On the widths of regular and context free languages, with an application to information flow
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Given a partially-ordered finite alphabet Σ and a language L ⊆Σ^*, how large can an antichain in L be (where L is ordered by the lexicographic ordering)? This fundamental property of partial orders is known as the width and its computation is the central extremal problem for partially ordered sets. More precisely, since L will in general be infinite, we should ask about the rate of growth of maximum antichains consisting of words of length n. In this work, we show that if L is regular or context free then there is a dichotomy between polynomial and exponential antichain growth. For regular languages we give a polynomial time algorithm to distinguish the two cases, even if L is specified as an NFA. On the other hand for context free languages we show that the problem of distinguishing the two cases is undecidable. We generalise the lexicographic order to tree languages, and show that for regular tree languages there is a trichotomy between polynomial, exponential and doubly exponential antichain growth. Finally we discuss the motivating problem for this work, which is related to information flow in the context of computer security.
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