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<Paper uid="C96-1044">
  <Title>Extended Dependency Structures and their Formal Interpretation</Title>
  <Section position="15" start_page="259" end_page="260" type="concl">
    <SectionTitle>
TYPE SENSITIVI~ COMPOSITION R.UI,ES If
</SectionTitle>
    <Paragraph position="0"> we apply the vat+table-binding rules to the snbtree PH = ((peter,-hl),hate(hi,h2)) of Fig. 8, we lind that we mtisl compose the semantic transhttions peter and %h 1.hate(h I ,h2) in &amp;quot;con+lplement&amp;quot; (+) mode. The litst ftmction is of type e, while the second function is oi type e~t (lor hate(hi,h2) is of type t, and hl of type e).</Paragraph>
    <Paragraph position="1"> ltow do we compose two such functions? A first so lution, in the spirit of Lambek calculus (Morrill, 1994) or of linear logic (Dah'ymple et al.. 1995), would tie to define a general computational mechanism which would be able, through a systematic discipline of type-changing operations, to &amp;quot;adapt&amp;quot; atttomatically to the types of the functions undergoing composition.</Paragraph>
    <Paragraph position="2"> Such mechanisms are powerful, but they tend to be algorithmically complex, to be non-lncal, and also to give rise to spurious antbiguities (superficial variations in the proof process which do not correspond to di fferent semantic readinos) t&amp;quot; &amp;quot; Here, we will prefer to use a less general tnecha+ nism, but one which has two advantages. First, it is local, simple, and efficient. Second, it is flexible attd can tie extended to handle the semantics of sentences extracted fiom a real corpus of texts, which it might he perilous to constrain too strongly fi'om the starc The mechanism is the following. We establish a list of acceptable &amp;quot;type-sensitive composition rules&amp;quot;, which tell us how to compose two flmctions according to their types. Such a (provisory) list is given be ow: e</Paragraph>
    <Paragraph position="4"> The entries in this list have the following fornmt.</Paragraph>
    <Paragraph position="5"> The lflst argtuncnt indicates tile type of composition (++' fl)r complement incorporation. &amp;quot;-&amp;quot; for modilier incorl)orath'm. &amp;quot;++let&amp;quot; for deter+miner incorporation): the second argument is of the Iklrln Lelt:l+eftTypc, where Left is the left translation entering the composition, and LeftTypc is its type: similarly, the second argument Right:RightType corresponds to the right sub-tree entclin ~r~. the composition: linallv+ the third atELl-, ment ~ives the resuh l,?,esuh:l,P, esultType of the composition, where the notation A(B) has been used to indi cate slandard functional application of function A on arguntent B. Uppercase letters indicate unifiable vari ahles.</Paragraph>
    <Paragraph position="6"> It may be remarked thai if, in these rules, we neglect the functions themselves (1 +eft, Right, Resnlt) and con.</Paragraph>
    <Paragraph position="7"> centrate on their types (l+eflType, RightType, Result + l'ype), then the rules can be seen as itnl)osin,,+ constraints on what can count :is validly typed trees: these constrahlts can flow from nlother to daugthers as ,.veil as in the opposite direction. Thus. through these rules.</Paragraph>
    <Paragraph position="8"> knowing thai the head-line functions projecting l\]tlnl it verbal head must he of type t imposes some constraints on wlmt are the possible types for the det)endents: this can be usefttl in partict, lar for constraining the types nf semantically ambiguot, s lexical elements.</Paragraph>
    <Paragraph position="9"> If we now go back to our example, we have to con&gt; pose in complement mode (+) the function peter, ol type e. with the ftmction th I .hate(h I .h2). of type e--t. Consnlting the list of composition rules, we see that the only applicable rnle is (C2). and that the result is Ahl.hate(h l,h2) (peter) = hate(peter.h2), of type t.</Paragraph>
    <Paragraph position="10"> Now that we have the semamic translation hate(peter, h2) for the subtree Pit, we can compute the translation for the suhtree ((PH,-h2).woman). By the variable-binding rnle for modiliers, we need lirst to form the abstraction Xh2.hate(peter.lt2). of type e~t.</Paragraph>
    <Paragraph position="11"> and compose it in '-' mode with wonmn, of type e--t.</Paragraph>
    <Paragraph position="12"> Consnlting the list of composition rules, we find that the only applicable rule is (C5). and that the result of this application is Ah2.woman(h=)Ahate(l~eter, h=).: e It is a matter for further research to propose principles lk)l&amp;quot; ploducing such Ill\]ON. SotllC t)t&amp;quot; them can be seen as special cases of general type-raising principles, others (such as C5) are necessary it one accepts that the type of intersectivc adjectives and restrictive relative clauses has to be e -t.</Paragraph>
    <Paragraph position="13"> :~Thc rule (C5) differs from the previous rules in ll~e list in that it introduces the logical connective A which does lint originate in functional material already present in either of the arguments. A possible justilication for the rule. however,  l:igurC/ 10: B form interpretation, l&amp;quot;or &amp;quot;cvcry', we make use of the gcner;tlized quantilier notation qmm l( n'.st ri cl i ou.SCOl)e ).</Paragraph>
    <Paragraph position="14"> Fhe process of semantic translalion tin&gt;coeds in this way bottom Ul-~on the B form. The end restth is; shown in Fig. 10.</Paragraph>
  </Section>
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