First Order Formulas with Modular Ppredicates

Chaubard, Laura; Pin, Jean-Éric; Straubing, Howard

doi:10.1109/lics.2006.24

Cited by 23 publications

(46 citation statements)

References 22 publications

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“…In earlier papers [5,8,11,12,17,18,23,25,26,27,28], I used the opposite order, but this one is considered more intuitive by many people. There are also strong topological arguments, which cannot be detailed in this paper, in favour of this order.…”

Section: Eilenberg's Variety Theoremmentioning

confidence: 99%

“…Let us give two instances of this theorem. The first example is taken from [12]. Let us call modular simple a language of the form…”

Section: C-varietiesmentioning

confidence: 99%

“…It is known [38] that FO[< + MOD] captures exactly the regular languages of AC 0 and that Σ 1 [< + MOD] captures the finite unions of modular simple languages [12]. Therefore these two logic fragments can be defined by the midentities above-mentioned.…”

Section: C-varietiesmentioning

confidence: 99%

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Equational Descriptions of Languages

2012

Int. J. Found. Comput. Sci.

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This paper is a survey on the equational descriptions of languages. The first part is devoted to Birkhoff's and Reiterman's theorems on equational descriptions of varieties. Eilenberg's variety theorem and its successive generalizations form the second part. The more recent results on equational descriptions of lattices of languages are presented in the third part of the paper.Equations have been used for a long time in mathematics to provide a concise description of various mathematical objects. This article roughly follows a historical approach to present such equational descriptions for formal languages, ranging over a period of 45 years: from Schützenberger's characterization of star-free languages [36] to the following recent result of [18]: Every lattice of languages admits an equational description.This evolution was made possible by a gradual abstraction of the notion of equation. The story really starts in 1935 with Birkhoff's theorem on equational classes [6]. It holds for any kind of universal algebra, but I will present it only for monoids.

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Section: Eilenberg's Variety Theoremmentioning

confidence: 99%

“…Let us give two instances of this theorem. The first example is taken from [12]. Let us call modular simple a language of the form…”

Section: C-varietiesmentioning

confidence: 99%

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Equational Descriptions of Languages

2012

Int. J. Found. Comput. Sci.

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“…This kind of restriction is formalized in classes of homomorphisms called C-varieties [70]; see also [14,35,52]. An application of C-varieties to first-order logic can be found in [15].…”

Section: More Logicsmentioning

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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“…The extra expressive power afforded by the unary predicate ≡ i,m x (which is true at x if x ≡ i (mod m)) has also been considered [CPS06,Str02]. More generally, a numerical predicate R ⊆ N t is said to be regular if it is definable in FO + MOD [<].…”

Section: Sentences With Regular Predicatesmentioning

confidence: 99%

Logic Meets Algebra: the Case of Regular Languages

Thérien¹

2007

Logical Methods in Computer Science

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Abstract. The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment.We survey the existing results relating the expressibility of regular languages in logical fragments of monadic second order logic with successor with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoids. We also explain the impact of these connections on circuit complexity theory.

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First Order Formulas with Modular Ppredicates

Cited by 23 publications

References 22 publications

Equational Descriptions of Languages

Equational Descriptions of Languages

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

Logic Meets Algebra: the Case of Regular Languages

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