We examine the role that atoms of regular languages play in boolean automata. We observe that the size of a minimal boolean automaton of a regular language is directly related to the number of atoms of the language. We present a method to construct minimal boolean automata, using the atoms of a given regular language. The "illegal" cover problem of the Kameda-Weiner method for NFA minimization implies that using the union operation only to construct an automaton from a cover - as is the case with NFAs -, is not sufficient. We show that by using the union and the intersection operations (without the complementation operation), it is possible to construct boolean automata accepting a given language, for a given maximal cover.
@InProceedings{tamm:LIPIcs.MFCS.2021.86, author = {Tamm, Hellis}, title = {{Boolean Automata and Atoms of Regular Languages}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {86:1--86:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.86}, URN = {urn:nbn:de:0030-drops-145267}, doi = {10.4230/LIPIcs.MFCS.2021.86}, annote = {Keywords: Boolean automaton, Regular language, Atoms} }
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