The Dot-Depth Hierarchy, 45 Years Later

Abstract

In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy.This paper is a follow-up of the survey article Open problems about regular languages, 35 years later [57]. The dot-depth hierarchy, also known as Brzozowski hierarchy, is a hierarchy of star-free languages first introduced by Cohen and Brzozowski [25] in 1971. It immediately gave rise to many interesting questions and an account of the early results can be found in Brzozowski's survey [20] from 1976. Terminology, notation and backgroundMost of the terminology used in this paper was introduced in [57]. We just complete these definitions by giving the ordered versions of the notions of syntactic monoid and variety of finite monoids. Syntactic order and positive varietiesAn ordered monoid is a monoid equipped with an order compatible with the multiplication: x y implies zx zy and xz yz.The syntactic preorder 1 of a language L of A * is the relation L defined on A * by u L v if and only if, for every x, y ∈ A * , xuy ∈ L ⇒ xvy ∈ L. * The author was funded from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 670624).1 Unfortunately, the author used the opposite order in earlier papers (from 1995 to 2011). 1The syntactic congruence of L is the associated equivalence relation ∼ L , defined by u ∼ L v if and only if u L v and v L u.The syntactic monoid of L is the quotient M (L) of A * by ∼ L and the natural morphism η : A * → A * /∼ L is called the syntactic morphism of L. The syntactic preorder L induces an order on the quotient monoid M (L). The resulting ordered monoid is called the syntactic ordered monoid of L.For instance, the syntactic monoid of the language {a, aba} is the monoid M = {1, a, b, ab, ba, aba, 0} presented by the relations a 2 = b 2 = bab = 0. Its syntactic order is given by the relations 0 < ab < 1, 0 < ba < 1, 0 < aba < a, 0 < b.The syntactic ordered monoid of a language was first introduced by Schützenberger [86] in 1956, but thereafter, he apparently only used the syntactic monoid.A positive variety of languages is a class of languages closed under finite unions, finite intersections, left and right quotients and inverses of morphisms. A variety of languages is a positive variety closed under complementation.Similarly, a variety of finite ordered monoids is a class of finite ordered monoids closed under taking ordered submonoids, quotients and finite products. Varieties of finite (ordered) semigroups are defined analogously. If V is a variety of ordered monoids, let V d denote the dual variety, consisting of all ordered monoids (M, ) such that (M, ) ∈ V. We refer the reader to the books [2, 28, 62] for more details.Eilenberg's variety theor...

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

“…In this paper, we only focus on some specific aspects. For more details and a complete bibliography, we invite the reader to refer to the papers surveying the subject, e.g., by Brzozowski [1976], Eilenberg [1976], Weil [1989a], Thomas [1997] and Pin [1995a;1998;2015b;2016a] and to the literature cited in these papers.…”

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“…Even if some of these questions are now well understood, a few others remain wide open, despite a wealth of research work spanning several decades. This is the case for the fascinating dotdepth problem [Pin17a], which has two elementary formulations: a language-theoretic one and a logical one. The language-theoretic one is the older of the two.…”

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