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<Paper uid="C96-1045">
  <Title>Direct and Underspecified Interpretations of LFG f-structures</Title>
  <Section position="3" start_page="0" end_page="262" type="metho">
    <SectionTitle>
PRED features:
&amp;quot;PRED ~CANDIDATI,~ ~
NUM SG L OBJ
</SectionTitle>
    <Paragraph position="0"> LsPEc. A Unresolved QLF gives the basic predicate-argument structure of a sentence, mixed with some syntactic information encoded in the categories of QLF terms and forms: 1 ?Scope : support (term (+r, &lt;nmn=sg, spec=every&gt;, representative, ?Q, ?X) , term (+g, &lt;num=sg, spec=a&gt;, candidate, ?P, ?R) ) While there is difference in approach and emphasis unresolved QLFs and f-structures bear a striking similarity and it ix easy to see how to get from one to the other: pn~:, n($ r~,.,? r,~&gt; ~ .*Scope : n(~,.,~,~)</Paragraph>
    <Paragraph position="2"> The core of a mapping taking us from fstructures to QLFs places the values of subcategorizable grammmatieal fnnctions into their argument positions in the governing semantic form and recurses on those arguments. I,\]'om this rather general perspective the difference between f-structures mid l'l'he motivation for including tiffs syntactic information in QLFs is that resolution of such things as anaphora, ellipsis or quantifier scope may be constrained by syntactic factors (Alshawi, 1990).</Paragraph>
    <Paragraph position="3">  QLF is one of information packaging rather than mGthing else. We tbrmalise this intuition in terms of translation functions r. The precise fln'm of these mappings depends on whether the Q1,Fs and f-structures to be, mapI)ed contain comparable levels of syntactic information, and in the case, of QLF how this inforination is distributed between term and form categories and the recursive structure of the QLF. The QLF formalisln delitmratcly leaves entirely open the amounl; of syntactic information that should be encoded within a QLF the decision rests on how much syntactic intbrmation is required for successful contextual resolution. The architecture of the LFG and QLF formalism are described at length elsewhere (Ksplan &amp; Bresnan, 1982; Alshawi &amp; Crouch, 1992; Cooper et al., 1994a). l/eh)w we detine, a language of wJ\]:s (well-formed f-struct'tm:s), a (family of) translation function(s) r fi:om {-stru(:tures to (unresolved) QLFs and an inverse flmction r ~ Dom uin'esolved QLFs hack to f-structures, r and r -~ determine isolnorphic subsets of the QLF and LFG formalism. We eliminate r and give a direct and underspecified interpretation in terms of adapting QLF interpretation rules to fstrueture representations. While the initial definition of'r is designed to maxilnally exploit 1;he contextual resolution of QLF, later ve, rskms nfininfise resolution efl'ecl;s. A simph; version of ~ where the QLF COill;extual resolution component is &amp;quot;swil;ched off&amp;quot; is truth preserving with respect 1;o an independelfl, ly given semantics (DMrymple el; al., 1995).</Paragraph>
  </Section>
  <Section position="4" start_page="262" end_page="262" type="metho">
    <SectionTitle>
2 Well-formed f-structures
</SectionTitle>
    <Paragraph position="0"> We define a language of wff-s (we, ll-fornmd fstructures). The basic vocabulary consists of five disjoint set;s: GFs = {SUIU, OBJ, OlU2, ore,0,...} (subcategorizable grmnnml;ical flmctions), Gl&amp;quot;~, := {AmS, I, MODS, AMOI)S,...} (noIlsubcategorizahle gralnmatical ftlnctioIlS), SI,': {candidate0, marY0, support(j&amp;quot; suns, j&amp;quot; oB,,},...} (semantic forms), A'/&lt;: {SI'I4C, NUM, 1H,;II,...} (a/,;tributes) and AV= {~,;vl,mY, MOST, el,, FEM,...} (atomic values). The tirst two forlnation clauses pivot on the semantic form PILED values. The two tinM clmlses cow;r non-subcategorizMfle granmml;ical fulmtions aim what we call alomic attrilmtewdue pairs. Tags i\[i\] are used to repre, sent reentrancies and often appear vacuously, q'he side condition in the second and third clause ensure, s that only identical substructures can have identical tags:</Paragraph>
    <Paragraph position="2"> and tbr any C/~ and qS\[i~ occurring in ~\[~, 1 ~- m except possibly where 'gJ =- @.</Paragraph>
    <Paragraph position="4"> and for any C/\[1\] and XI~ occurring in (\[~, 1 -C/ m except possibly where 4) ~ X.</Paragraph>
    <Paragraph position="6"> Proposition: tim detinition specilies f-structures that are (',omph%e, coherent and consistent. 2</Paragraph>
  </Section>
  <Section position="5" start_page="262" end_page="264" type="metho">
    <SectionTitle>
3 How to QLF an f-structure:
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="262" end_page="263" type="sub_section">
      <SectionTitle>
3.1 A Basic Mapping
</SectionTitle>
      <Paragraph position="0"> Non-recursive f-structures are mapped to QLF terms and recursive f-structm'es to QI,F forms by metals of a two place flmction r detined below:</Paragraph>
      <Paragraph position="2"> where ?Scope is a new QLF mete-variable,, P ~ new w~riable and ~i 6_ AT ~Prool': induction on the formation rules for wff-s using the definitions of completeness, coherence atttl consistency (KalJan &amp; lbesmm, 1982), The not;ions of a'u, bst'r'u, ct'wre occwrrin.q in an f-structure al|d dom,,in of an f-struct'urc can easily be spelleA out fol'ntally. ~ is syntactic identity modulo permm;ation. The dciinition of w\]..f~s uses graphical rel)resen{;ations of t'-struct;ure.s. It can e.asily be recast in l;erlns of hierarchical sets, finite functions, directed graphs etc.</Paragraph>
      <Paragraph position="3">  To translate an f-structure, we call on r with the first argument set to a dummy grammatical flmction, SIGMA. The reader may check that given</Paragraph>
      <Paragraph position="5"> ?F_f).</Paragraph>
      <Paragraph position="6"> The truth conditions of the resulting underspecified QLF formula are those defined by the QLF evaluation rules (Cooper et al., 1994a). The original f-structure and its component parts inherit the QLF semantics via r. r defines a simple homomorphic embedding of f-structures into QLFs. It comes as no surprise that we can eliminate r and provide a direct underspecified interpretation for f-structures.</Paragraph>
      <Paragraph position="7"> Note that r as defined above maximises tile use of tile QLF contextual resolution component: quantifier meta-variables allow for resolution to logical quantifiers diflbxent fl'om surface form (e.g. to cover generic readings of indefinites), as do predicate variables (in e.g. support verb constructions) etc. A definition of r along these lines is useful in a reusability scenario where an existing LFG grammar is augmented with the QLF contextual resolution component. Alternative definitions of r &amp;quot;resolve&amp;quot; to surface form, i.e. minimise QLF contextual resolution. Such definitions are useflfl in showing basic results such as preservation of truth. Below we outline how r can be extended in order to capture more then just the basic LFG constructs and to allow for different styles of QLF construction.</Paragraph>
    </Section>
    <Section position="2" start_page="263" end_page="263" type="sub_section">
      <SectionTitle>
3.2 F-structure reentrancies
</SectionTitle>
      <Paragraph position="0"> r respects f-structure reentrancies (indicated in terms of identical tag annotations ~\]) without fllrther stipulation. Consider e.g. the f-st;ructure qo associated with the the control construction Most representatives persuaded a manager to support every subsidiary:  where the object \[~ of the matix clause is token identical with tile controlled subject \[~ of the embedded clause. ~o translates into  where the f-structure reentrancy surfaces in terms of identical QLF term indices ++- and metaw~riables ?0_+-,?R i as required.</Paragraph>
    </Section>
    <Section position="3" start_page="263" end_page="264" type="sub_section">
      <SectionTitle>
3.3 Non-Subcategorizable Grammatical
l%mctions
</SectionTitle>
      <Paragraph position="0"> The treatment of modification in both f-structure and QLF is open to some flexibility and variation.</Paragraph>
      <Paragraph position="1"> Here we can only discuss some exemplary cases such as LFG analyses of N and NP pre- and postmodification. We assume an analysis involving the restriction operator in the LFG description language (Wedekind &amp; Kaplan, 1993) and selnantic form indexing (II&lt;...&gt; @) e.g. by string position) as introduced by (Kaplan &amp; Bresnan, 1982). The f-structure associated with The company which sold APCOM started a new subsidiary is a aHere attd in tile following we will sometimes omit tags in the f-structure representations.</Paragraph>
      <Paragraph position="3"> The f-strneture associated with our example sentence translate.s into ?SO: form(+f, &lt;gf=sigma, pred=start (subj, obj ) &gt;,</Paragraph>
      <Paragraph position="5"> ?F_~).</Paragraph>
      <Paragraph position="6"> as required. Note, however, that the translation inay overspecify the range. In the f-structure domain modifiers are collected in an unordered set while in the range we impose some arbitrary ordering. For intensional adjectives (compare a former famous president with a famous former president), this ordering may well be incorrect. Hence ordering information should be codable in (or recoverable from) the representations. In LFC this is available in terms of f-precedence. A more satisfactory translation into QLF complicates the treatment of (nominal) Inodification as abstracted QLF forms. Modifiers are represente(1 as extra arguments in the body of the form and take the form index of dm restriction as one of their argmnents: 4 x- ?Scp : form (+r, &lt;gf=np-re str, pred=subs idiar y&gt;, P^P(x, form(+a, &lt;gf=am, pred=new&gt;,</Paragraph>
      <Paragraph position="8"> Modifier ordering can then be transferred to resolution, or encoded in the categories of the rest, r|(&gt; Lion and modifiers to filrther constrain the order of application selected by resolution.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="264" end_page="266" type="metho">
    <SectionTitle>
4 Direct interpretation
</SectionTitle>
    <Paragraph position="0"> The core of tile direct interpretation clauses for wff-s involve~s a simple variation of the quantifier rule and the t)redieation rule of the QLF sentanlies (Cooper et el., 1994a). Consider tile flagmeat without N and NP modification. As before, t;he semant;ics is detined in terms of a supervaluat|on construction on sets of disambiguated representations. Models, variable assignment flmclions, generalized quantifier interpretations and the QLF definitions for the connectives, abstraction and application etc. (see Appendix) carry  over unchanged. The Ile.W quantification rule D14 non-det;erlninistically retrieves non-recursive Sll|)~ categorizable grammatical fiulctions and entploys the vahle of a SI'EC feature in a generalized quaIb tiller interpretation:</Paragraph>
    <Paragraph position="2"> dmn V,(% v) if V,,(~o\[~/,(~\]) &lt; Ill), v) The new predication rule 1)10 is defined in terms of a notion of nuclear scope f-structure: '~' 4See (Cooper et al., 1994b) for examples of this style of treating VP modification.</Paragraph>
    <Paragraph position="3"> r)A nuclear scope f-structure ~ C nf-s is is an f-structure resulting from exhaustive at)plicatiou of D14. It can be defined inducdwdy as follows: * if 3`i a variable or a constant symbol then I F1 3'1 ~ I'RI';I) II(? Pl,..., ? Pn) tK @s</Paragraph>
    <Paragraph position="5"> then 12~(~, v) if Vg(II(v~ , .,3,~), v) To give an example, under the direct interpretation the f-structure associated with most representatives supported two candidates is interpreted as an underspecified semantic representation in terms of the supervaluation over the two generalized quantifier representations most (repr, Ax. two ( cand, .~y. support (x, y))) two ( cand, Ay. most( repr , Ax. support ( x, y) ) ) as required. The direct underspecified interpretation schema extends to the modification cases discussed above in the obvious fashion.</Paragraph>
    <Paragraph position="6"> 5 How to f-structure a QLF The reverse mapping from QLFs to LFG f-structures ignores information conveyed by resolved recta-variables in QLF (e.g. quantifier scope, pronouns antecedents), just as the mapping froIn f-structure to QLF did not attempt to fill in values for these recta-variables. For QLF terms with simple restrictions (i.e. no modifiers),</Paragraph>
    <Paragraph position="8"> As an example the reader may verify that r-~ retranslates the QLF associated with Most representatives persuaded a candidate to support every subsidiary back into the f-structure associated with the sentence as required. Again, 7 --1 can be extended to the non-subcategorizable grammatical functions discussed above. The extension is straightforward but messy to state in full generality and for reasons of space not given here.</Paragraph>
    <Paragraph position="10"> The result establishes isomorphic subsets of the QLF and LFG formalisms. For an arbitrary QLF C/, however, the reverse does not hold  w assigns a meaning to an f-structure that depends on the f-structure and QLF contextual resolution. We define a restricted version T' of ~- which &amp;quot;switches off&amp;quot; the QLF contextual resolution component, w' maps logical quantifiers to their surface form and semantic forms to QLF formulas (or resolved QLF forms):  O~ 1 ZVl, . * . ~ ~m zVm &gt; , i~(T(r,, ~,~\]),..., T(rn, ~nli~)),m Proposition: T' is truth preserving with respect to an independent semantics, e.g. the glue language semantics of (Dalrymple et al., 1995)* Preservation of truth, hence correctness of the translation, is with respect to sets of disambiguations. The proof is by induction on the complexity of ~7 The correctness result carries over to the direct interpretation since what is eliminated is T'. s 6Proof: induction on the complexity of y;.</Paragraph>
    <Paragraph position="11"> 7Proof sketch: refer to the set of disambiguated QLFs resulting from w'(~o) through application of the QLF interpretation clauses as \])(T'(~)) and to the set of conclusions obtained trough linear logic deduction from the premisses of the (r projections of ~p as (a(~o))F. Consider the fragment without modification. Base case: for So with nonrecursive values of grammatical functions show Y(T'(~)) = (a(W))e. Induction: for ~ with possibly reeursive values ~i of grammatical functions on the assumption that for each i:</Paragraph>
    <Paragraph position="13"> sIf the results of linear logic deductions are interpreted in terms of the supervaluation construction we have preservation of truth directly with respect to underspecified representations, QLFs and sets of linear logic premisses.</Paragraph>
  </Section>
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