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In Search of Most Complex Regular Languages

  • Conference paper

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

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.

Keywords

  • combined operation
  • finite automaton
  • operation
  • regular language
  • state complexity
  • syntactic semigroup
  • witness

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

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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

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