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    "description": "We consider a problem of hyper-minimisation of an automaton [2,3]: given a DFA M we want to compute a smallest automaton N such that the language L(M) \u0394L(N) is finite, where \u0394 denotes the symmetric difference. We improve the previously known \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\mathcal O (|\\Sigma|n^2)$\\end{document} solution by giving an expected \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\mathcal O (|\\delta|\\log n)$\\end{document} time algorithm for this problem, where |\u03b4| is the size of the (potentially partial) transition function. We also give a slightly slower deterministic \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\mathcal O(|\\delta|\\log^2 n)$\\end{document} version of the algorithm.Then we introduce a similar problem of k-minimisation: for an automaton M and number k we want to find a smallest automaton N such that L(M) \u0394L(N)\u2009\u2286\u2009\u03a3<\u2009k, i.e. the languages they recognize differ only on words of length less than k. We characterise such minimal automata and give algorithm with a similar complexity for this problem.", 
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