Co-lexicographically Ordering Automata and Regular Languages - Part I
نویسندگان
چکیده
The states of a finite-state automaton \(\mathcal {N} \) can be identified with collections words in the prefix closure regular language accepted by . But ordered, and among many possible orders very natural one is co-lexicographic order. Such naturalness stems from fact that it suggests transfer order to automaton’s states. This suggestion is, fact, concrete number papers automata admitting total ( co-lex for brevity) ordering have been proposed studied. class ordered — Wheeler turned out require just constant bits per transition represented enable expression matching queries time matched character. Unfortunately, not all totally as previously outlined. In present work, we lay new theory showing always partially an intrinsic measure their complexity defined effectively determined, namely, minimum width p admissible partial —dubbed here We first show this captures at once several seemingly-unrelated hard problems on automata. Any NFA : (i) has equivalent powerset DFA whose size exponential rather than (as classic analysis shows) NFA’s size; (ii) encoded using Θ (log ) transition; (iii) admits linear-space data structure solving proportional 2 Some consequences parameterization are PSPACE-hard such equivalence FPT , quadratic lower bounds problem do hold sufficiently small Having established fundamental measure, proceed determining its computational extending notion languages studying smallest-width accepting NFAs DFAs. work focus deterministic case prove canonical minimum-width {L} Hasse {H} —can exhibited. provides, precise sense, best way (partially) any long want maintain operational link (co-lexicographic) ’s prefixes. Finally, explore relationship between two conflicting objectives: minimizing DFA. context, provide analogue Myhill-Nerode Theorem co-lexicographically languages.
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ژورنال
عنوان ژورنال: Journal of the ACM
سال: 2023
ISSN: ['0004-5411', '1557-735X']
DOI: https://doi.org/10.1145/3607471