Co-lexicographically ordering automata and regular languages. Part I
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In the present work, we lay out a new theory showing that all automata can always be co-lexicographically partially ordered, and an intrinsic measure of their complexity can be defined and effectively determined, namely, the minimum width p of one of their admissible co-lex partial orders - dubbed here the automaton's co-lex width. We first show that this new measure captures at once the complexity of several seemingly-unrelated hard problems on automata. Any NFA of co-lex width p: (i) has an equivalent powerset DFA whose size is exponential in p rather than (as a classic analysis shows) in the NFA's size; (ii) can be encoded using just Θ(log p) bits per transition; (iii) admits a linear-space data structure solving regular expression matching queries in time proportional to p^2 per matched character. Some consequences of this new parameterization of automata are that PSPACE-hard problems such as NFA equivalence are FPT in p, and quadratic lower bounds for the regular expression matching problem do not hold for sufficiently small p. Having established that the co-lex width of an automaton is a fundamental complexity measure, we proceed by (i) determining its computational complexity and (ii) extending this notion from automata to regular languages by studying their smallest-width accepting NFAs and DFAs. In this work we focus on the deterministic case and prove that a canonical minimum-width DFA accepting a language ℒ - dubbed the Hasse automaton ℋ of ℒ - can be exhibited. Finally, we explore the relationship between two conflicting objectives: minimizing the width and minimizing the number of states of a DFA. In this context, we provide an analogous of the Myhill-Nerode Theorem for co-lexicographically ordered regular languages.
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