Hierarchies and reducibilities on regular languages related to modulo counting

Selivanov, Victor L.

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 95-132

We discuss some known and introduce some new hierarchies and reducibilities on regular languages, with the emphasis on the quantifier-alternation and difference hierarchies of the quasi-aperiodic languages. The non-collapse of these hierarchies and decidability of some levels are established. Complete sets in the levels of the hierarchies under the polylogtime and some quantifier-free reducibilities are found. Some facts about the corresponding degree structures are established. As an application, we characterize the regular languages whose balanced leaf-language classes are contained in the polynomial hierarchy. For any discussed reducibility we try to give motivations and open questions, in a hope to convince the reader that the study of these reducibilities is interesting for automata theory and computational complexity.

Classification : 03D05, 03C13, 68Q15
Keywords: regular language, quasi-aperiodic regular language, quantifier-alternation hierarchy, difference hierarchy, polylogtime reducibility, quantifier-free reducibility, forbidden pattern

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