Which Regular Languages can be Efficiently Indexed?
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In the present work, we study the hierarchy of p-sortable languages: regular languages accepted by automata of width p. In this hierarchy, regular languages are sorted according to the new fundamental measure of NFA complexity p. Our main contributions are the following: (i) we show that the hierarchy is strict and does not collapse, (ii) we provide (exponential) upper and lower bounds relating the minimum widths of equivalent NFAs and DFAs, and (iii) we characterize DFAs of minimum p for a given ℒ via a co-lexicographic variant of the Myhill-Nerode theorem. Our findings imply that in polynomial time we can build an index breaking the worst-case conditional lower bound of Ω(π m), whenever the input NFA's width is at most ϵlog_2 m, for any constant 0 ≤ϵ < 1/2.
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