On Separation by Locally Testable and Locally Threshold Testable Languages

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

doi:10.2168/lmcs-10(3:24)2014

Cited by 21 publications

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“…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%

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Place

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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Going Higher in First-Order Quantifier Alternation Hierarchies on Words

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2019

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“…However, the motivations are disconnected from our own, and the proofs rely on deep, purely algebraic arguments. Recently, a research effort has been made to investigate this problem from a different perspective, with the aim of finding new and self-contained proofs relying on elementary ideas and notions from language theory only [8,16,19,17]. This paper is a continuation of this effort: we solve the separation problem for Σ 2 , and use our solution as a basis to obtain decidable characterizations for BΣ 2 , ∆ 3 and Σ 3 .…”

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Going Higher in the First-Order Quantifier Alternation Hierarchy on Words

Place

Zeitoun

2014

Lecture Notes in Computer Science

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We investigate the quantifier alternation hierarchy in firstorder logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels BΣ2 (boolean combination of formulas having only 1 alternation) and Σ3 (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.The connection between logic and automata theory is well known and has a fruitful history in computer science. It was first observed when Büchi, Elgot and Trakhtenbrot proved independently that the regular languages are exactly those that can be defined using a monadic second-order logic (MSO) formula. Since then, many efforts have been made to investigate and understand the expressive power of relevant fragments of MSO. In this field, the yardstick result is often to prove decidable characterizations, i.e., to design an algorithm which, given as input a regular language, decides whether it can be defined in the fragment under investigation. More than the algorithm itself, the main motivation is the insight given by its proof. Indeed, in order to prove a decidable characterization, one has to consider and understand all properties that can be expressed in the fragment.The most prominent fragment of MSO is first-order logic (FO) equipped with a predicate "<" for the linear-order. The expressive power of FO is now well-understood over words and a decidable characterization has been obtained. The result, Schützenberger's Theorem [20,10], states that a regular language is definable in FO if and only if its syntactic monoid is aperiodic. The syntactic monoid is a finite algebraic structure that can effectively be computed from any representation of the language. Moreover, aperiodicity can be rephrased as an equation that needs to be satisfied by all elements of the monoid. Therefore, Schützenberger's Theorem can indeed be used to decide definability in FO.In this paper, we investigate an important hierarchy inside FO, obtained by classifying formulas according to the number of quantifier alternations in their prenex normal form. More precisely, an FO formula is Σ i if its prenex normal form has at most (i − 1) quantifier alternations and starts with a block of existential quantifiers. The hierarchy also involves the classes BΣ i of boolean combinations of Σ i formulas, and the classes ∆ i of languages that can be defined ⋆ Supported by ANR 2010 BLAN 0202 01 FREC Our solution works as follows: given two regular languages, one can easily construct a monoid morphism α : A * → M that recognizes both of them. We then design an algorithm that computes, inside the monoid M , enough Σ 2 -related information to answer the Σ 2 -separation question for any pair of languages that are recognized by α. It turns out that it is also possible (though much more difficult) to use thi...

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“…Since V * L1 = V * D [21] (where D is the pseudovariety of definite semigroups that satisfies se = e for all e 2 = e) the problem is slightly simplified. There are two main approaches to solving this problem, the first one is using the Derived Category theorem [18], [22], [24], [25] and the other is using Presburger arithmetic [2], [19]. Both these approaches crucially depend on the fact that a semigroup S divides a semigroup in V * D if and only if it divides a semigroup in V * D n (pseudovariety D n satisfies the identity yx 1…”

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An Algebraic Characterisation of First-Order Logic with Neighbour

Manuel¹,

Nevatia²

2021

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We give an algebraic characterisation of first-order logic with the neighbour relation, on finite words. For this, we consider languages of finite words over alphabets with an involution on them. The natural algebras for such languages are involution semigroups. To characterise the logic, we define a special kind of semidirect product of involution semigroups, called the locally hermitian product. The characterisation theorem for FO with neighbour states that a language is definable in the logic if and only if it is recognised by a locally hermitian product of an aperiodic commutative involution semigroup, and a locally trivial involution semigroup. We then define the notion of involution varieties of languages, namely classes of languages closed under Boolean operations, quotients, involution, and inverse images of involutory morphisms. An Eilenberg-type correspondence is established between involution varieties of languages and pseudovarieties of involution semigroups.

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On Separation by Locally Testable and Locally Threshold Testable Languages

Cited by 21 publications

References 56 publications

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Going Higher in the First-Order Quantifier Alternation Hierarchy on Words

An Algebraic Characterisation of First-Order Logic with Neighbour

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