Mathematical Aspects of Command Relations

Mathematical Aspects of Command Relations Marcus Kracht II. Mathematisches Institut ArnimaUee 3 D - 1000 Berlin 33 GERMANY email: kracht~ath, fu-berlin, de Abstract In GB, the importance of phrase-structure rules has dwindled in favour of nearness conditions. Today, nearness conditions play a major role in defining the correct linguis- tic representations. They are expressed in terms of special binary relations on trees called command relations. Yet, while the formal theory of phrase-structure gram- mars is quite advanced, no formal investi- gation into the properties of command re- lations has been done. We will try to close this gap. In particular, we will study the in- trinsic properties of command relations as relations on trees as well as the possibil- ity to reduce nearness conditions expressed by command relations to phrase-structure rules. 1 Introduction 1.1 Historic Origin Early transformational grammar consisted of a rather complex generative component and an equally complex and equally imperspicuous transformational component. But since the aim always has been to understand languages rather than describing them, there has been a need for a reduction of these rule systems into preferably few and simple principles. The analysis of transformations as series of move- ments - an analysis made possible by the introduc- tion of empty categories - was one step. This in- deed drastically simplified the transformational com- ponent. A second step consisted in simplifying the generative component by reducing the rules in favour of well-formedness conditions, so-called filters. While this turned transformational grammar into a real theory now known as GB, the relationship of GB with other syntactic formalisms such as GPSG, LFG, cate- gorial grammar etc. became less and less clear. This in addition to Noam Chomsky's often repeated scep- ticism with respect to formalizations has led to the common attitude that GB is simply gibberish, unfor- malizable or hopelessly untractable at best. How- ever, since it is possible to evaluate predictions of theories of GB and have constructive debates over them these theories are if not formal then at least rigorous. Hence, it must be possible to formalize them. Formalizations of GB have been offered, e. g. in [Stabler, 1989] hut in a manner that makes 6B even less comprehensible. So if formalization means providing as complete as possible intellectual access to the formal consequences of an otherwise rigor- ously defined theory the project has failed if ever begun. More or less the same criticism applies to [Gazdar et al., 1985]. Even if 6PsG is rigorously de- fined the formalism as laid out in this book does not lead to an understanding of it's properties. More or less the same applies to categorial grammar which might have the advantage that it's formal proper- ties are well-studied but which suffers from the same ill-suitedness to the human intellect. The situation can be compared with computer science. While it is perfectly possible to reduce programs in PASCAL to programs in machine language, hardly is anyone in- terested in doing so. Even if machine language suits the machine, …

Content truncated to 500 words