Showing papers on "Chomsky hierarchy published in 2001"

Languages for analysis and testing of event sequences

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A Characterization of Minimalist Languages

Abstract: In this paper we will fix the position of Minimalist Grammars as defined in Stabler (1997) in the hierarchy of formal languages. Michaelis (1998) has shown that the set of languages generated by Minimalist Grammars is a subset of the set of languages generated by Multiple Context-Free Grammars (Seki et al., 1991). In this paper we will present a proof showing the reverse. We thus conclude that Minimalist Grammars are weakly equivalent to Multiple Context-Free Grammars.

Nonterminal Complexity of Programmed Grammars

Abstract: We show that, in the case of context-free programmed grammars with appearance checking working under free derivations, three nonterminals are enough to generate every recursively enumerable language. This improves the previously published bound of eight for the nonterminal complexity of these grammars. This also yields an improved nonterminal complexity bound of four for context-free matrix grammars with appearance checking. Moreover, we establish nonterminal complexity bounds for context-free programmed and matrix grammars working under leftmost derivations.

Local economy and Generalized Pied-Piping

Abstract: In developing and elaborating the theory of Attract-F, Chomsky (1995) proposes the mechanism of Generalized Pied-Piping. According to Chomsky, when a formal feature of a head H attracts a matching feature of a category Ψ, other formal features of a category Ψ can be moved automatically, without any cost in terms of the general economy condition, to the target H together with the feature that is attracted to H. In this paper it will be demonstrated, contra Chomsky's (1995) proposal, that the application of Generalized Pied-Piping should be somehow constrained by the economy condition as long as the computation involved in the grammar of the human language (what Chomsky 1995 calls C HL ) is conducted only in a strictly derivational manner. This paper, therefore, claims that Generalized Pied-Piping, just like Move/Attract, is a kind of syntactic operation subject to the general economy condition under the view that C HL is strictly derivational.

Lambek Calculus and Formal Grammars

Abstract: The question about the position of categorial grammars in the Chomsky hierarchy arose in late 1950s and early 1960s. In 1960 Bar-Hillel, Gaifman, and Shamir [1] proved that a formal language can be generated by some basic categorial grammar if and only if the language is context-free. They conjectured (see also [7]) that the same holds for Lambek grammars, i. e., for categorial grammars based on a syntactic calculus introduced in 1958 by J. Lambek [10] (this calculus operates with three connectives: multiplication or concatenation of languages, left division, and right division). The proof of one half of this conjecture (namely, that every context-free language can be generated by some Lambek grammar) in fact coincides with the proof

Computing with Shapes

Abstract: Visual languages represent a response to the communicational challenges posed by end-user computing, but lack established computability frameworks for evaluating their computational power. In this paper, we introduce a computability model?called shape completion system?for the restricted, but important, case in which the visual representation of the concepts to be communicated is built as a puzzle. Shape completion systems are based on adjoining polyominoes, shapes from a basic set. A description in the form of a string on some alphabet can be associated with each basic shape. A computation in a shape completion system is correct when: (1) it starts by using a specified polyomino; (2) it ends when a rectangle is obtained (without holes); (3) at any step the current picture is connected; and (4) a sequencing mapping is given, so that at every step (except the first one) we use a polyomino depending on the previously used polyomino, as specified by this mapping (such a condition is essential for interactive visual languages, as formalized in 1, 2). We also establish how symbols associated with the polyominoes are concatenated to form strings in a string language associated with the computation. Surprisingly enough, in these circumstances we can characterize the recursively enumerable languages (hence the power of Turing machines). If we preserve only conditions (1), (2) and (3) above, then we cannot generate all linear languages but we can generate all regular languages and strictly more: also some one-letter non-regular languages can be obtained. In particular, we can obtain as correct computations squares only, which is often a difficult task in picture languages (see, e.g. 3).

On the Relationship between the McNaughton Families of Languages and the Chomsky Hierarchy

Abstract: By generalizing the Church-Rosser languages the McNaughton families of languages are obtained. Here we concentrate on those families that are defined by monadic or special string-rewriting systems. We investigate the relationship of these families to each other and to the lower classes of the Chomsky hierarchy and present some closure and some non-closure properties for them. Moreover, we address some complexity issues for their membership problems.

Extending Elementary Formal Systems

Abstract: An elementary formal system (EFS) is a logic program such as a Prolog program, for instance, that directly manipulates strings. Arikawa and his co-workers proposed elementary formal systems as a unifying framework for formal language learning.In the present paper, we introduce advanced elementary formal systems (AEFSs), i.e., elementary formal systems which allow for the use of a certain kind of negation, which is nonmonotonic, in essence, and which is conceptually close to negation as failure.We study the expressiveness of this approach by comparing certain AEFS definable language classes to the levels in the Chomsky hierarchy and to the language classes that are definable by EFSs that meet the same syntactical constraints.Moreover, we investigate the learnability of the corresponding AEFS definable language classes in two major learning paradigms, namely in Gold's model of learning in the limit and Valiant's model of probably approximately correct learning. In particular, we show which learnability results achieved for EFSs extend to AEFSs and which do not.

The Processing & Recognition of Symbol Sequences

Abstract: The Processing & Recognition of Symbol Sequences Mark W. Andrews (mwa1@cornell.edu) Department of Psychology; Uris Hall Ithaca, NY 14853 USA Abstract It is proposed that learning a language (or more gener- ally, a sequence of symbols) is formally equivalent to reconstructing the state-space of a non-linear dynamical system. Given this, a set of results from the study of nonlinear dynamical systems may be especially relevant for an understanding of the mechanisms underlying lan- guage processing. These results demonstrate that a dy- namical system can be reconstructed on the basis of the data that it emits. They imply that with minimal assump- tions the structure of an arbitrary language can be inferred entirely from a corpus of data. State-Space reconstruc- tion can implemented in a straightforward manner in a model neural system. Simulations of a recurrent neural network, trained on a large corpus of natural language, are described. Results imply that the network sucessfully recognizes temporal patterns in this corpus. Introduction Complex pattern recognition is often characterized by means of a simple geometric analogy. Any object or pattern may be described as a single point in a high- dimensional space. For example, a square grayscale im- age that is 256 pixels in length, may be described as a point in the 256 2 dimensional space of all possible im- ages. A collection of such images is a set of points in this space. If these patterns are not entirely random, this set will reside in a subspace of lower dimensionality. To learn the structure of these images, an organism or ma- chine must discover a compact parametric representation of this subspace. This might take the form of, for exam- ple, finding a reasonably small set of basis vectors that will span the subspace and projecting each image onto these vectors. Having done this, each image can be clas- sified in terms of a new and more meaningful coordinate system. You effectively describe ’what is there’ in terms of ’what is known’. This geometric approach is routinely employed in the study of visual object recognition, but may easily be extended to a wide range of categorization and classi- fication tasks. In almost all cases, however, the pat- terns under study have been multi-dimensional static pat- terns. In contrast, the study of temporal pattern recogn- tion using this or related approaches has not been well- developed. For example, one of the most widely em- ployed techniques for temporal pattern recognition, Hid- den Markov Models are limited in their generality due to their fundamental inability to handle patterns above a certain complexity. This absence of general models for temporal pattern recognition is evident in the study of human language processing, which traditionally has es- chewed serious consideration of statistical learning and pattern recognition. This paper aims to introduce a general framework for the study of temporal pattern recognition. This is devel- oped in the context to language processing, but it could be extended in a straightforward manner to most other cases of temporal patterns. First, a general characteriza- tion of the problem of language learning and language processingis proposed. Then, some recent results in the study of nonlinear dynamical systems are described. These are seen as being especially relevant for an under- standing of the mechanisms underlying temporal pattern recognition, especially with regard to language process- ing. Finally, simulations with a recurrent neural network are described, which suggest successful pattern recogni- tion of English sentences. The processing of symbol sequences A paradigm for the study temporal pattern processing, especially language processing, has developed as a re- sult of the deep relationship between formal languages and abstract automata (Chomsky 1963) 1 . Any language (or more generally, any sequence of symbols), can be de- scribed as the product of a particular automaton. By this account, learning a language is equivalent to identifying a particular automaton on the basis of a sample of the language that it generates. More formally, an automa- ton A is specified by the quadruple X Y F G . X and Y are sets known as the state and the output spaces, re- spectively. The functions F : X X and G : X Y 1 The correspondence between formal languages and ab- stract automata can be summarized by the so-called Chomsky hierarchy: Classes of automata that are increasingly restrictive versions of the Turing machine produce classes of languages described by increasingly restrictive generative grammars. The regular languages R are produced by strictly finite automata, the context-free languages CF are produced by pushdown stack automata, the context-sensitive languages CS are produced by linear bounded automata and the recursively enumerable lan- guages RN are produced by unrestricted Turing machines. R CF CS RN, and likewise for their corresponding automata.

A New Approach to Formal Language Theory by Kolmogorov Complexity

Abstract: We present a new approach to formal language theory using Kolmogorov complexity. The main results presented here are an alternative for pumping lemma(s), a new characterization for regular languages, and a new method to separate deterministic context-free languages and nondeterministic context-free languages. The use of the new `incompressibility arguments' is illustrated by many examples. The approach is also successful at the high end of the Chomsky hierarchy since one can quantify nonrecursiveness in terms of Kolmogorov complexity. (This is a preliminary uncorrected version. The final version is the one published in SIAM J. Comput., 24:2(1995), 398-410.)