Contextual Grammars versus Natural Languages

Mathematical Structures of Computer Science
TUCS Research Group
1 Introduction
The contextual grammars produce languages starting from a nite set of axioms and adjoining contexts, iteratively, according to a selection procedure: each context (presented as a pair of strings) has an associated set of strings (called a selector set) to which the context may be adjoined. As introduced in 13], the adjoining of the contexts is done at the ends of the strings. In the internal contextual grammars, introduced in 21], the contexts can be adjoined to the selector strings appearing as substrings of the derived strings. Details about these basic variants of contextual grammars can be found in 19] (see also 20]). Recently, a variant has been considered, which turned out to be able to cover all usual non-context-free constructions in natural (and arti cial) languages: internal contextual grammars with maximal use of selectors, see 16]. Speci cally, a string selector may be used only if no superstring of it can be used. We shall recall here the results in 16], 14], 15], about the generative capacity of these grammars and about their relevance for natural language modelling, then we shall continue the study of the second topic (covering speci c constructions, semilinearity and related properties, ambiguity). In the last section of the paper we introduce contextual grammars with the selectors presented as patterns in the sense of 1]. Speci cally, we consider as selector language the language identi ed by a pattern (that is, a string consisting of constants and variables), by consistently interpreting the variables. In order to derive a string, we identify an interpretation of the pattern selector as a substring of it and we adjoin the associated context to this interpretation. As for usual contextual grammars, we consider free, minimal and maximal modes of interpreting the pattern selectors.
Байду номын сангаасbstract
We continue here the investigations concerning the relevance of certain classes of contextual grammars (internal contextual grammars with maximal use of selectors) as generative models of natural languages. We discuss the mildly context-sensitiveness property (especially the semilinearity and related properties), the possibility to cover certain constructions speci c to natural languages, as well as the ambiguity of languages generated by our grammars. Then, contextual grammars with the selectors presented as patterns are introduced and brie y investigated from the point of view of covering the mentioned constructions in natural languages.
Institute of Mathematics of the Romanian Academy PO Box 1-764, 70700 Bucuresti, Romania
Turku Centre for Computer Science TUCS Technical Report No 44 September 1996 ISBN 951-650-837-5 ISSN 1239-1891
Contextual Grammars versus Natural Languages
Solomon Marcus Carlos Martin-Vide Gheorghe Paun
University of Bucharest, Faculty of Mathematics Str. Academiei 14, 70109 Bucuresti, Romania Research group in Mathematical Linguistics and Language Engineering Rovira i Virgili University, Pca. Imperial Tarraco 1 43005 Tarragona, Spain
2 Internal contextual grammars with maximal use of selectors
We denote: A = the free monoid generated by the alphabet A, = the empty word, A+ = A ? f g, jxj = the length of x 2 A , mi(x) = the mirror image (the reversal) of x 2 A . The families of nite, regular, contextfree, context-sensitive, and recursively enumerable languages are denoted by FIN; REG; CF; CS; RE , respectively. For the elements of formal language theory we use, we refer to 26]. A contextual grammar is a construct
G = (A; B; (S1; C1); : : :; (Sn; Cn )); n 1;
1
where A is an alphabet, B is a nite language over A, S1; : : : ; Sn are languages over A, and C1; : : :; Cn are nite subsets of A A . The elements of B are called axioms, the sets Si are called selectors, and the elements of sets Ci, written in the form (u; v), are called contexts. The pairs (Si; Ci) are also called productions. The contexts in Ci may be adjoined only to words in the associated set Si. Formally, we de ne the direct derivation relation on A as follows: x =)in y i x = x1x2x3; y = x1ux2vx3; where x2 2 Si; (u; v) 2 Ci; for some i; 1 i n: Remark. The above de ned relation (introduced in 21]) has been denoted by =)in in order to distinguish it from the external derivation de ned for G (this type is considered in 13]), where the context is adjoined at the ends of the derived word: x =)ex y i y = uxv for (u; v) 2 Ci; x 2 Si, for some i; 1 i n. We do not investigate here the external derivation. Two natural variants of the relation =)in de ned above are (see 16]): x =)Ml y i x = x1x2x3; y = x1ux2vx3; for x2 2 Di; (u; v) 2 Ci; for some 1 i n; and there are no x01; x02; x03 2 V such that x = x01x02x03; x02 2 Di; jx01j jx1j; jx03j jx3j; jx02j > jx2j; x =)Mg y i x = x1x2x3; y = x1ux2vx3; for x2 2 Di; (u; v) 2 Ci; for some 1 i n; and there are no x01; x02; x03 2 V such that x = x01x02x03; x02 2 Dj ; for some 1 j n; jx01j jx1j; jx03j jx3j; jx02j > jx2j: We say that =)Ml is a derivation in the maximal local mode (the wordselector x2 is maximal in Si) and =)Mg is a derivation in the maximal global mode (the word-selector x2 is maximal with respect to all S1; : : : ; Sn). For 2 fin; Ml; Mgg; we denote L (G) = fz 2 A j w =) z; for some w 2 B g; where =) is the re exive and transitive closure of =) . If in a grammar G as above, all selectors S1; : : : ; Sn are languages in a given family F , then we say that G is a contextual grammar with F choice (or with F selection). The families of languages L (G), for G a contextual grammar with F choice, is denoted by CL (F ). Here we consider F one of the families FIN; REG only. 2