by Christian Glaer, Heinz Schmitz
In Proceedings 17th Symposium on Theoretical Aspects of Computer Science
http://www-info4.informatik.uni-wuerzburg.de/person/mitarbeiter/glasser/publications/ddh32stacs.ps.gz
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Abstract:
Abstract. 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 u0L1u1L2u2 \Delta \Delta \Delta Lnun, where u i 2 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 \Sigma 2--formulas of the logic FO[!; min; max; S; P] follows. We give an algebraic interpretation of our result. 1
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