Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Place, Thomas; Rooijen, Lorijn van; Zeitoun, Marc

doi:10.1007/978-3-642-40313-2_64

Cited by 46 publications

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References 22 publications

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“…A direct proof for FO(<) has been obtained recently by the authors [2014b; 2016]. Separation is also known decidable up to ∆ 2 in both hierarchies [Czerwiński et al 2013;Place et al 2013b;Place and Zeitoun 2015a]. In this paper, we present a solution for Σ 2 and Π 2 in both hierarchies.…”

Section: The Membership and Separation Problemsmentioning

confidence: 82%

“…Theorem 6.7 reduces the separation problem for BΣ i (<) to finding an algorithm which, given a morphism α, computes all pairs (s 1 , s 2 ) ∈ M 2 such that C i [α]∩{s 1 , s 2 } * has bounded alternation. The problem has been solved when i = 1 in [Place et al 2013b]. Above i = 1, the problem remains open, even when i = 2.…”

Section: Separation and Membership For Bσi(mentioning

confidence: 99%

“…Recently, a research effort has been made to investigate this problem from a radically different perspective, with the aim of finding new and self-contained proofs relying on elementary ideas and notions from language theory only. Such proofs were obtained for several results already known in the algebraic framework [Czerwiński et al 2013;Place et al 2013b;Place and Zeitoun 2014b;Place et al 2013a;Place and Zeitoun 2016]. This paper is a continuation of this effort for classes that were not solved even in the algebraic setting: we solve the separation problem for Σ 2 (<), and we use our solution as a basis to obtain decidable characterizations for the classes BΣ 2 (<), ∆ 3 (<) and Σ 3 (<).Our proof works as follows: given two regular languages, one can easily construct a morphism α from A * into a finite monoid M that recognizes both languages.…”

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confidence: 93%

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Going Higher in First-Order Quantifier Alternation Hierarchies on Words

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Zeitoun

2019

J. ACM

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We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language in the levels BΣ 2 (finite boolean combinations of formulas having only one alternation) and Σ 3 (formulas having only two alternations and beginning with an existential block). Our proofs work by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.A:3 to be decidable. The class BΣ 1 (<) consists exactly of all piecewise testable languages, i.e., such that membership of a word only depends on its scattered subwords up to a fixed size. These languages were characterized by Simon [1975] as those whose syntactic monoid is J-trivial. A decidable characterization of Σ 2 (<)-hence of ∆ 2 (<) as well-was obtained by Arfi [1987;, a problem revisited and clarified by Pin and Weil [1995;, who also set up a generic algebraic framework to work with. For ∆ 2 (<), the literature is very rich, see the survey by Tesson and Thérien [2002]. For example, the ∆ 2 (<) definable languages are exactly the ones definable in the two-variable restriction of FO(<) [Thérien and Wilke 1998]. These are also the languages whose syntactic monoid belongs to the class DA, as shown again by Pin and Weil [1995; (see also [Schützenberger 1976]). For higher levels in the hierarchy, getting decidable characterizations remained a major open problem. In particular, the case of BΣ 2 (<) has a very abundant history and a series of combinatorial, logical, and algebraic conjectures have been proposed over the years. We refer to Section 3 and to several surveys cited in this section for a bibliography. So far, the only known effective result was partial, working only when the alphabet is of size 2 [Straubing 1988].Contributions.. In this paper, we establish decidable characterizations for the fragments BΣ 2 (<), ∆ 3 (<) and Σ 3 (<) of first-order logic. These new results are based on a deeper decision problem than membership: the separation problem. Fix a class C of languages. The C-separation problem amounts to deciding whether, given two input regular languages, there exists a third language in C containing the first language while being disjoint from the second one. Solving the C-separation problem is more general than obtaining a decidable characterization for the class C. Indeed, since regular languages are effectively closed under complement, testing membership in C can be achieved by testing whether the input is C-separable from its complement. While this reduction immediately transfers decision procedures for one problem to the other, this is not our primary motivation for looking at separation. Although intrinsically more challenging, a solution to the separation problem requires more understanding than just getting a decidable characterization. This understanding for a given fragment can then be exploited in order to ob...

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Section: The Membership and Separation Problemsmentioning

confidence: 82%

Section: Separation and Membership For Bσi(mentioning

confidence: 99%

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confidence: 93%

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Going Higher in First-Order Quantifier Alternation Hierarchies on Words

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Zeitoun

2019

J. ACM

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“…Separation is known to be decidable for the lower levels in the Straubing-Thérien and dot-depth hierarchies. This was first shown to be decidable for the levels 1 2 , 1, 3 2 and 2 in the Straubing-Thérien hierarchy [CMM13, PvRZ13,PZ14,PZ17c]. Let us point out that while these results were first formulated independently, it was recently proved in [PZ17c] that the four of them are corollaries of only two generic theorems: • For any finite class C, P ol(C)-separation is decidable.…”

Section: Introductionmentioning

confidence: 99%

Separating Regular Languages with Two Quantifiers Alternations

Place

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second. Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C. This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes.Here, we are interested in classes within concatenation hierarchies. Such hierarchies are built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations. The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages definable in first-order logic. Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks. Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory.Our main theorem is generic: we show that separation is decidable for the level 3 2 of any concatenation hierarchy whose basis is finite. Furthermore, in the special case of the dot-depth, we push this result to the level 5 2 . In logical terms, this solves separation for Σ3: first-order sentences having at most three quantifier blocks starting with an existential one. CC Creative Commons THOMAS PLACEWe use separation as a mathematical tool whose purpose is to investigate classes of languages: given a fixed class C, obtaining a C-separation algorithm usually requires a solid understanding of C. A typical objective when considering separation is to not only get an algorithm that decides it, but also a generic method for computing a separator, if it exists.Remark 2.1. C-separation generalizes another classical decision problem: C-membership which asks whether a single regular language L belongs to C. Indeed, it is simple to verify that asking whether L ∈ C is equivalent to asking whether L is C-separable from its complement (in this case, the only candidate for being a separator is L itself ).Covering. Our second problem is more general and was originally defined in [PZ16,PZ18a]. Given a language L, a cover of L is a finite set of languages K such that, L ⊆ K∈K K Moreover, given a class C, a C-cover of L is a cover K of L such that all K ∈ K belong to C.Covering takes as input a language L and a finite set of languages L. A separating cover for the pair (L, L) is a cover K of L such that for every K ∈ K, there exists L ∈ L which satisfies K ∩ L = ∅. Finally, given a class C, we say that the pair (L, L) is C-coverable when there exists a separating C-cover. The C-covering problem is now defined as follows:INPUT: A regular language L and a finite set of regular languages L. OUTPUT: Is (L, L) C-coverable ?Remark 2.2. Thi...

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“…The decidability of the separation problem re-proves a result of Place, van Rooijen and Zeitoun [21]. The algorithms for the word problem for ω-terms are more efficient than those of Moura [18].…”

Section: Introductionmentioning

confidence: 66%

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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Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Cited by 46 publications

References 22 publications

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Separating Regular Languages with Two Quantifiers Alternations

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

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