On the Minimal Uncompletable Word Problem for Unambiguous Automata

This paper deals with nite (possibly not complete) unambiguous automata, not necessarily deterministic. In this setting, we investigate the problem of the minimal length of the uncompletable word. This problem is associated with the well-known conjecture formulated by A. Restivo. We introduce the concept of relatively maximal row for a suitable set of matrices, and show the existence of a relatively maximal row of length of quadratic order with respect to the number of the states of the treated automaton. We give some estimates of the maximal length of the minimal uncompletable word in connection with the number the states of the involved automaton and the length of a suitable relatively maximal but not maximal word, provided that it exists. In the general case, we establish an estimate of the length of the minimal uncompletable word in terms of the number of states of the studied automaton, the length of a suitable relatively maximal word and the minimal length of the uncompletable word of the automaton formed by all associated maximal rows.