Logic Meets Algebra: the Case of Regular Languages

Thérien, Denis

doi:10.2168/lmcs-3(1:4)2007

Cited by 34 publications

(34 citation statements)

References 118 publications

(166 reference statements)

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“…Several surveys related to concatenation hierarchies can be found in the literature [20,31,60,64,67,81,110]. Moreover, the study of concatenation hierarchies is not limited to words and similar hierarchies were considered for infinite words, for traces, for data words [15] and even for tree languages.…”

Section: Resultsmentioning

confidence: 99%

The Dot-Depth Hierarchy, 45 Years Later

2017

The Role of Theory in Computer Science

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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...

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Section: Resultsmentioning

confidence: 99%

The Dot-Depth Hierarchy, 45 Years Later

2017

The Role of Theory in Computer Science

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“…In this section, we summarize the information on monoid and variety theory that will be relevant for our purpose. For more detailed information and proofs, we refer the reader to [20,2,31,32,30], among other sources.…”

Section: On Varieties and Pseudovarietiesmentioning

confidence: 99%

On logical hierarchies within FO^2-definable languages

Kufleitner¹,

Weil²

2012

Logical Methods in Computer Science

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Abstract. We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of quantifier alternations yields an infinite hierarchy whose levels are varieties of languages (and hence admit an algebraic characterization). Using this algebraic approach, we show that the quantifier alternation hierarchy inside FO 2 [<] is decidable within one unit. For this purpose, we relate each level of the hierarchy with decidable varieties of languages, which can be defined in terms of iterated deterministic and co-deterministic products. A crucial notion in this process is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle languages of Schwentick, Thérien and Vollmer.Many important properties of systems are modeled by finite automata. Frequently, the formal languages induced by these systems are definable in first-order logic. Our understanding of its expressive power is of direct relevance for a number of application fields, such as verification.The first-order logic we are interested in, in this paper, is the first-order logic of the linear order, written FO[<], interpreted on finite words. It is well-known that the languages that are definable in this logic are exactly the star-free languages, or equivalently the regular languages whose syntactic monoid is aperiodic (that is: satisfies an identity of the form x n+1 = x n for some integer n) [24,18] (see also [5,20,27,29]); and that deciding whether a finite automaton accepts such a language is PSPACE-complete [3].Fragments of first-order logic defined by the limitation of certain resources have been studied in detail. For instance, the quantifier alternation hierarchy, with its close relation with the dot-depth hierarchy of star-free languages, offers one of the oldest open problems in formal language theory: we know that the hierarchy is infinite and that its levels are

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“…Tesson and Therien have presented an algebraic characterization of the quantifier depth hierarchy [78]. In the same article they have also given an algebraic counterpart of the quantifier depth hierarchy within FO [<].…”

Section: Quantifier Depthmentioning

confidence: 99%

A Survey on Small Fragments of First-Order Logic Over Finite Words

Diekert

Gastin

Kufleitner

2008

Int. J. Found. Comput. Sci.

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We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose. PreambleThere are many brilliant surveys on formal language theory [36,41,48,85,86]. Quite many surveys cover first-order and monadic second-order definability. But there are also nuggets below. There are deep theorems on proper fragments of firstorder definability. The most prominent fragment is FO 2 ; it is the class of languages which are defined by first-order sentences which do not use more than two names for variables. Although various characterizations are known for this class, there seems to be little knowledge in a broad community. A reason for this is that the proofs are spread over the literature and even in the survey [76] many proofs are referred to the original literature which in turn is sometimes quite difficult to read. This is our starting point. We restrict our attention to fragments strictly below first-order definability. We concentrate on algebraic and formal language theoretic characterizations for those fragments where decidability results are known, although we do not discuss complexity issues here. We give a clear preference to full proofs rather than to state all results. In our proofs we tried to be minimalistic. All technical concepts which are introduced are also used in the proofs for the main results.

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Logic Meets Algebra: the Case of Regular Languages

Cited by 34 publications

References 118 publications

The Dot-Depth Hierarchy, 45 Years Later

The Dot-Depth Hierarchy, 45 Years Later

On logical hierarchies within FO^2-definable languages

A Survey on Small Fragments of First-Order Logic Over Finite Words

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