Languages of Dot-Depth 3/2
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 i ∈ A* 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.
KeywordsRegular Language Transition Graph Membership Problem Deterministic Finite Automaton Algebraic Interpretation
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