Separating regular languages with two quantifier alternations

Separating regular languages with two quantifier alternations

Blog Article

We investigate a famous decision problem in automata theory: separation.Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second.Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C.This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes.

Here, we are interested in classes within concatenation hierarchies.Such hierarchies are read more built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations.The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages better waters xl7000 definable in first-order logic.Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks.

Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory.Our main theorem is generic: we show that separation is decidable for the level 3/2 of any concatenation hierarchy whose basis is finite.Furthermore, in the special case of the dot-depth, we push this result to the level 5/2.In logical terms, this solves separation for $Sigma_3$: first-order sentences having at most three quantifier blocks starting with an existential one.