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<Paper uid="E93-1004">
  <Title>Talking About Trees</Title>
  <Section position="2" start_page="0" end_page="21" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> In this paper we introduce a modal language L 7 for talking about trees, and an extension L'r(L F) for talking about trees decorated with feature structures. From a logical point of view this is a natural thing to do. After all, the trees and feature structures used in linguistics are simple graphical objects. To put it another way, they are merely rather simple Kripke models, and modal languages are probably the simplest languages in which non-trivial constraints can be imposed on such structures. Moreover the approach is also linguistically natural: many of the things linguists need to say about trees (and feature structures) give rise to modal operators rather naturally, and indeed our choice of modalities has been guided by linguistic practice, not logical convenience.</Paragraph>
    <Paragraph position="1"> There are several reasons why we think this path is an interesting one to explore, however two are of particular relevance to the present paper. 1 First, we believe that it can lead to relatively simple and natural formalisations of various grammatical frameworks. In our view, neither simplicity nor naturalness are luxuries: unless a formalisation possesses a high degree of clarity, it is unrealistic to hope that it can offer either precise analyses of particular systems or informative comparisons of different frameworks. We believe our approach has the requisite clarity (largely because it arose by abstracting from linguistic practice in a rather direct manner) and much of this paj?er is an attempt to substantiate this. Second, L r can be combined in a very natural way with feature logics to yield simple systems which deal with configurational concepts, complex categories and their interaction. The key idea is to perform this combination of logics in a highly constrained way which we have called layering. Layering is a relatively new idea in modal logic (in fact the only paper devoted exclusively to this topic seems to be \[Finger and Gabbay 1992\]), and it seems to provide the right level of expressive power needed to model many contemporary grammar formalisms.</Paragraph>
    <Paragraph position="2"> The paper is structured as follows. In section 2 we define the syntax and semantics of L T, our modal language for imposing constraints on tree structure.</Paragraph>
    <Paragraph position="3"> ~Lurking in the background are two additional, rather more technical, reasons for our interest. First, we believe that being ezplicit about tree structure in our logical object languages (instead of, say, coding tree structure up as just another complex feature) may make it easier to find computationally tractable logics for linguistic processing. Second, we believe that logical methods may interact fruitfully with the mathematical literature on tree admissibility (see \[Peters and Ritchie 1969\], \[Rounds 1970\] and \[Joshi and Levy 1977\]). However we won't explore these ideas here.</Paragraph>
    <Paragraph position="4">  In section 3 we put L T to work, showing how it can be used to characterise the parse trees of context free phrase structure grammars. In section 4 we consider how the structured categories prevalent in modern linguistic formalisms are dealt with. Our solution is to introduce a simple feature logic L F for talking about complex categories, and then to layer L T across L F. The resulting system LT(L F) is capable of formulating constraints involving the interaction of con~igurational and categorial information. Section 5 illustrates how one might use this expressive power by formulating some of the leading ideas of (~PS(~ in LT(LF). We conclude the paper with some general remarks on the use of modal languages as constraint formalisms in linguistics.</Paragraph>
  </Section>
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