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In Search of Most Complex Regular Languages

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Implementation and Application of Automata (CIAA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7381))

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Abstract

Regular languages that are most complex under common complexity measures are studied. In particular, certain ternary languages U n (a,b,c), \(n\geqslant 3\), over the alphabet {a,b,c} are examined. It is proved that the state complexity bounds that hold for arbitrary regular languages are also met by the languages U n (a,b,c) for union, intersection, difference, symmetric difference, product (concatenation) and star. Maximal bounds are also met by U n (a,b,c) for the number of atoms, the quotient complexity of atoms, the size of the syntactic semigroup, reversal, and 22 combined operations, 5 of which require slightly modified versions. The language U n (a,b,c,d) is an extension of U n (a,b,c), obtained by adding an identity input to the minimal DFA of U n (a,b,c). The witness U n (a,b,c,d) and its modified versions work for 14 more combined operations. Thus U n (a,b,c) and U n (a,b,c,d) appear to be universal witnesses for alphabets of size 3 and 4, respectively.

This work was supported by the Natural Sciences and Engineering Research Council of Canada under grant No. OGP0000871.

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Authors and Affiliations

  1. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Janusz Brzozowski

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  1. Janusz Brzozowski
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Editors and Affiliations

  1. Departamento de Ciência de Computadores, Universidade do Porto, Faculdade de Ciências, Rua do Campo Alegre 1021-1055, 4169-007, Porto, Portugal

    Nelma Moreira  & Rogério Reis  & 

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Brzozowski, J. (2012). In Search of Most Complex Regular Languages. In: Moreira, N., Reis, R. (eds) Implementation and Application of Automata. CIAA 2012. Lecture Notes in Computer Science, vol 7381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31606-7_2

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  • DOI: https://doi.org/10.1007/978-3-642-31606-7_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31605-0

  • Online ISBN: 978-3-642-31606-7

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