On the Height of Towers of Subsequences and Prefixes

A tower is a sequence of words alternating between two languages in such a way that every word is a subsequence of the following word. The height of the tower is the number of words in the sequence. If there is no infinite tower (a tower of infinite height), then the height of all towers between the languages is bounded. We study upper and lower bounds on the height of maximal finite towers with respect to the size of the NFA (the DFA) representation of the languages. We show that the upper bound is polynomial in the number of states and exponential in the size of the alphabet, and that it is asymptotically tight if the size of the alphabet is fixed. If the alphabet may grow, then, using an alphabet of size approximately the number of states of the automata, the lower bound on the height of towers is exponential with respect to that number. In this case, there is a gap between the lower and upper bound, and the asymptotically optimal bound remains an open problem. Since, in many cases, the constructed towers are sequences of prefixes, we also study towers of prefixes.