<?xml version="1.0" standalone="yes"?>
<Paper uid="C94-1083">
  <Title>CATEGORIAL GRAMMAR AND DISCOURSE REPRESENTATION THEORY</Title>
  <Section position="4" start_page="508" end_page="509" type="metho">
    <SectionTitle>
2. FROM LAMBEK PROOFS TO
SEMANTICAL I~I,~CI PES
</SectionTitle>
    <Paragraph position="0"> Proof theory teaches us that there is a close co,respondence between proofs and lanlt)da terms.</Paragraph>
    <Paragraph position="1"> The lambda term which corresponds to a given proof can be obtained with the help of the so-called Curry-ltoward correspondence. Van Bcnthem \[1986\] observed that the lambda term that we get in this way also gives us a COlTCSpOndence between L:unbek proofs on the one hand and the intended meanings of the resulting expressions on the other. In the present exposition of the Curry-Iloward-Van Benthem correspondence I sh;dl follow the set-up and also tile notational conventions of I lendriks 111993\]. For more Cxl)lanation, the reader is rcferred to this work, to Van Benthenl \[1986, 1988, 1991\[ and to Moortgat \[198811.</Paragraph>
    <Paragraph position="2"> The idea behind the correspondence is that we match each rule in the Lambek calcuhis with a corresponding senvintic rnle and that, for each p,'oof, we build an isomorphic tree of semantic sequenls, which we define as expressions &amp;quot;/&amp;quot;F- Y, where 7&amp;quot; is a seqt,ence of variables and y is a hunbda term with exactly the variables in 7&amp;quot; free. The semantic rules that are to match the rules of</Paragraph>
    <Paragraph position="4"> v,R, QFQ(3.v,.R(v,)iv)) , \[\L\] \[\R\] R, 0 F Zv. O(Zv'. R(v'Xv)) Q',&amp; Q F Q'(Zv.Q(Xv'. R(v')(v))) p&amp;quot;1- p&amp;quot; \[/L\] \[/L\] P'1- P' D,P,R,Q 1- D(P)(Xv.Q(Xv'. R(v')(v))) l/L\] D, P, R, D', P' 1- D(P)(Zv. D'(l&amp;quot;)(Zv'. R (v')(v))) fig. 2. Semantic tree for 'a man adores a woman' the Lambek calculus above are as follows. (Tile term y\[u := w(fl)\] is meant to denote the result of substituting w(fl) for u in 7.) ~,\[AX\] x1- x ~'Ffl v',,~v'1-r ,...., , ~, ' L/L\] U,w, 7, V'1- flu:= wq/)\] 7f'k~ u',.,v'k r &amp;quot;\L\] u', T',w, V' 1- r\[;,:= ,,,qj)\] t 7&amp;quot;,v1-a \[IRI )&amp;quot;1- Zv.</Paragraph>
    <Paragraph position="5"> v,T'1- a \[\R\] T'1- Zv.a  Note that axioms and the rules l/L\] and \[~L\] introduce new free variables. With respect to these some conditions hold. The first of these is that only variables that do not already occur elsewhere in the tree may be introduced. To state the second condition, we assume that some fixed function TYPE from categories to semantic types is given, such that TYPE(a / b) = &amp;quot;rYPg(b \ a) = (TYPE(b), TYPI~a)). The condition requires that the variablc x in an axiom x 1- x must be of TYPE(c) if x 1- x corresponds to c ~ c in the Lambek proof. Also, the variable w that is introduced in l/L\] (\[\L\]) must be of (TYPE(b), TYPE(a)), where a / b (b \ a) is the active category in the corresponding sequent. null With the help of these rules we can now build a tree of semantic sequents that is isomorphic to the Lambek proof in fig. I; it is shown in fig. 2. The semantic sequent at the root of this tree gives us a recipe to compute the meaning of 'a man adores a woman' once we are given the meanings of its constituting words. Let us suppose momentarily that the translation of the determiner 'a' is given as the term ZI&amp;quot;XP3x(P'(x) ^ P(x)) of type (et)((et)t) and that the remaining words are translated as the terms man, adores and woman of types el, e(et) and et respectively, then substituting ZP'ZP3x(P'(x) ^ P(x)) for D and for D' in the succedent and substituting man, adores and woman for P, R and 1&amp;quot; gives us a lambda term that readily ,'educes to the sentence 3x(man(x) ^</Paragraph>
    <Paragraph position="7"> The same recipe will assign a meaning to any sentence that consists of a determiner followed by a noun, a transitive verb, a determiner and a noun (in that order), provided that meanings for these words are given. For example, if we translate the word 'no' as ZP'XP~qx(l&amp;quot;(x) ^ P(x)) and 'every' as ZPgvPVx(P'(x) ---, P(x)), substitute tile first term for D, the second for D', and man, adores and woman for P, R and P' as before, we get a term that is equivalent to --3x(man(x) ^ Vy(woman(y) --, adores(y)(x))), the translation of 'no mall adores every womiul'.</Paragraph>
  </Section>
  <Section position="5" start_page="509" end_page="510" type="metho">
    <SectionTitle>
3. BOXES IN TYPE LOGIC
</SectionTitle>
    <Paragraph position="0"> In this section I will show that there is a natural way to emtflate tile DRT language ill tile first-ordcr part of type logic, provided that we adopt a few axioms. This possibility to view DP, T as being a fragnaent of ordinary type logic will enable us to define our interface between Catcgorial Grammar and DRT in the next section, We shall have four types of primitive objects in our logic: apart from the ordinary cabbages and kings sort of entities (type e) and the two truth values (type t) we shall also allow for what i woukl like to call pigeon-holes or registers (type n) and for states (type s). Pigeon-holes, which are the things that are denoted by discourse referents, may be thought of as small chunks of space that can contain exactly one object (whatever its size). States may be thought of as a list of the current inhabitants of all pigeot&gt;holcs. States arc very much like the program states that theoretical  computer scientists talk about, which are lists of the current values of all variables in a given program at some stage of its execution.</Paragraph>
    <Paragraph position="1"> In order to be able to impose the necessary structure on our m~xlels, we shall let V be some fixed non-logical constant of type ~(se) and denote the inhabitant of pigeon-hole u in state i with the type e term V(u)(0. We define i\[u I... unl \] to be short for Vv((li IrC/ V A... A It n * V) ~ V(v)(i) = V('e)(\]')), a term which expresses that states i and j differ at most in u I ..... un; i\[\]j will stand for tile formula</Paragraph>
    <Paragraph position="3"> for each two different diseonrse referents (constants of type ~) u and u' AX1 requires that for each state, each pigeon-hole and each object, there must be a second state that is just like the first one, except that the given object is an occupant of the given pigeon-hole. AX2 says that two states cannot be different if they agree in all pigeon-holes. AX3 makes sure that different discourse referents refer to different pigeon-holes, so that an update on one discourse referent will not result in a change in some other discourse referent's value.</Paragraph>
    <Paragraph position="4"> Type logic enriched with these three first-order non-logical axioms has the very useful property that it allows us to have a tk~rm of the 'unselective binding' that seems to be omnipresent in natural language (see Lewis \[1975\]). Since states correspond to lists of items, quantifying over states corresponds to quantifying over such lists. The following lemma gives a precise formulation of this phenomenon; it has an elementary proof.</Paragraph>
    <Paragraph position="5"> UNSELF, CI'IVE BINDING LEMMA. Let Ul ..... un be constants of type ~, let xl ..... x n be distinct variables of type e, let q~ be a formula that does not contain j and let qo'be the result of the simultaneous substitution of V(ttl)(j ) for Xl and ... and V(un)(j) for xn in ep, then: I=Ax Vi(3j(i\[to ..... un Y AtD) ~ ~Xl... ~Xnq~ ) I=Ax Vi(Vj(i\[u, ..... u,,\]j-,&amp;quot; q)) -,-,&amp;quot; Vxl... Vx,,q)) We now come to the enmlation of the DRT language in type logic. Let us fix some type s variable i and define (tO t = V(u)(i) for each discourse referent (constant of type J~) u and (/)i = t for each type e term t, and let us agree to write l'w for X/l'(@, &amp;quot;rlRT 2 for t~..i( R ( 171l '~ (&amp;quot;f2) \]&amp;quot; )i ~) is v 2 for )d((v/)&amp;quot;=(v2) ), if 1' is it term of type et, R is a term of type e(et) and the z's are either discourse referents or terms of type e. This gives us our basic conditions of the DRT language as terms of type st. In order to have complex conditions and boxes as well, we shall write not * for ,a.i-,3jO(O0), * or 'I t for M3j(O(i)(j) v ff*(O(J)), q' ~ lit for )dVj(O(i)(\]) --+ 3k~P(j)(k)), \[ul...u,, Ib ..... y,,,\[ for ZiZj(itu, ..... u,,lj A yIQ/) A...A 'gin(J)), O; q,r for MZf\]k(O(i)(k) ^ ql(k)(\])).</Paragraph>
    <Paragraph position="6"> Ilere * and qJ stand for any tc,'m of type s(st), which shall be the type we associate with boxes, :rod the y's stand for conditions, terms of type st. \[ttl.&amp;quot;It,, \]Yl ..... Ym\] will be our linear notation for standard DRT boxes and the last clause elnbodies an addition to the standard DRT language: in order to be able to give conlpositional translations tO natural lilngu'lge expressions and texts, we borrow the sequencing operator ';' from the usual imperative programming hmguages and stipulate that a sequence of boxes is again a box. The following useful lemma is easily seen to hold.</Paragraph>
    <Paragraph position="7"> MI-k~GING LEMMA. If /~' do not occur in any of ~,&amp;quot; then I=^x \[/i I g\[ :\]/r I g'\] = I/i if' I g ?\] Tim present emulation of DRT in type logic should be compared with tile semantics for DRT given in Groenendijk &amp; Stokhof \]199l\]. While Groenendijk &amp; Stokhof giw; a Tarski definition for DRT in terms of set theory and thus interpret the object DRT language in a metalanguage, the clauses given above are simply abbreviations on the object level of standard type logic. Apart from this difference, tile chmses given above and tile clauses given by Oroenendijk &amp; Stokhof are tnueh the same.</Paragraph>
  </Section>
  <Section position="6" start_page="510" end_page="512" type="metho">
    <SectionTitle>
4. FROM SEMANTIC RECIPES TO
BOXES
</SectionTitle>
    <Paragraph position="0"> Now that we have the DRT language as it part of type logic, connecting l~ambck proofs for sentenccs and texts with Discourse Representation  Structures is just plain sailing. All that needs to be done is to define a function TYPE of the kind described in section 3 and to specify a lexicon for some fragment of English. The general mechanism that assigns meanings to proofs will then take care of the rest. The category-to-type function TYPE is defined as follows. WYPE(txt)</Paragraph>
    <Paragraph position="2"> z(s(st)), while TYPE(a / b) = TYPE(b \ a) = (TYPE(b), TYPE(a)) in accordance with our previous requirement, It is handy to abbreviate a type of the form at(... (ctn(s(st))... ) as \[a,... a,,\], so that the type of a sentence now becomes \[1 (a box!), the type of a common noun \[or\] and so on.</Paragraph>
    <Paragraph position="3"> In Table 1 the lexicon for a limited fragment of English is given. The sentences in this fragment are indexed as in Barwise \[1987\]: possible antecedents with superscripts, anaphors with subscripts. The second column assigns one or two categories to each word in the first column, the third column lists the types that correspond to these categories according to the function TYPE and the last column gives each word a translation of this type. Here P is a variable of type \[or\], P and q are variables of type \[\], and v is a variable of type ~r.</Paragraph>
    <Paragraph position="4"> Let us see how this immediately provides us with a semantics. We have seen before that our Lambek analysis of (1) provides us with a semantic recipe that is reprinted as (2) below. If we substitute the translation of a 1, AP'ZP(\[u I 1\] ;</Paragraph>
    <Paragraph position="6"> substitute Av\[ \[ man v\] for P, we get a lambda term that after a few conversions reduces to (3).</Paragraph>
    <Paragraph position="7"> This can be reduced somewhat further, for now the merging lemma applies, and we get (4).</Paragraph>
    <Paragraph position="8"> Proceeding further in this way, we obtain (5), the desired translation of (1).</Paragraph>
    <Paragraph position="9">  (I) A I man adores a = woman (2) D,P,R,D&amp;quot; I&amp;quot; F D(l')(Av.D'(l&amp;quot;)(Zv'.le(v3(v))) (3) ZP(\[Ul\[ \] ; \[Iman it1\] ;D'(P')(Zv'.R(v)(ul)) ) (4) ZP(\[ul lman Ul\] ; D'(P')(Zv'.R(v3(u~))) (5) \[U 1 tt 2 I man u 1, woman u&gt; u I adores u2\] (6) Every ~ man adores a 2 woman (7) \[I \[,tl \[ ,,,a. ,1,1 \[a= I woma,, u&gt; u\] adores u2\] \] (8) D,P,R,D ;1&amp;quot;~- D'(l&amp;quot;)(Xv'.D(P)(Zv.R(v')(v))) (9) \[U2 \[ woman 112, \[ttl l man 11l\] :=C/&amp;quot; \[ I ul adores u2\]\] (10) A ~ man adores a 2 woman. Sh% abhors him1 (1 l) \[It I It 2 \[ matt It1, womalt tt 2, It I adores u 2, u 2 abhors 111\] (12) If a ~ man bores a 2 woman she= ignores him I (13) \[I \['11 u21 '''a&amp;quot; ul, woma, u&gt; u~ bores' u2\]  \[ tu2 ignores ul\]\] The same semantical recipe can be used to obtain a translation fo,&amp;quot; sentence (6). we find it in (7). But (1) and (6) have alternative derivations in the Lambek calculus too. Some of these lead to semantical recipes equivalent to (2). but others lead  to recipes that are equivalent to (8) (for lnore explanation consult Hendriks \[1993\]). If we apply this recipe to the translations of the words in (6), we obtain (9). the interpretation of the sentence in which a = woman has a wide scope specific reading and is available for anaphoric reference from positions later in the text.</Paragraph>
    <Paragraph position="10"> I leave it to the reader to verify that the little text in (10) translates as (11) by the same method (note that the stop separating tile first and second sentences is lexicaliscd as an item of category s \ (txt/ s)), and that (12) translates as (13). A reader who has worked himself through one or two of these examples will be happy to learn from Moortgat \[11988\] that there are relatively fast Prolog programs that automatically find all semantic recipes for a given sentence.</Paragraph>
  </Section>
  <Section position="7" start_page="512" end_page="573" type="metho">
    <SectionTitle>
5. FROM BOXES TO TRUTll
CONDITIONS
</SectionTitle>
    <Paragraph position="0"> Wc now have a way to provide the expressio,ls of our fragmcnt automatically with Discourse Representation Structures which denote relations between states, but of course we arc also interested in the truth conditions of a given text. These we equate with the domain of the relation that is denoted by its box translation (as is done in Groenendijk &amp; Stokhof \[11991\]).</Paragraph>
    <Paragraph position="1"> Theoretically, if we are in the possession of a box (/), we also have its truth conditions, since these are denoted by the first-oMer terln xiqj(q)(i)(j)).</Paragraph>
    <Paragraph position="2"> but in practice, reducing the last term to some manageable first-order term may be a less than trivial task. Therefore we define an algorithmic function that can do the job for us. The function given will in fact be a slight extension of a sinlilar function defined in Kamp &amp; Reyle \[1993\].</Paragraph>
    <Paragraph position="3"> First some technicalities. Define adr(~), the set of active discourse referents of a box ~1), by adr(\[ii I }'-\]) = {/i} and adr((D ; Ill) = adr(qO U adr(~lO. Let us define llt/u\]l, tile substitution of the type e term t for the discourse referent u in the construct of the tx)x hmguage F,, by letting it / ttlu = t and \[t/u\]u' = u' if a', u; for type e terlns t' We let \[t/U\]t'= t'. For complex constructs It/I\[1I' is defined as follows.</Paragraph>
    <Paragraph position="4"> I.t / u\]l'v : Pit~ u\]v \[t I ulvlRv 2 : llt I u\]vlR\[t I u\]v 2 \[t / u\](v 1 is ~c2) = \[t / Ill'IT 1 iS llt / u\]*'2 \[t l u\]not @ = not \[t l u\]C/</Paragraph>
    <Paragraph position="6"> if u ~ adr( q 0 The next definition gives our translation function 1&amp;quot; from boxes and coMitions to first-order formulae. The wuiable x that is appearing in the sixth and eighth chmscs is supposed to be fresh in both cases, i.e. it is defined to be the first variable in sonic fixed oMering that does not occur (at all) in q) or in tl*. Note that the sequencing operation ; is associative: q, ; (q/; ~) is equivalent with (q~ ; lit) ; E for all q), q/and ~. This lneans that wc may assume that all boxes are either of the form \[Ft \] \]\] ; (P or of tim form \[ii\] ?'\]. We shall use the form I F/: q) to cover both cases, thus allowing the possibility that q~ is clnply, if ~1~ is elnpty,</Paragraph>
    <Paragraph position="8"> (not q))f = -~(q))l (4) or list q)t' v Ipl ((\[. ii \[ ~; 1 ; 40 ~ q,)r Vx(lx / ul((\[Ft \] ~; \] ; @ ~ qO) t ((\[Irz ..... ~,,1 ; C/):~ qO* = (),j t. ^ ... ^ h .r)_,. ((b ::&gt; ~I/) t (I.//I~-\]; q,)-r = 3x(lx/.\](\[//l ~7\] ; q,))l (\[\[Ys ..... 7,,,1; q,)l = y/l A ... A ym r n rill  It is clear that the function ~ is algoritlunic: at each stage in tile reduction of a box or condition it is determined what step should be taken. The folk)wing tl~eorem, which has a surprisingly te~ dious proof, says that the function does what it is intended to do.</Paragraph>
    <Paragraph position="9">  TIIEOREM. For all conditions y and boxes 05:</Paragraph>
    <Paragraph position="11"/>
  </Section>
class="xml-element"></Paper>