Relationship between strong and weak generative power of formal systems

The relationship between strong and weak generative powers of formal systems is explored, in particular, from the point of view of squeezing more strong power out of a formal system without increasing its weak generative power. We examine a whole range of old and new results from this perspective. However, the main goal of this paper is to investigate the strong generative power of Lambek categorial grammars in the context of crossing dependencies, in view of the recent work of Tiede (1998). Introduction Strong generative power (SGP) relates to the set of structural descriptions (such as derivation trees, dags, proof trees, etc.) assigned by a formal system to the strings that it specifies. Weak generative power (WGP) refers to the set of strings characterized by the formal system. SGP is clearly the primary object of interest from the linguistic point of view. WGP is often used to locate a formal system within one or another hierarchy of formal grammars 1. Clearly a study of the relationship between WGP and SGP is highly relevant, both formally and linguistically. Although there has been interest in the study of this relationship, almost from the beginning of the work in mathematical linguistics, the results are few, as this relationship is quite complex and not always easy to study mathematically (see Miller (1969) for a recent comprehensive discussion of SGP). Our main goals in this paper are (1) to look at some old and recent results and try to put them in a general framework, a framework that can best be described by the slogan–How to squeeze more strong generative power out of a grammatical system?– and (2) to present a new result concerning Lambek categorial grammars. Our general discussion of the relationship of SGP and WGP will be in the context of context-free grammars, categorial grammars and lexicalized tree-adjoining grammars. 1. Context-Free Grammar (CFG) McCawley(1967) was the first person to point out that the use of context-sensitive rules by linguists was really for checking structural descriptions (thus related to SGP) and not for generating strings (i.e., WGP), suggesting that this use of context-sensitive rules possibly does not This work was partially supported by NSF Grant SBR8920230 1SGP is also relevant in the context of annotated corpora. The annotations reflect aspects of SGP and not of the rules of a grammar and therefore WGP. give more SGP than CFG’s. Peters and Ritchie (1969) showed that this was indeed the case. These results are closely related to the notion of recognizable sets of trees (structural descriptions) as explained below. In a CFG,G, the derivation trees of G correspond to the possible structural descriptions assignable by G. It is easily shown that there are tree sets whose yield language is context-free but the tree sets are not the tree sets of any CFG. That is, we are able to squeeze more strong power out of CFG’s indirectly. Here is a simple example.