is a deterministic automaton, which is complete if and only if AA is. We prove that the minimal length of the uncompletable words of AA is polynomially bounded by the number of states of AA and the minimal length of the uncompletable words of the associated maximal row automaton.