## In Search Of Most Complex Regular Languages

dc.contributor.author | Brzozowski, Janusz | |

dc.date.accessioned | 2017-09-29 14:03:10 (GMT) | |

dc.date.available | 2017-09-29 14:03:10 (GMT) | |

dc.date.issued | 2013-09-01 | |

dc.identifier.uri | http://dx.doi.org/10.1142/S0129054113400133 | |

dc.identifier.uri | http://hdl.handle.net/10012/12514 | |

dc.description | Electronic version of an article published as International Journal of Foundations of Computer Science, 24(06), 2013, 691–708. http://dx.doi.org/10.1142/S0129054113400133 © World Scientific Publishing Company http://www.worldscientific.com/ | en |

dc.description.abstract | Sequences (L-n vertical bar n >= k), called streams, of regular languages L-n are considered, where k is some small positive integer, n is the state complexity of L-n, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: (1) the state complexity n of L-n, that is the number of left quotients of L-n (used as a reference); (2) the state complexities of the left, quotients of L-n; (3) the number of atoms of L-n; (4) the state complexities of the atoms of L-n; (5) the size of the syntactic semigroup of L; and the state complexities of the following operations: (6) the reverse of L-n; (7) the star; (8) union, intersection, difference and symmetric difference of and L-n; and the concatenation of L-m and L-n. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (U-n (a, b, c) vertical bar n >= 3 is defined by the deterministic finite automaton with state set {0, 1, ..., n-1}, initial state 0, set {n-1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the;a states, b transposes states 0 and 1, and c maps state n - 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use U-n (a, b, c) and U-n(a, b, c), where the roles of a and b are interchanged in the second language. In this sense, U-n (a, b, c) is a universal witness This witness and its extensions also apply to a large number of combined regular operations. | en |

dc.description.sponsorship | Natural Sciences and Engineering Research Council of Canada [OGP0000871] | en |

dc.language.iso | en | en |

dc.publisher | World Scientific Publishing | en |

dc.subject | Atom | en |

dc.subject | complexity of operation | en |

dc.subject | finite automaton | en |

dc.subject | quotient complexity | en |

dc.subject | regular language | en |

dc.subject | state complexity | en |

dc.subject | syntactic semigroup | en |

dc.subject | Witness | en |

dc.title | In Search Of Most Complex Regular Languages | en |

dc.type | Article | en |

dcterms.bibliographicCitation | Brzozowski, J. (2013). IN SEARCH OF MOST COMPLEX REGULAR LANGUAGES. International Journal of Foundations of Computer Science, 24(06), 691–708. https://doi.org/10.1142/S0129054113400133 | en |

uws.contributor.affiliation1 | Faculty of Mathematics | en |

uws.contributor.affiliation2 | David R. Cheriton School of Computer Science | en |

uws.typeOfResource | Text | en |

uws.peerReviewStatus | Reviewed | en |

uws.scholarLevel | Faculty | en |