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Chomsky hierarchy

About: Chomsky hierarchy is a research topic. Over the lifetime, 601 publications have been published within this topic receiving 31067 citations. The topic is also known as: Chomsky–Schützenberger hierarchy.


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Book ChapterDOI
07 Jul 2003
TL;DR: A survey on the various models and their properties is given, their relationships to the language classes of the Chomsky hierarchy are described, and some open problems are presented as mentioned in this paper, where the authors present a set of open problems.
Abstract: The restarting automaton, introduced by Jancar et al in 1995, is motivated by the so-called 'analysis by reduction, ' a technique from linguistics. By now there are many different models of restarting automata, and their investigation has proved very fruitful in that they offer an opportunity to study the influence of various kinds of resources on their expressive power. Here a survey on the various models and their properties is given, their relationships to the language classes of the Chomsky hierarchy are described, and some open problems are presented.

55 citations

Journal ArticleDOI
TL;DR: This review brings together accounts of the principles of structure building in music and animal song to corresponding models in formal language theory, the extended Chomsky hierarchy (CH), and their probabilistic counterparts.
Abstract: Human language, music and a variety of animal vocalizations constitute ways of sonic communication that exhibit remarkable structural complexity. While the complexities of language and possible parallels in animal communication have been discussed intensively, reflections on the complexity of music and animal song, and their comparisons, are underrepresented. In some ways, music and animal songs are more comparable to each other than to language as propositional semantics cannot be used as indicator of communicative success or wellformedness, and notions of grammaticality are less easily defined. This review brings together accounts of the principles of structure building in music and animal song. It relates them to corresponding models in formal language theory, the extended Chomsky hierarchy (CH), and their probabilistic counterparts. We further discuss common misunderstandings and shortcomings concerning the CH and suggest ways to move beyond. We discuss language, music and animal song in the context of their function and motivation and further integrate problems and issues that are less commonly addressed in the context of language, including continuous event spaces, features of sound and timbre, representation of temporality and interactions of multiple parallel feature streams. We discuss these aspects in the light of recent theoretical, cognitive, neuroscientific and modelling research in the domains of music, language and animal song.

54 citations

Journal ArticleDOI
TL;DR: The brain represents grammars in its connectivity, and its ability for syntax is based on neurobiological infrastructure for structured sequence processing, and the acquisition of this ability is accounted for in an adaptive dynamical systems framework.
Abstract: The human capacity to acquire language is an outstanding scientific challenge to understand. Somehow our language capacities arise from the way the human brain processes, develops and learns in interaction with its environment. To set the stage, we begin with a summary of what is known about the neural organization of language and what our artificial grammar learning (AGL) studies have revealed. We then review the Chomsky hierarchy in the context of the theory of computation and formal learning theory. Finally, we outline a neurobiological model of language acquisition and processing based on an adaptive, recurrent, spiking network architecture. This architecture implements an asynchronous, event-driven, parallel system for recursive processing. We conclude that the brain represents grammars (or more precisely, the parser/generator) in its connectivity, and its ability for syntax is based on neurobiological infrastructure for structured sequence processing. The acquisition of this ability is accounted for in an adaptive dynamical systems framework. Artificial language learning (ALL) paradigms might be used to study the acquisition process within such a framework, as well as the processing properties of the underlying neurobiological infrastructure. However, it is necessary to combine and constrain the interpretation of ALL results by theoretical models and empirical studies on natural language processing. Given that the faculty of language is captured by classical computational models to a significant extent, and that these can be embedded in dynamic network architectures, there is hope that significant progress can be made in understanding the neurobiology of the language faculty.

54 citations

Book ChapterDOI
25 Sep 2006
TL;DR: An arc-consistency algorithm for context-free grammars, an investigation of when logic combinations of grammar constraints are tractable, and when the boundaries run between regular, context- free, and context-sensitive grammar filtering are studied.
Abstract: By introducing the Regular Membership Constraint, Gilles Pesant pioneered the idea of basing constraints on formal languages. The paper presented here is highly motivated by this work, taking the obvious next step, namely to investigate constraints based on grammars higher up in the Chomsky hierarchy. We devise an arc-consistency algorithm for context-free grammars, investigate when logic combinations of grammar constraints are tractable, show how to exploit non-constant size grammars and reorderings of languages, and study where the boundaries run between regular, context-free, and context-sensitive grammar filtering.

51 citations

Journal ArticleDOI
01 Jan 1984
TL;DR: A version of the Ehrenfeucht-Fraisse game is used to obtain a new proof of a hierarchy result in formal language theory: It is shown that the concatenation hierarchy ("dot-depth hierarchy") of star-free languages is strict.
Abstract: A version of the Ehrenfreucht-Fraisse game is used to obtain a new proof of a hierarchy result in formal language theory: It is shown that the concatenation hierarchy ("dot-depth hierarchy") of star-free languages is strict. Resume Une version du jeu de Ehrenfeucht-Fraisse est appliquee pour obtenir une nouvelle preuve d'un theoreme dans la theorie des-langages formels: On montre que la hierarchic de concatenation ("dot-depth hierarchy") deslangages sans etoile est stricte. 1 . Introduction. The present paper is concerned with a connection between formal language theory and model theory. We study a hierarchy of formal languages {namely, the dot-depth hierarchy of star-free regular languages) using logical notions such as quantifier complexity of first-order sentences. In this context we apply a form of the Ehrenfeucht-Fraisse game which serves to establish the elementary equivalence between structures with respect to sentences of certain prefix types. The class of star-free regular languages is of a very basic nature: It consists of all languages (= word-sets) over a given alphabet A which can be obtained from the finite languages by finitely many applications of boolean operations and the concatenation product. (For technical reasons we consider only nonempty words over A , i . e . 0037-9484/8403 11 11/S 3.10/ © Gauthier-Villars 11 + languages L c A ; in particular, the complement operation is applied w . r . t . A^) General references on the star-free regular languages are McNaughton-Papert ( 1 9 7 1 ) , Chapter IX of Eilenberg ( 1 9 7 6 ) , or Pin ( 1 9 8 4 b ) . A natural classification of the star-free regular languages is obtained by counting the "levels of concatenation" which are necessary to build up such a language: For a fixed alphabet A , let B.. = {LcA'^lL finite or cofinite), B = { L c A |L is a boolean combination of languages of the form L • . . . L (n > 1 ) with L ^ , . . . , L ^ € B^} . The language classes B , B , , . « . form the so-called dot-depth hierarchy (or: Brzozowski hierarchy), introduced by Cohen/Brzozowski ( 1 9 7 1 ) . In the framework of semigroup theory, Brzozowski/Knast ( 1 9 7 8 ) showed that the hierarchy is infinite ( i . e . that B^ B^_^ for k > 1 ) . The aim of the present paper is to give a new proof of this result, based on a logical characterization of the hierarchy that was obtained in Thomas ( 1 9 8 2 ) . The present proof does not rely on semigroup-theory; instead, an intuitively appealing model-theoretic technique is applied: the Ehrenfeucht-Fraisse game. Let us first state the mentioned characterization.result, taking A = { a , b } . One identifies any word W G A 4 ' , say of length n , with a "word model" w = ( { 1 , . . . , n } , < , m i n , m a x , S , P , Q ^ , Q ^ ) where the domain { 1 , . . . , n } represents the set of positions of letters in the word w , ordered by < , where min and max are the first and the last position, i . e . min = 1 and max=n, S and P are the successor and predecessor function on { 1 , . . . , n } with the convention that S(max) =max and P(min) =min, and Q^Q^ are unary predicates over { 1 , . . . , n } containing the positions with letter a , b respectively. (Sometimes it is convenient to assume that the position-sets of two words u, v are disjoint; then one takes any two nonoverlapping segments of the integers as the position-sets of u and v . ) Let L be the first-order language with equality and nonlogical symbols <,min, m a x , S , P , Q , Q . . Then the satisfaction of an Lsentence tp in a word w a D * • 12 EHRENFEUCHT-FRAISSE GAME (written: w t= ip) can be defined in a natural way, and we say that L c: A is defined by the L-sentence tp if L == {w € A Iw^ ip} . For example, the language L= (ab) is defined by Q min A Q,max A Vy (y < max -• (Q y ̂ Q,S ( y ) ) ) . As usual, a I,-formula is a formula in pr.enex normal form with a prefix consisting of k alternating blocks of quantifiers, beginning with a block of existential quantifiers. A B (£,)-formula is a boolean combination of £,-formulas. 1 .1 Theorem. (Thomas ( 1 9 8 2 ) ) . Let k>0. A language L <= A belongs to B iff L is defined by a B(£)-sentence of L. For the formalization of properties of words the symbols min,max,S,P are convenient. But of course they are definable in the restricted first-order language L,. with the nonlogical constants <,Q ,Q, alone. u a JD Indeed, we have: 1 .2 Lemma. Let k>0 . If L <= A is defined by a B (I,)-sentence of L, then L is defined by a B (Z,^ )-sentence of L . Proof. The quantifier-free kernel of a Z,-formula tp of L can be expressed both by a £..and a n,--formula of L/.. For example, Q S(min) is expressible in the following two ways: (+) 3y(y=S(min) A Q^y) , Vy(y=S(min) -» Q^y) where y = S (min) is rewritten as a II.-formula of L.. using x = min —^ V z ( x = z v x < z ) , x = max -* V z ( z = x v z < x ) S (x) = y -^ (x = max A x = y ) v ( x < y A Vz~l (x < z A z < y) ) . Hence we obtain a £, .-sentence of Lwhich is equivalent (in all word-models) to tp by applying one of the two definitions in (+), depending on the case whether the innermost quantifier-block of cp is existential or universal. We mention without proof that (for k >0) the B(Z,)-sentences of L.. define exactly those languages L c: A which occur on the k-th level of another hierarchy of star-free regular languages, introduced by

51 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20232
20223
20219
20208
201912
201810