On Upper and Lower Bounds on the Length of Alternating Towers

Abstract

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.

On the height of towers of subsequences and prefixes

“…For a binary alphabet, the upper bound of Theorem 1 gives n 2 + n + 1 and it is known to be tight up to a linear factor [5]. Namely, for every odd positive integer n, there are two binary NFAs with n − 1 and n states having a tower of height n 2 − 4n + 5 and no infinite tower.…”

“…For a four-letter alphabet and for every n ≥ 1, there are two NFAs with at most n states having a tower of height Ω(n 3 ) and no infinite tower [5,Theorem 3]. We now improve this bound by generalizing Theorem 3.…”

“…Considering the lower bound, we first improve in Theorem 3 an existing bound for binary regular languages [5]. Theorem 5 and its corollaries then show that the upper bound is asymptotically tight if the alphabet is fixed.…”

“…The height of maximal finite towers is closely related, for instance, to the complexity of an algorithm computing a piecewise testable separator of two regular languages [5]. Namely, the algorithm requires at least as many steps as is the height of the maximal tower.…”

Piecewise Testable Languages and Nondeterministic Automata

“…Piecewise testable languages form a strict subclass of star-free languages, that is, of the limit of the above-mentioned hierarchies or, in other words, of the languages definable by LTL logic. They are investigated in natural language processing [10,27], in cognitive and sub-regular complexity [28], in learning theory [11,20], and in databases in the context of XML schema languages [7,13,14]. They have been extended from words to trees [4,12].…”