On the Size Complexity of Deterministic Frequency Automata

  • Rūsiņš Freivalds
  • Thomas Zeugmann
  • Grant R. Pogosyan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7810)

Abstract

Austinat, Diekert, Hertrampf, and Petersen [2] proved that every language L that is (m,n)-recognizable by a deterministic frequency automaton such that m > n/2 can be recognized by a deterministic finite automaton as well. First, the size of deterministic frequency automata and of deterministic finite automata recognizing the same language is compared. Then approximations of a language are considered, where a language L′ is called an approximation of a language L if L′ differs from L in only a finite number of strings. We prove that if a deterministic frequency automaton has k states and (m,n)-recognizes a language L, where m > n/2, then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more than k states.

Austinat et al. [2] also proved that every language L over a single-letter alphabet that is (1,n)-recognizable by a deterministic frequency automaton can be recognized by a deterministic finite automaton. For languages over a single-letter alphabet we show that if a deterministic frequency automaton has k states and (1,n)-recognizes a language L then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more that k states. However, there are approximations such that our bound is much higher, i.e., k!.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Rūsiņš Freivalds
    • 1
    • 2
  • Thomas Zeugmann
    • 2
  • Grant R. Pogosyan
    • 3
  1. 1.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia
  2. 2.Division of Computer ScienceHokkaido UniversitySapporoJapan
  3. 3.Division of Natural SciencesInternational Christian UniversityMitakaJapan

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