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<?xml version="1.0" standalone="yes"?> <Paper uid="C96-1044"> <Title>Extended Dependency Structures and their Formal Interpretation</Title> <Section position="5" start_page="255" end_page="255" type="metho"> <SectionTitle> WELL-FORMEDNESS CONIilTIONS (iN U- </SectionTitle> <Paragraph position="0"> FORMS In order to be well-formed, a U-iorm Uf: has to respect tile following condition. For ;.lily node A of L!F. the predication tree T i must be such that: 1. \[No hoh, s comliti(m\] If (A.i.B) is an edge of 1&quot; i. then for any number j between I and i, T.I nlttst contain a node of form (A,i,C)+ 2. \[No rdpe'tilioll comlimml No two edges of T.t can have tile salne label i.</Paragraph> </Section> <Section position="6" start_page="255" end_page="256" type="metho"> <SectionTitle> MORE ON U-FORMS Negative labels are a device </SectionTitle> <Paragraph position="0"> which permits to reconcile the notalien of predicateargnnlenl structnre with the notation of syntactic depondoilcy, So, in the i..\]-fornl considered above, while &quot;semantically&quot; tile 'wellqan' node is an ;.irglnl\]oni of tile &quot;hate&quot; node, &quot;syntactically&quot; tile hate' node is a dependent of tile 'woman' node. Cases such as this one.</Paragraph> <Paragraph position="1"> where there is a conflict between predicate-argument directionality and dependency directionality are no tated ill the U form throngh negative labels, and correspond tO #llodifie#',','. Cases where tile directionality is parallel correspond to complement.s.</Paragraph> <Paragraph position="2"> When used as interlingual representations in machine translation systems, U-forms have several advantages. The first is ttmt they neutralize certain details of syntactic structure that de not carry easily between languages. For instance. French and English expiess negation in syntactically different ways: &quot;Rachel does net like Claude&quot; vs. deg'Rachel n'aime pas Claude&quot;: this difference is neutralized in the U-fornl representation, for both negations are expressed through a single negation predicate in the U-feral.</Paragraph> <Paragraph position="3"> A second advantage is that they represent a good compromise between paraphrasing potential and semantic precision. So. for instance, in tile CRITTEI,I system, the three sentences: John does not like every woman that Peter hates John does not like every woman hated by Peter Every woman whom Peter hates is not liked by John would be assigned the U-form of Fig. 1. On the other hand, the sentence: Peter hates every woman that John does not like would be assigned tile t!-form or' f:i~,. 3, which is different from the previous U-form, although tile predicate-argument rehitions are exactly tile sanie in hoth cases.</Paragraph> <Paragraph position="4"> ()he can take advantage of such paral)hrasing potential in cerlain cases of synlaclJc divergence belween languages, l:or instance, French does not have a syn tactic equivalent to the dative-lnoven/etlt + passive configuration o1: Rachel was given a book by Claude so that a direct syntactic translation is not possible. However, at tile level of U-form, this sentence is equivalent to the French sentence: Claude a donne un livre ~t Rachel and this equivalence can he exploited it) provide a translation of the first sentence.</Paragraph> <Paragraph position="5"> One serious problenl with 1.\]. \[ornis, however, is tilat they do not have unainbiguous readhigs in cases where the rehliive scopes of constituents can result in clifl'erent semantic ii~terpretations. So, in the case of sen fence (S l), tile two readings: &quot;it is not the case thai John likes every woman hated by Peter&quot;, and 'Lh)tl.n dislikes every woman thai Peter hates&quot; are not distinguished by tile t l-l:oi+nl Of l&quot;ig+ 1,</Paragraph> </Section> <Section position="7" start_page="256" end_page="256" type="metho"> <SectionTitle> 3 S-FORMS INTI,I()I)UCIN(; SC()I'E Lefs consider the trec </SectionTitle> <Paragraph position="0"> represented in Iqg. 4.</Paragraph> <Paragraph position="1"> The only differeiice hetween this tree aml the l,Jform of l:ig. l is thai the nodes of our new tree are considered ordered whereas they were considered tinordered in the I!-lorm+ The convention is now that (tepetlttent sister nodes are interpreted :is having ttil'l\]2retlt scopes, with llarrower scope correspondillg to a position iilore It) tile right.</Paragraph> <Paragraph position="2"> The tree of l:'ig. 4 can he glossed in the following way:</Paragraph> <Paragraph position="4"> ter hates I f we consider tile six l)mmulalions of lhe nodes under like. we can produce six differenl scopings. Be cause John teleis to an individual, not a quantified NP. these six pernmtations really corrcsl)ond to only the two interl)relaiiens given ahove. The tree of Fig. 4 corresponds to the lirst of Ihese interpretations, which is the preferred interpretation Ik)l sentence (S I).</Paragraph> <Paragraph position="5"> Our discussion of scope being represented by node order has been infornml so far. In order to nlake it \['Oi'lllal, we need to encode our representation into a binary-tree fornmt ell which a compositioiml senlan tics can he delined. To tie that. in a lirst step we rephtce the at'gunlent nunibers of l:ig. 4 hy exl)lici! argument haines; ill a seColld slep we encode the resulting or dered mary free inh)a himuy forumt which makes explicit the order in which dependents are incorlmrated inlo their head.</Paragraph> <Paragraph position="6"> S-I&quot;ORMS Consider tile mary tree of Fig. 4. For any node A in this tree, take the set of predication edges associated with A, that is the set of edges (a,+i,B,) and (Bi, i,A). By renaming each such node A into A(XI .... X,+). where X I ..... X, are hesh identiliers, and by renaming each such htbel +i (resp. +i) into +X,: (resp. -X,:), one obtains a flew tree where argunmnl numbers have been replaced by argument haines. \[:or instance the previous representation now becomes tile tree of l&quot;ig. 5.</Paragraph> <Paragraph position="7"> This representation is called a scopeU depemh'm3' ,lotto, or Sqbrm.</Paragraph> </Section> <Section position="8" start_page="256" end_page="257" type="metho"> <SectionTitle> BINARY TREE FNCODING OF S-FORMS: B- </SectionTitle> <Paragraph position="0"> FORMS In order to encode tile ordered n-ary tree into a binary tree, we need to apply recursively the transfotnmtiou ilhlstrated in Fig. 6, which consists in forming a &quot;head-line&quot;, projecting in a north-west direction from tile head 11, and in &quot;attaching&quot; to this line</Paragraph> <Paragraph position="2"> forms.</Paragraph> <Paragraph position="3"> Applying this encoding to our example, we obtain the binary tree of Fig. 7, which is called a B-form. The B-form makes explicit the order of incorporation of dependents into the head-line. By permuting several dependent-lines along their head-line, this incorporation order is changed and gives rise to different scopings.</Paragraph> <Paragraph position="4"> S-forms and B-forms are completely equivalent representations. Cle~ly, the encoding, called the Sform/B:fi~rm encoding, which has just been defined is reversible. The S-form is more compact ,and makes the dependency relations more conspicuous, whereas the B-form makes the compositionality more explicit.</Paragraph> </Section> <Section position="9" start_page="257" end_page="257" type="metho"> <SectionTitle> WELL-FORMEDNESS CONDITIONS ON B- FORMS AND S-FORMS Stm'ting fromthe U-form </SectionTitle> <Paragraph position="0"> and enriching it, we have informally introduced the notions of S-form and B-form. We now define them formally.</Paragraph> <Paragraph position="1"> We start by giving a rect, rsive definition of IBFs (incomplete B-forms), that is, B-forms which may contain unresolved flee variables. We use the notation ((D,Label),H) the labelled binary tree obtained by taking H as the right subtree, D as the left subtree, and by labelling the left edge with Label. We ,also use the notation fv(IBF) for the set of the free variables in IBF.</Paragraph> </Section> <Section position="10" start_page="257" end_page="257" type="metho"> <SectionTitle> DEFINITION OF INCOMPLETE B-FORMS </SectionTitle> <Paragraph position="0"> 1. A node N of the form Pmd(xl,..,xn) is an IBF with the set of free variables fv(N) = { x I ,..,xn }; 2. If D and H am IBFs, fv(D) and fv(H) ale disjoint, and x ~ fv(H) then H'=((D,+x),H) is an IBF with fv(ll') = fv(D) U fv(H) \ {x}; 3. If D and H are 1BFs, fv(D) and fv(H) ,are disjoint, and x C fv(D) then H'=((D,-x),H) is an IBF with fv(H') = fv(D) U fv(H) \ {x}; 4. If D and H are IBFs, and fv(D) and D(H) are disjoint, then H'=((D,det),H) is an 1BF with fv(H') = fv(D) U fv(H).</Paragraph> </Section> <Section position="11" start_page="257" end_page="257" type="metho"> <SectionTitle> DEFINITION OF B-FORMS A B-form is an IBF </SectionTitle> <Paragraph position="0"> with an empty set of free variables.</Paragraph> <Paragraph position="1"> The notion of S-form cart now be delined through the nse of the S-form/B-form encoding.</Paragraph> </Section> <Section position="12" start_page="257" end_page="258" type="metho"> <SectionTitle> DEFINITION OF S-FORMS A S-form is an of </SectionTitle> <Paragraph position="0"> demd labelled n-ary tree which can be obtained from a B-form through the inverse application of the S-form/Bform encoding.</Paragraph> <Paragraph position="1"> It can be easily verified that the representation of resentation of Fig. 5 is a valid S-form. More generally, it can be easily verified that enriching a U-form by ordering its nodes, and then replacing argument variables by argument names always results in a valid Sform) null tThe converse is not true: not all S-forms can be obtained in this way from a U-form. For instance, there exists a S-fonn corresponding to the prefelTed reading for &quot;Fido visited most trashcans on every street&quot;, which has &quot;every street&quot;</Paragraph> </Section> <Section position="13" start_page="258" end_page="258" type="metho"> <SectionTitle> 4 THE INTERPRETATION PROCESS </SectionTitle> <Paragraph position="0"> We now describe the interpretation process on B-fl)rms.</Paragraph> <Paragraph position="1"> lnlerpretation proceeds by propagating semantic translations and their types bottom-up.</Paragraph> <Paragraph position="2"> The first step consists in typing the leaves of the tree, while keeping track of the types of fl'ee variables, as in Fig. 8.</Paragraph> <Paragraph position="4"> peter: e haleChl,h2).' t their types are indicated in brackets. The types given to the leaves of the tree are the usual functional types formed starting with e (entities) and t (truth values). In the case where the leaf entity cot> tains flee variable arguments, the types of these free variables are indicated, and the type of the leaf takes into account the fact that these free variables have already been included in the functioned form of the leaf. Thus hate(h l,h2), which can be glossed as: &quot;hi hates h2&quot;, is given type t, while hl and h2 are constrained to be free variables of type e.</Paragraph> </Section> <Section position="14" start_page="258" end_page="259" type="metho"> <SectionTitle> VARIABLE-BINDING RULES According to the </SectionTitle> <Paragraph position="0"> well-formedness conditions tot B-forms, a complement incorporation ((D,+x),It) is only possible when H contains x among its fi:ee variables; the &quot;syntactic dependent&quot; D is seen as semantically &quot;filling&quot; the place that x occttpies in the +'syntactic head&quot; H. In the same way, a modifier incorporation ((D,-x),H) is only possible when D contains x among its fiee wuiables; outscoping &quot;most trashcans&quot;, and which is not obtained from a U-form in this simple way. However, thet+e exists a mapping fiom S-forms to U-h>rms, the scope-fi)rgetting mapping, which permits to deiine equiwtlence chtsses among Storms &quot;sharing&quot; the same U-form. This relation between S-R~rms and Uqbrms can be used to give a (not>deterministic) lbrmal interpretation to U-forms, by considering the interpretations of the various S-forms associated with it (see the technical report eolnpanion to this paper.) in this case the &quot;syntactic&quot; head I I is seen as semantically &quot;filling&quot; the place that x occupies in the &quot;syn-tactic dependent&quot; D. (This difference corresponds to the opposition which is sometimes made between syntactic and semantic heads and dependents: complements are dependents both syntactically and semantically, while modiliers are syntactically dependents but semantically heads.) In order to make formal sense of the informal notion &quot;filling the place of x in A.,,&quot; (where the notation A,: means that A contains the free variable x), we introduce the variable-binding rules of Fig. 9.</Paragraph> <Paragraph position="1"> cotnllh, ntenl tntldilict dot c t L\]IiI/CI i\[iC~)l i)Ol \[it{t,ll illCt )t pol \[t\[iOll { lit'o\[ pOl ~It iOll spond to the senmntic translation of the subtrees rooted in 1)and 11 respectively.</Paragraph> <Paragraph position="2"> These rules tell t,s how to &quot;get rid&quot; of the free vailable being bound during complement or tnodifier incorporation, namely by forming the abstraction ,Xx.A,: before actually performing the semantic composition between tile dependent and tile head. For completeness, detemainer incorporation, which does not inw)lve vmiable binding, is given along with complement and rnodifier incorporation.</Paragraph> <Paragraph position="3"> Two things should be noted about this way of &quot;delaying&quot; variable-binding until the relevant dependent is incorporated: * Suppose that we had bound the variables appearing in the head predicate locally, that is to say, that, in the style of Montague grammar (Gamut, 1991 ), we had written )d21 l.like(l1,12) instead of like(11,12), and so forth, in Fig. 7. Then each incorporation of a dependent into the &quot;head-line&quot; would have changed the type of the head; thus 'not' would have had to combine either with a head of type e--+e~t, or e--t, or t, depending on its scope relative to the other dependents; with the scheme adopted here, the type of the head renmins invariant along the head-line; * tinder the same hypothesis, the incorporation of the second mgnment first and of the first argt,ment second would have been much simpler than the reverse incorporation order, and some mechanism would have had to be found to distinguish the two orders. Then permuting the relative order of two dependents along the head-line -- corre* O ' &quot;&quot; spondm~ to dttferent scope possibililies-- wonld have had complex computational conseqttences+ In the scheme adopted here, these cases are handled in a tiniforna way.</Paragraph> <Paragraph position="4"> The way free wu'iables are used in our scheme is somewhat remi n i scent of the nse ol:.vvitlaC/'liC/' variables he,, in Montague glanlliiar. Montague gl+anlmar hits the general requirement that only closed lambda-tetms (lanibda terms containing only bound variables) are composed together. This requirement, however, is di fficult to reconcile with the flexibility needed for handling quantilier scope ambiguities. Syntactic variables are zt device which pertnit to &quot;'quantify Jim'>&quot; clauses at an arbitrary time, hypassing the normal functional compc~sition of lambda-terms, which requires a strict management of incorporation order. In our scheme.</Paragraph> <Paragraph position="5"> by contrast, this secondary mechanism of Montague o\]ammar is graduated to a central position. Composition is always done between two lambda-terms one of which at least contains a free variable which gets bound at the time of incorporation.</Paragraph> </Section> class="xml-element"></Paper>