Some classifications of context-free languages (original) (raw)
Oxford Handbooks Online, 2005
This article introduces the preliminaries of classical formal language theory. It outlines the main classes of grammars as language-generating devices and automata as language-recognizing devices. It offers a number of definitions and examples and presents the basic results. It classifies grammar according to several criteria. The most widespread one is the form of their productions. This article presents a systematic study of the common properties of language families has led to the theory of abstract families of languages. It shows that a context-free grammar generates not only a set of strings, but a set of trees too: each one of the trees is associated with a string and illustrates the way this string is derived in the grammar.
On the size of context-free grammars
Kybernetika, 1972
In papers [2] and [3] four criteria of complexity of context-free grammars (CFG's), denoted by Var, Lev, Lev„, and Depth, have been studied. These criteria reflect the intrinsic complexity of CFG's and they induce the criteria of complexity of contextfree languages (CFL's) which reflect the intrinsic complexity of the description of CFL's by CFG's. The criterion Prod (G) = the number of rules of a CFG G, studied in [3] represents the size of CFG's. In the present paper one more criterion of complexity of CFG's, namely Symb (G) = = the number of all occurrences of all symbols in the rules of G, is defined and some results concerning the criteria Prod and Symb are derived.
Complexity and unambiguity of context-free grammars and languages
Information and Control, 1971
Four of the criteria of complexity of the description of context-free languages by context-free grammars are considered. The unsolvability of the basic problems is proved for each of these criteria. For instance, it is unsolvable to determine the complexity of the language generated by a given grammar, or to find out the simplest grammar, or to decide whether a given grammar is the simplest one and so on. Next, it is shown that in some cases one can obtain unambiguity only by increasing complexity. Namely, for each of the four criteria, in any complexity class there are unambiguous languages, all simplest grammars of which are ambiguous. As one would expect, it is unsolvable whether for an arbitrary grammar G there are unambiguous grammars within the simplest grammars for the language generated by G.
On Vertical Grammatical Restrictions that Produce an Infinite Language Hierarchy
International Conference on Information System Implementation and Modeling, 2007
This paper introduces deriuation table.s that represent a complete grammatical derivations as whole in a vertical way. These tables are obtained by writing the consecutive sentential forms of grammatical derivations vertically one by one. The present paper places and discusses some restrictions on the columns of these tables. IVIore specifically, these restrictions constrain the order of context-sensitive derivations on boundaries of columns. It is demonstrated, that grammars restricted in this way generate an infinite language hierarchy.
2011
Stabler proposes an implementation of the Chomskyan Minimalist Program, [1] with Minimalist Grammars-MG, [2]. This framework inherits a long linguistic tradition. But the semantic calculus is more easily added if one uses the Curry-Howard isomorphism. Minimalist Categorial Grammars-MCG, based on an extension of the Lambek calculus, the mixed logic, were introduced to provide a theoreticallymotivated syntax-semantics interface, [3]. In this article, we give full definitions of MG with algebraic tree descriptions and of MCG, and take the first steps towards giving a proof of inclusion of their generated languages. The Minimalist Program-MP, introduced by Chomsky, [1], unified more than fifty years of linguistic research in a theoretical way. MP postulates that a logical form and a sound could be derived from syntactic relations. Stabler, [2], proposes a framework for this program in a computational perspective with Minimalist Grammars-MG. These grammars inherit a long tradition of generative linguistics. The most interesting contribution of these grammars is certainly that the derivation system is defined with only two rules: merge and move. The word Minimalist is introduced in this perspective of simplicity of the definitions of the framework. If the merge rule seems to be classic for this kind of treatment, the second rule, move, accounts for the main concepts of this theory and makes it possible to modify relations between elements in the derived structure. Even if the phonological calculus is already defined, the logical one is more complex to express. Recently, solutions were explored that exploited Curry's distinction between tectogrammatical and phenogrammatical levels; for example, Lambda Grammars, [4], Abstract Categorial Grammars, [5], and Convergent Grammars [6]. First steps for a convergence between the Generative Theory and Categorial Grammars are due to S. Epstein, [7]. A full volume of Language and Computation proposes several articles in this perspective, [8], in particular [9], and Cornell's works on links between Lambek calculus and Transformational Grammars, [10]. Formulations of Minimalist Grammars in a Type-Theoretic way have also been proposed in [11], [12], [13]. These frameworks were evolved in [14], [3], [15] for the syntax-semantics interface. Defining a syntax-semantics interface is complex. In his works, Stabler proposes to include this treatment directly in MG. But interactions between syntax