Springer Nature number S en https://doi.org/10.1007/978-3-642-22993-0_31 2011-01-01 length finite languages number of states time transition function results best algorithm words similar results Minimising Automata number algorithm number k 327-338 2011 On Minimising Automata with Errors minimisation automata total DFAs problem computation chapter comparison state calculations error minimal DFA least number smallest DFA chapters partial transition functions true language previous algorithms parallel 2022-01-01T19:11 DFA words of length NP https://scigraph.springernature.com/explorer/license/ new algorithm The problem of k-minimisation for a DFA M is the computation of a smallest DFA N (where the size |M | of a DFA M is the size of the domain of the transition function) such that L(M) ΔL(N) ⊆ Σ< k, which means that their recognized languages differ only on words of length less than k. The previously best algorithm, which runs in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(\mid M \mid{\rm log}^{2} n)$\end{document} where n is the number of states, is extended to DFAs with partial transition functions. Moreover, a faster \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(\mid M \mid\log n)$\end{document} algorithm for DFAs that recognise finite languages is presented. In comparison to the previous algorithm for total DFAs, the new algorithm is much simpler and allows the calculation of a k-minimal DFA for each k in parallel. Secondly, it is demonstrated that calculating the least number of introduced errors is hard: Given a DFA M and numbers k and m, it is NP-hard to decide whether there exists a k-minimal DFA N with |L(M) ΔL(N) ≤ m. A similar result holds for hyper-minimisation of DFAs in general: Given a DFA M and numbers s and m, it is NP-hard to decide whether there exists a DFA N with at most s states such that |L(M) ΔL(N) ≤ m. function Institute for Natural Language Processing, Universität Stuttgart, Azenbergstraße 12, 70174, Stuttgart, Germany Institute for Natural Language Processing, Universität Stuttgart, Azenbergstraße 12, 70174, Stuttgart, Germany Artur Jeż Mathematical Foundations of Computer Science 2011 978-3-642-22993-0 978-3-642-22992-3 pub.1018296198 dimensions_id Gawrychowski Paweł Springer Nature - SN SciGraph project Filip Murlak doi 10.1007/978-3-642-22993-0_31 Information and Computing Sciences Maletti Andreas Piotr Sankowski 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 Computation Theory and Mathematics