From algebra to logic: there and back again -- the story of a hierarchy

Formal language theory historically arose from the definition of models of computation (automata, grammars, etc) and relied for its first step on combinatorial reasoning, especially combinatorics on words. Very quickly however, algebra and logic were identified as powerful tools for the classification of rational languages, e.g. with the definition of the syntactic monoid of a language and Büchi’s theorem on monadic second-order logic. It didn’t take much time after that to observe that, conversely, formal language theory is itself a tool for algebra and logic. The results which we will present are an illustration of this back-and-forth movement between languages, algebra and logic. They deal with a hierarchy of classes of rational languages which arises in different contexts and turned out to solve a problem in logic, namely the decidability of the quantifier alternation hierarchy within the two-variable fragment of first-order logic FO[<]. The full picture uses a collection of results in logic, combinatorics on words and algebra which were obtained independently of the quantifier alternation hierarchy by various authors over several decades. Let R0 be the class of piecewise testable languages, which is natural from a combinatorial and automata-theoretic point of view, and corresponds to the first level of the quantifier alternation hierarchy within FO[<] (and within FO[<]) as well). This class is rather simple and reasonably well understood, see [10,1]. We first consider the hierarchies of classes of languages obtained from R0 by alternatingly closing it under deterministic and co-deterministic closure: we let L0 = R0, Rk+1 (resp. Lk+1) be the deterministic (resp. co-deterministic) closure of Lk (resp. Rk). Results from the 1970s and 1980s [12,9] show that the classes Rk and Lk are varieties (whether a language L belongs to one of these classes depends only on its syntactic monoid) and describe the corresponding varieties of finite monoids Rk and Lk. Results from the 1960s [5] (see also [17,11,6] shows that their membership problems are decidable and they form an infinite hierarchy. A first view of the structure of the lattice formed by these varieties can be obtained by using purely algebraic results from the 1970s on a seemingly different hierarchy, that of varieties of idempotent monoids [2]. The theory of the latter varieties is particularly well understood, and one can exhibit for each of them structurally elegant identities and solutions of the word problem (of the corresponding relatively free object) [3].