On logical hierarchies within FO^2-definable languages

Kufleitner, Manfred; Weil, Pascal

doi:10.2168/lmcs-8(3:11)2012

Cited by 15 publications

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References 32 publications

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“…Finally, several subhierarchies of ∆ 2 = FO 2 were considered in the recent years [30,48,49,50,51,53,52,106,119].…”

Section: Theorem 71 a Language Is Locally Testable If Its Syntactic mentioning

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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“…Finally, several subhierarchies of ∆ 2 = FO 2 were considered in the recent years [30,48,49,50,51,53,52,106,119].…”

Section: Theorem 71 a Language Is Locally Testable If Its Syntactic mentioning

confidence: 99%

The Dot-Depth Hierarchy, 45 Years Later

2017

The Role of Theory in Computer Science

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“…We will define it using Mal'cev products. Though this approach is different to the original one used by Trotter and Weil, both are equivalent [15]. We define:…”

Section: Figure 1: Trotter-weil Hierarchymentioning

confidence: 99%

“…In this paper, we present the following results.• Our main tool for studying a variety V of the Trotter-Weil Hierarchy is a family of finite index congruences ≡ V,n for n ∈ N. These congruences have the property that a monoid M is in V if and only if there exists n for which M divides a quotient by ≡ V,n . The congruences are not new but they differ in some details from the ones usually found in the literature, where they are introduced in terms of rankers [12,14,15]. Unfortunately, these differences necessitate new proofs.• We lift the combinatorics from finite words to ω-terms using the "linear order approach" introduced by Huschenbett and the first author [10].…”

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confidence: 99%

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

Kufleitner

Wächter

2017

Theory Comput Syst

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For two given ω-terms α and β, the word problem for ω-terms over a variety V asks whether α = β in all monoids in V. We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by ω-terms (this property is also known as ω-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable. * The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-017-9763-z. † The first author was supported by the German Research Foundation (DFG) under grants DI 435/5-2 and KU 2716/1-1.in a finite monoid is easy: one can simply substitute each variable by all elements of the monoid. For each substitution, this yields a monoid element on the left hand side and one on the right hand side. The equation holds if and only if they are always equal. Often, the question whether an equation holds is not only interesting for a single finite monoid but for a (possibly infinite) class of such monoids. For example, one may ask whether all monoids in a certain class are aperiodic. This is trivially decidable if the class is finite. But what if it is infinite? If the class forms a variety (of finite monoids, sometimes also referred to as a pseudo-variety), i. e. a class of finite monoids closed under (possibly empty) direct products, submonoids and homomorphic images, then this problem is called the variety's word problem for ω-terms. Usually, the study of a variety's word problem for ω-terms also gives more insight into the variety's structure, which is interesting in its own right. McCammond showed that the word problem for ω-terms of the variety A of aperiodic finite monoids is decidable [17]. The problem was shown to be decidable in linear time for J, the class of J -trivial finite monoids, by Almeida [3] and for R, the class of R-trivial monoids, by Almeida and Zeitoun [4]. For the variety DA, Moura adapted and expanded those ideas to show decidability in time O((nk) 5 ) where n is the length of the input ω-terms and k is the maximal nesting depth of the ω-power (which can be linear in n) [18]. Remember that DA is the class of finite monoids whose regular D-classes form aperiodic semigroups. This variety received a lot of attention due to its many different characterizations; see e.g. [5,26]. Most notably is its connection to two-variable first-order logic [27]. This logic is a natural restriction of first-order logic over finite words, which in turn is the logic characterization of A.In this paper, we consider the word problem for ω-terms over the varieties in the Trotter-Weil Hierarchy. It was introduced by Trot...

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“…With these definitions in place, we define for m, n ∈ N the relations ≡ X m,n , ≡ Y m,n and ≡ WI m,n of accessible words. 5 The idea is that 5 The presented relations could also be defined by (condensed) rankers (as it is done in [14] and [15]). Rankers were introduced by Weis and Immerman [30] (thus, the WI exponent in ≡ WI m,n ) who reused these relations hold on two words u and v if both words allow for the same sequence of factorizations at the first or last occurrence of a letter.…”

Section: Relations For the Trotter-weil Hierarchymentioning

confidence: 99%

“…Now, we are prepared to prove the characterization of the Trotter-Weil Hierarchy stated in Theorem 1. This is done in the following two theorems (see also [15] for the corners and [14] for the intersection levels). We use the notations…”

Section: A Proof For Theoremmentioning

confidence: 99%

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

Kufleitner

Wächter

2016

Computer Science – Theory and Applications

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Abstract. For two given ω-terms α and β, the word problem for ω-terms over a variety V asks whether α = β in all monoids in V. We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by ω-terms (this property is also known as ω-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.

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On logical hierarchies within FO^2-definable languages

Cited by 15 publications

References 32 publications

The Dot-Depth Hierarchy, 45 Years Later

The Dot-Depth Hierarchy, 45 Years Later

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

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