Separating regular languages with two quantifier alternations

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 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 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

A Characterization for Decidable Separability by Piecewise Testable Languages

The separability problem for word languages of a class $\mathcal{C}$ by languages of a class $\mathcal{S}$ asks, for two given languages $I$ and $E$ from $\mathcal{C}$, whether there exists a language $S$ from $\mathcal{S}$ that includes $I$ and excludes $E$, that is, $I \subseteq S$ and $S\cap E = \emptyset$. In this work, we assume some mild closure properties for $\mathcal{C}$ and study for which such classes separability by a piecewise testable language (PTL) is decidable. We characterize these classes in terms of decidability of (two variants of) an unboundedness problem. From this, we deduce that separability by PTL is decidable for a number of language classes, such as the context-free languages and languages of labeled vector addition systems. Furthermore, it follows that separability by PTL is decidable if and only if one can compute for any language of the class its downward closure wrt. the scattered substring ordering (i.e., if the set of scattered substrings of any language of the class is effectively regular). The obtained decidability results contrast some undecidability results. In fact, for all (non-regular) language classes that we present as examples with decidable separability, it is undecidable whether a given language is a PTL itself. Our characterization involves a result of independent interest, which states that for any kind of languages $I$ and $E$, non-separability by PTL is equivalent to the existence of common patterns in $I$ and $E$

A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

A ``hybrid method'', dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions--this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.Comment: 31 page

Algebraic Birkhoff decomposition and its applications

Central in the Hopf algebra approach to the renormalization of perturbative quantum field theory of Connes and Kreimer is their Algebraic Birkhoff Decomposition. In this tutorial article, we introduce their decomposition and prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give some applications of this decomposition in the study of divergent integrals and multiple zeta values.Comment: 39 pages. To appear in "Automorphic Forms and Langlands Program