Separating Regular Languages with Two Quantifiers Alternations

Place, Thomas

doi:10.1109/lics.2015.28

Cited by 26 publications

(93 citation statements)

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“…Combining with the results of Place and Zeitoun [12], one obtains that our result implies decidability of ∆ 4 , a fact that was independently discovered by Place [11]. Furthermore, from the decidability of Σ 4 also proved by Place [11], our result yields the decidability of ∆ 5 .…”

Section: Introductionsupporting

confidence: 79%

“…Using decidability of the variety V7 /2 proved by Place [11], this theorem gives Corollary 16. It is algorithmically decidable whether a given regular language is definable in ∆ 5 .…”

Section: Proposition 14mentioning

confidence: 96%

“…In particular, the first algorithm for deciding membership of languages in the second level of the hierarchy has been announced only as late as 2014 by Place and Zeitoun [12]. In the same paper, it is also shown that definability in Σ 3 is decidable (to which, for shortness, we will refer by saying Σ 3 is decidable) and decidability of Σ 4 was later also proved by Place [11]. Place and Zeitoun [12] further provided, for each n, a non-effective description of languages definable in ∆ n based on inequalities valid in syntactic ordered monoids of languages definable in Σ n−1 .…”

Section: Introductionmentioning

confidence: 99%

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On Decidability of Intermediate Levels of Concatenation Hierarchies

Almeida¹,

Bartoňová

Klíma

et al. 2015

Developments in Language Theory

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Abstract. It is proved that in every concatenation hierarchy of regular languages, decidability of one of its half levels, obtained by polynomial closure, implies decidability of the intersection of the following half level with its complement. In terms of the quantifier-alternation hierarchy of sentences in the first-order logic of finite words, this means that decidability of (definability in) the Σn fragment implies that of ∆n+1. In particular, the decidability of ∆5 is obtained.

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Section: Introductionsupporting

confidence: 79%

“…Using decidability of the variety V7 /2 proved by Place [11], this theorem gives Corollary 16. It is algorithmically decidable whether a given regular language is definable in ∆ 5 .…”

Section: Proposition 14mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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On Decidability of Intermediate Levels of Concatenation Hierarchies

Almeida¹,

Bartoňová

Klíma

et al. 2015

Developments in Language Theory

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“…The latest result, due to Place [77] states that the separation problem is decidable for Σ 3 [<] and Π 3 [<]. New decidability results follow, as a corollary of Theorem 6.15 and 6.10.…”

Section: Theorem 615 If the Separation Problem Formentioning

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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“…It has then been shown to be decidable for the first fragments of first-order logic in the quantifier alternation hierarchy, namely the ones consisting of Σ 2 (<) [23], respectively of Σ 3 (<) sentences [19] (i.e, first order sentences of the form ∃ * ∀ * ϕ, respectively of the form ∃ * ∀ * ∃ * ϕ, with ϕ quantifier-free). In view of the aforementioned transfer result, this yields decidability of membership for the next level, Σ 4 (<).…”

Section: Introductionmentioning

confidence: 99%

Separating regular languages with first-order logic

Place

Zeitoun

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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Abstract. Given two languages, a separator is a third language that contains the first one and is disjoint from the second one. We investigate the following decision problem, called separation: given two regular languages of finite words, decide whether there exists a firstorder definable separator. A more general problem was solved in an algebraic framework by Henckell in 1988, although the connection with separation was pointed out only in 1996, by Almeida. The result was then generalized by Henckell, Steinberg and Rhodes in 2010. In this paper, we present a new, self-contained and elementary proof of it, which actually covers the original result of Henckell.We prove that in order to answer this question, sufficient information can be extracted from semigroups recognizing the input languages, using a fixpoint computation, similar to that originally proposed by Henckell. Given as input a morphism recognizing both languages to be separated, this yields an Exptime algorithm for checking first-order separability. Moreover, the correctness proof of this algorithm yields a stronger result, namely a description of a possible separator. More precisely, one can compute a bound on the quantifier rank of potential separators, as well as a first-order formula that describes a separator, if there exists one. Finally, we prove that this technique can be generalized to answer the same question for regular languages of infinite words.

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Separating Regular Languages with Two Quantifiers Alternations

Cited by 26 publications

References 27 publications

On Decidability of Intermediate Levels of Concatenation Hierarchies

On Decidability of Intermediate Levels of Concatenation Hierarchies

The Dot-Depth Hierarchy, 45 Years Later

Separating regular languages with first-order logic

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