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<Paper uid="E93-1029">
  <Title>Mathematical Aspects of Command Relations</Title>
  <Section position="2" start_page="0" end_page="240" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="240" type="sub_section">
      <SectionTitle>
1.1 Historic Origin
</SectionTitle>
      <Paragraph position="0"> 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.</Paragraph>
      <Paragraph position="1"> The analysis of transformations as series of movements - an analysis made possible by the introduction of empty categories - was one step. This indeed drastically simplified the transformational component. 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, categorial grammar etc. became less and less clear. This in addition to Noam Chomsky's often repeated scepticism with respect to formalizations has led to the common attitude that GB is simply gibberish, unformalizable or hopelessly untractable at best. However, 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.</Paragraph>
      <Paragraph position="2"> 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 rigorously 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 defined 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 properties 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 interested in doing so. Even if machine language suits the machine, we need to provide a higher language and a translation to make computers really useful for practical tasks. However, as long as we do not know in linguistics what the 'machine language' of the human mind is, the best we can do at the moment is to provide means to translate in between all these syntactical formalisms. So, even if from the point of  view of universal grammar this gets us no closer to the language faculty of the human mind, the need to understand the formal properties of Gs and the relationship between all these approaches remains and must be satisfied in order to achieve real progress.</Paragraph>
      <Paragraph position="3"> The theory of command relations forms part of an investigation that should ultimately lead to such an understanding. The present paper will sketch the theory of command relation and is a distilled version of \[Kracht, 1993\].</Paragraph>
    </Section>
    <Section position="2" start_page="240" end_page="240" type="sub_section">
      <SectionTitle>
1.2 Relevance of Command Relations
</SectionTitle>
      <Paragraph position="0"> The idea to study the formal properties of command relations is due to \[Barker and Pullum, 1990\]. There we find a definition of command relations as well as many illustrations of command relations from linguistic theory. In that paper the origins of the notions are also discussed. I guess it is fair to attribute to \[l~inhart, 1981\] the beginning of the study of domains. Moreover, \[Koster, 1986\] presents a impressive and thorough study of the role of domains in grammar. Yet all this work is either too specific or too vague to lead to a proper understanding of nearness conditions in grammar. In \[Kracht, 1992\] I took the case of \[Barker and Pullum, 1990\] further and proved some more results concerning these relations especially the structure of the heyting algebra of command relations. The latter proved to be of little significance in the light of the questions raised in SS 1.1. Instead, it emerged that it is more fruitful to study the properties of command relations under intersection, union and relational composition. They form an algebraic structure called a distributoid. The structure of this distributoid can be determined. If the grammar is enriched with enough labels, this distributoid contains enough command relations to express all known nearness conditions. This being so, it becomes an immediate question whether the effect of a nearness condition expressed via command relations can be incorporated into the syntax. This is discussed at length in \[Kracht, 1993\]. The result is that indeed all such conditions are implementable, but this often requires a lot more basic features. The explosion of the size grammars when translating from GB to GPSG can be explained namely by the necessity to add auxiliary features that secure that the grammar obeys certain nearness restrictions. A typical example is the SLASH-feature which has been invented to guarantee a gap for a displaced filler.</Paragraph>
      <Paragraph position="1"> With such proof that implementations of nearness conditions into cfg's can always be given (maybe on certain other harmless conditions) one is in principle dispensed from writing GVSG-type grammars in order to make available the rich theory of context-free grammars. Now it is possible to transfer this theory to grammars which consist both of a generative context-free component and a set of well-formedness conditions based on command relations. In particular, it is perfectly decidable whether two such grammars generate the same bracketed strings and hence effective comparison between two different theories of natural language - if given in that format - is possible.</Paragraph>
    </Section>
  </Section>
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