Languages of Dot-Depth 3/2

  • Christian Glaßer
  • Heinz Schmitz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1770)


We prove an effective characterization of languages having dot-depth 3/2. Let B 3/2 denote this class, i.e., languages that can be written as finite unions of languages of the form u 0 L 1 u 1 L 2 u 2 ...L n u n, where u iA* and L i are languages of dot-depth one. Let F be a deterministic finite automaton accepting some language L. Resulting from a detailed study of the structure of B 3/2, we identify a pattern P (cf. Fig. 2) such that L belongs to B 3/2 if and only if F does not have pattern P in its transition graph. This yields an NL-algorithm for the membership problem for B 3/2.

Due to known relations between the dot-depth hierarchy and symbolic logic, the decidability of the class of languages definable by Σ 2-formulas of the logic FO[<, min, max, S, P] follows. We give an algebraic interpretation of our result.


Regular Language Transition Graph Membership Problem Deterministic Finite Automaton Algebraic Interpretation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Christian Glaßer
    • 1
  • Heinz Schmitz
    • 1
  1. 1.Theoretische InformatikUniversität WürzburgWürzburgGermany

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