pub.1010336549 dimensions_id chapter information false 2011 length size short words Hopcroft’s algorithm computation 2022-01-01T19:09 Given a language L and a number ℓ, an ℓ-cover automaton for L is a DFA M such that its language coincides with L on all words of length at most ℓ. It is known that an equivalent minimal ℓ-cover automaton can be constructed in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n \log n)$\end{document}, where n is the number of states of M. This is achieved by a clever and sophisticated variant of Hopcroft’s algorithm, which computes the ℓ-similarity inside the main algorithm. This contribution presents an alternative simple algorithm with running time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n \log n)$\end{document}, in which the computation is split into three phases. First, a compact representation of the gap table is created. Second, this representation is enriched with information about the length of a shortest word leading to the states. These two steps are independent of the parameter ℓ. Third, the ℓ-similarity is extracted by simple comparisons against ℓ. In particular, this approach allows the calculation of all the sizes of minimal ℓ-cover automata (for all valid ℓ) in the same time bound. gap same time calculations comparison https://scigraph.springernature.com/explorer/license/ phase algorithm words of length Computing All ℓ-Cover Automata Fast sophisticated variants number of states automata number 2011-01-01 language approach https://doi.org/10.1007/978-3-642-22256-6_19 simple comparison main algorithm en parameters alternative simple algorithm time compact representation table variants DFA chapters representation state words contribution simple algorithm step 203-214 978-3-642-22256-6 Implementation and Application of Automata 978-3-642-22255-9 Artur Jeż Springer Nature Andreas Maletti Béatrice Bouchou-Markhoff Pascal Caron Institute for Natural Language Processing, Universität Stuttgart, Azenbergstraße 12, 70174, Stuttgart, Germany Institute for Natural Language Processing, Universität Stuttgart, Azenbergstraße 12, 70174, Stuttgart, Germany Institute of Computer Science, University of Wrocław, ul. Joliot-Curie 15, 50–383, Wrocław, Poland Institute of Computer Science, University of Wrocław, ul. Joliot-Curie 15, 50–383, Wrocław, Poland doi 10.1007/978-3-642-22256-6_19 Springer Nature - SN SciGraph project Champarnaud Jean-Marc Psychology and Cognitive Sciences Maurel Denis Psychology