In [2], a short and elegant proof was presented showing that a binary word of length n contains at most n − 3 runs. Here we show, using the same technique and a computer search, that the number of runs in a binary word of length n is at most 22 23 n < 0.957n.1991 Mathematics Subject Classification. 68R15.
a b s t r a c tA word w is a fixed point of a nontrivial morphism h if w = h(w) and h is not the identity on the alphabet of w. The paper presents the first polynomial algorithm deciding whether a given finite word is such a fixed point. The algorithm also constructs the corresponding morphism, which has the smallest possible number of non-erased letters.
Abstract.A tower between two regular languages is a sequence of strings such that all strings on odd positions belong to one of the languages, all strings on even positions belong to the other language, and each string can be embedded into the next string in the sequence. It is known that if there are towers of any length, then there also exists an infinite tower. We investigate upper and lower bounds on the length of finite towers between two regular languages with respect to the size of the automata representing the languages in the case there is no infinite tower. This problem is relevant to the separation problem of regular languages by piecewise testable languages.
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