We investigate the quantifier alternation hierarchy in firstorder logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels BΣ2 (boolean combination of formulas having only 1 alternation) and Σ3 (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.The connection between logic and automata theory is well known and has a fruitful history in computer science. It was first observed when Büchi, Elgot and Trakhtenbrot proved independently that the regular languages are exactly those that can be defined using a monadic second-order logic (MSO) formula. Since then, many efforts have been made to investigate and understand the expressive power of relevant fragments of MSO. In this field, the yardstick result is often to prove decidable characterizations, i.e., to design an algorithm which, given as input a regular language, decides whether it can be defined in the fragment under investigation. More than the algorithm itself, the main motivation is the insight given by its proof. Indeed, in order to prove a decidable characterization, one has to consider and understand all properties that can be expressed in the fragment.The most prominent fragment of MSO is first-order logic (FO) equipped with a predicate "<" for the linear-order. The expressive power of FO is now well-understood over words and a decidable characterization has been obtained. The result, Schützenberger's Theorem [20,10], states that a regular language is definable in FO if and only if its syntactic monoid is aperiodic. The syntactic monoid is a finite algebraic structure that can effectively be computed from any representation of the language. Moreover, aperiodicity can be rephrased as an equation that needs to be satisfied by all elements of the monoid. Therefore, Schützenberger's Theorem can indeed be used to decide definability in FO.In this paper, we investigate an important hierarchy inside FO, obtained by classifying formulas according to the number of quantifier alternations in their prenex normal form. More precisely, an FO formula is Σ i if its prenex normal form has at most (i − 1) quantifier alternations and starts with a block of existential quantifiers. The hierarchy also involves the classes BΣ i of boolean combinations of Σ i formulas, and the classes ∆ i of languages that can be defined ⋆ Supported by ANR 2010 BLAN 0202 01 FREC Our solution works as follows: given two regular languages, one can easily construct a monoid morphism α : A * → M that recognizes both of them. We then design an algorithm that computes, inside the monoid M , enough Σ 2 -related information to answer the Σ 2 -separation question for any pair of languages that are recognized by α. It turns out that it is also possible (though much more difficult) to use thi...

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 Σ3: first-order sentences having at most three quantifier blocks starting with an existential one. CC Creative Commons THOMAS PLACEWe use separation as a mathematical tool whose purpose is to investigate classes of languages: given a fixed class C, obtaining a C-separation algorithm usually requires a solid understanding of C. A typical objective when considering separation is to not only get an algorithm that decides it, but also a generic method for computing a separator, if it exists.Remark 2.1. C-separation generalizes another classical decision problem: C-membership which asks whether a single regular language L belongs to C. Indeed, it is simple to verify that asking whether L ∈ C is equivalent to asking whether L is C-separable from its complement (in this case, the only candidate for being a separator is L itself ).Covering. Our second problem is more general and was originally defined in [PZ16,PZ18a]. Given a language L, a cover of L is a finite set of languages K such that, L ⊆ K∈K K Moreover, given a class C, a C-cover of L is a cover K of L such that all K ∈ K belong to C.Covering takes as input a language L and a finite set of languages L. A separating cover for the pair (L, L) is a cover K of L such that for every K ∈ K, there exists L ∈ L which satisfies K ∩ L = ∅. Finally, given a class C, we say that the pair (L, L) is C-coverable when there exists a separating C-cover. The C-covering problem is now defined as follows:INPUT: A regular language L and a finite set of regular languages L. OUTPUT: Is (L, L) C-coverable ?Remark 2.2. Thi...

Abstract. Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from another class. We discuss the separation problem for regular languages. We give a Ptime algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a BΣ1(<) sentence can witness that the languages are disjoint. The proof refines an algebraic argument from Almeida and the third author. When separation is possible, we also express a separator by saturating one of the original languages by a suitable congruence. Following the same line, we show that one can as well decide whether two regular languages can be separated by an unambiguous language, albeit with a higher complexity.

In the theory of formal languages, the understanding of concatenation hierarchies of regular languages is one of the most fundamental and challenging topic. In this paper, we survey progress made in the comprehension of this problem since 1971, and we establish new generic statements regarding this problem.

Abstract. Given two languages, a separator is a third language that contains the first one and is disjoint from the second one. We investigate the following decision problem, called separation: given two regular languages of finite words, decide whether there exists a firstorder definable separator. A more general problem was solved in an algebraic framework by Henckell in 1988, although the connection with separation was pointed out only in 1996, by Almeida. The result was then generalized by Henckell, Steinberg and Rhodes in 2010. In this paper, we present a new, self-contained and elementary proof of it, which actually covers the original result of Henckell.We prove that in order to answer this question, sufficient information can be extracted from semigroups recognizing the input languages, using a fixpoint computation, similar to that originally proposed by Henckell. Given as input a morphism recognizing both languages to be separated, this yields an Exptime algorithm for checking first-order separability. Moreover, the correctness proof of this algorithm yields a stronger result, namely a description of a possible separator. More precisely, one can compute a bound on the quantifier rank of potential separators, as well as a first-order formula that describes a separator, if there exists one. Finally, we prove that this technique can be generalized to answer the same question for regular languages of infinite words.

Abstract. In this paper, we study boolean (not necessarily positive) combinations of open sets. In other words, we study positive boolean combinations of safety and reachability conditions. We give an algorithm, which inputs a regular language of infinite trees, and decides if the language is a boolean combination of open sets.

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level?Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation for looking at separation is threefold: (1) while more difficult, it is more rewarding, as solving it requires a better understanding; (2) being more general, it provides a more convenient framework, and (3) all recent membership algorithms are actually reductions to separation for lower levels.We present a separation algorithm for dot-depth two. A key point is that while dotdepth two is our most prominent application, our theorem is more general. We consider a family of hierarchies, which includes the dot-depth: concatenation hierarchies. They are built through a generic construction process: one first chooses an initial class, the basis, which serves as the lowest level in the hierarchy. Then, further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis consists of finitely many languages, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results.See [DGK08] for a survey. Following these results, membership for level 2 remained open for a long time and was named the "dot-depth 2 problem".Separation. Recently [PZ14, Pla15, Pla18], solutions were found for levels 2, 5/2 and 7/2. The key ingredient is a new problem stronger than membership: separation. Rather than asking whether an input language belongs to the class C under investigation, the Cseparation problem takes as input two languages, and asks whether there exists a third one from C containing the first and disjoint from the second. While the interest in separation is recent, it has quickly replaced membership as the central question. A first practical reason is that separation proved itself to be a key ingredient in obtaining all recent membership results. See [PZ15b, PZ18b] for an overview. A striking example is provided by a crucial theorem of [PZ14]. It establishes a generic reduction from P ol(C)-separation to C-membership which holds for any class ...

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results.