Quantifier Alternation for Infinite Words

We investigate the expressive power of the quantifier alternation hierarchy of first-order logic over words. This hierarchy includes the classes Σi (sentences having at most i blocks of quantifiers starting with an ∃) and BΣi (Boolean combinations of Σi sentences). So far, this expressive power has been effectively characterized for the lower levels only. Recently, a breakthrough was made over finite words, and decidable characterizations were obtained for BΣ2 and Σ3, by relying on a decision problem called separation, and solving it for Σ2. The contribution of this paper is a generalization of these results to the setting of infinite words: we solve separation for Σ2 and Σ3, and obtain decidable characterizations of BΣ2 and Σ3 as consequences. Regular word languages form a robust class, as they can be defined either by operational, algebraic, or logical means: they are exactly those that can be defined equivalently by finite state machines (operational view), morphisms into finite algebras (algebraic view) and monadic second order (“MSO”) sentences [4,27,8,5] (logical view). To understand the structure of this class in depth, it is natural to classify its languages according to their descriptive complexity. The problem is to determine how complicated a sentence has to be to describe a given input language. This is a decision problem parametrized by a fragment of MSO: given an input language, can it be expressed in the fragment? This problem is called membership (is the language a member of the class defined by the fragment?). The seminal result in this field is the membership algorithm for first-order logic (FO) over finite words, which is arguably the most prominent fragment of MSO. This algorithm was obtained in two steps. McNaughton and Papert [10] observed that the languages definable in FO are exactly the star-free languages: those that may be expressed by a regular expression in which complement is allowed while the Kleene star is disallowed. Furthermore, an earlier result of Schützenberger [23] shows that star-free languages are exactly the ones whose syntactic monoid is aperiodic. The syntactic monoid is a finite algebra that can be computed from any input regular language, and aperiodicity can be formulated as an equation that has to be satisfied by all elements of this algebra. Therefore, Schützenberger’s result makes it possible to decide whether a regular language is star-free (and therefore definable in FO by McNaughton-Papert’s result). ? This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the "Investments for the future" Programme IdEx Bordeaux CPU (ANR-10-IDEX-03-02). Following this first result, the attention turned to a deeper question: given an FO-definable language, find the “simplest” FO-sentences that define it. The standard complexity measure for FO sentences is their quantifier alternation, which counts the number of switches between blocks of ∃ and ∀ quantifiers. This measure is justified not only because it is intuitively difficult to understand a sentence with many alternations, but also because the nonelementary complexity of standard problems for FO [25] (e.g, satisfiability) is tied to quantifier alternation. In summary, we classify FO definable languages by counting the number of quantifier alternations needed to define them and we want to be able to decide the level of a given language (which amounts to solving membership for each level). This leads to define the following fragments of FO: an FO sentence is Σi if its prenex normal form has at most i blocks of ∃ or ∀ quantifiers and starts with a block of existential ones. Note that Σi is not closed under complement (the negation of a Σi sentence is called a Πi sentence). A sentence is BΣi if it is a Boolean combination of Σi sentences (cf. figure). Clearly, we have Σi ⊆ BΣi ⊆ Σi+1, and these inclusions are known to be strict [3,26]: Σi ( BΣi ( Σi+1.