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ACM"],"published-print":{"date-parts":[[2023,8,31]]},"abstract":"<jats:p>\n            The states of a finite-state automaton \ud835\udca9 can be identified with collections of words in the prefix closure of the regular language accepted by \ud835\udca9. But words can be ordered, and among the many possible orders a very natural one is the co-lexicographic order. Such naturalness stems from the fact that it suggests a transfer of the order from words to the automaton\u2019s states. This suggestion is, in fact, concrete and in a number of articles automata admitting a\n            <jats:italic>total<\/jats:italic>\n            co-lexicographic (\n            <jats:italic>co-lex<\/jats:italic>\n            for brevity) ordering of states have been proposed and studied. Such class of ordered automata \u2014\n            <jats:italic>Wheeler automata<\/jats:italic>\n            \u2014 turned out to require just a constant number of bits per transition to be represented and enable regular expression matching queries in constant time per matched character.\n          <\/jats:p>\n          <jats:p>\n            Unfortunately, not all automata can be totally ordered as previously outlined. In the present work, we lay out a new theory showing that all automata can always be\n            <jats:italic>partially<\/jats:italic>\n            ordered, and an intrinsic measure of their complexity can be defined and effectively determined, namely, the minimum width\n            <jats:italic>p<\/jats:italic>\n            of one of their admissible\n            <jats:italic>co-lex partial orders<\/jats:italic>\n            \u2013dubbed here the automaton\u2019s\n            <jats:italic>co-lex width<\/jats:italic>\n            . We first show that this new measure captures\n            <jats:italic>at once<\/jats:italic>\n            the complexity of several seemingly-unrelated hard problems on automata. Any NFA of co-lex width\n            <jats:italic>p<\/jats:italic>\n            : (i) has an equivalent powerset DFA whose size is exponential in\n            <jats:italic>p<\/jats:italic>\n            rather than (as a classic analysis shows) in the NFA\u2019s size; (ii) can be encoded using just \u0398(log\n            <jats:italic>p<\/jats:italic>\n            ) bits per transition; (iii) admits a linear-space data structure solving regular expression matching queries in time proportional to\n            <jats:italic>p<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            per matched character. Some consequences of this new parameterization of automata are that PSPACE-hard problems such as NFA equivalence are FPT in\n            <jats:italic>p<\/jats:italic>\n            , and quadratic lower bounds for the regular expression matching problem do not hold for sufficiently small\n            <jats:italic>p<\/jats:italic>\n            .\n          <\/jats:p>\n          <jats: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 \u2112\u2013dubbed the Hasse automaton \u210b of \u2112\u2013can be exhibited. \u210b provides, in a precise sense, the best possible way to (partially) order the states of any DFA accepting \u2112, as long as we want to maintain an operational link with the (co-lexicographic) order of \u2112\u2019s prefixes. 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 analogue of the Myhill-Nerode Theorem for co-lexicographically ordered regular languages.<\/jats:p>","DOI":"10.1145\/3607471","type":"journal-article","created":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T11:57:12Z","timestamp":1688731032000},"page":"1-73","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":["Co-lexicographically Ordering Automata and Regular Languages - Part I"],"prefix":"10.1145","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1402-5298","authenticated-orcid":false,"given":"Nicola","family":"Cotumaccio","sequence":"first","affiliation":[{"name":"Gran Sasso Science Institute, Italy and Dalhousie University, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8920-483X","authenticated-orcid":false,"given":"Giovanna","family":"D\u2019Agostino","sequence":"additional","affiliation":[{"name":"University of Udine, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8502-5896","authenticated-orcid":false,"given":"Alberto","family":"Policriti","sequence":"additional","affiliation":[{"name":"University of Udine, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3553-4953","authenticated-orcid":false,"given":"Nicola","family":"Prezza","sequence":"additional","affiliation":[{"name":"Ca\u2019 Foscari University of Venice, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2023,8,12]]},"reference":[{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/DCC52660.2022.00013"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975994.55"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2021.104820"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1109\/DCC.2019.00020"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/1322432.1322433"},{"key":"e_1_3_4_7_2","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2016.56"},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/btv603"},{"key":"e_1_3_4_9_2","doi-asserted-by":"crossref","unstructured":"Ruben Becker Davide Cenzato Sung-Hwan Kim Bojana Kodric Alberto Policriti and Nicola Prezza. 2023. 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