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...

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. 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.

Message sequence charts (MSC) andHigh-level MSC (HMSC) is a visual notation for asynchronously communicating processes and a standard of the ITU. They usually represent incomplete specifications of required or forbidden properties of communication protocols. We consider in this paper two basic problems concerning the automated validation of HMSC specifications, namely model-checking and synthesis. We identify natural syntactic restrictions of HMSCs for which we can solve the above questions. We show first that model-checking for globally cooperative (and locally cooperative) HMSCs is decidable within the same complexity as for the restricted class of bounded HMSCs. Furthermore, model-checking local-choice HMSCs turns out to be as efficient as for finite-state (sequential) systems. The study of locally cooperative and local-choice HMSCs is motivated by the synthesis question, i.e., the question of implementing HMSCs through communicating finite-state machines (CFM) with additional message data. We show that locally cooperative and ✩ local-choice HMSCs are always implementable. Furthermore, the implementation of a local-choice HMSC is deadlock-free and of linear size.

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.

This paper studies the pseudovariety R of all finite R-trivial semigroups. We give a representation of pseudowords over R by infinite trees, called R-trees. Then we show that a pseudoword is an ω-term if and only if its associated tree is regular (i.e. it can be folded into a finite graph), or equivalently, if the ω-term has a finite number of tails. We give a linear algorithm to compute a compact representation of the R-tree for ω-terms, which yields a linear solution of the word problem for ω-terms over R. We finally exhibit a basis for the ω-variety generated by R and we show that there is no finite basis. Several results can be compared to recent work of Bloom and Choffrut on long words.

We introduce new classes of weighted automata on words. Equipped with pebbles and a two-way mechanism, they go beyond the class of recognizable formal power series, but capture a weighted version of first-order logic with bounded transitive closure. In contrast to previous work, this logic allows for unrestricted use of universal quantification. Our main result states that pebble weighted automata, nested weighted automata, and this weighted logic are expressively equivalent. We also give new logical characterizations of the recognizable series.

Abstract. We propose an algorithm to find a counterexample to some property in a finite state program. This algorithm is derived from SPIN's one, but it finds a counterexample faster than SPIN does. In particular it still works in linear time. Compared with SPIN's algorithm, it requires only one additional bit per state stored. We further propose another algorithm to compute a counterexample of minimal size. Again, this algorithm does not use more memory than SPIN does to approximate a minimal counterexample. The cost to find a counterexample of minimal size is that one has to revisit more states than SPIN. We provide an implementation and discuss experimental results.