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<?xml version="1.0" standalone="yes"?> <Paper uid="C96-1073"> <Title>Focus and Higher-Order Unification</Title> <Section position="3" start_page="430" end_page="430" type="metho"> <SectionTitle> 3 The basic analysis </SectionTitle> <Paragraph position="0"> For computing the Focus Semantic Value, we propose to use t\[igher Order Unification. More specitically, given (part of) an utterance U with semantic representation Sern and loci F1... b ''~, we require that the following equation, the ground equation, be solved: Sere = Gd(l,'t)... (t,'&quot;) Assuming the typed A calculus as our semantic representation language, this equation can be solved by Huet's algorithm (cf. (ltuet, 1975)), thus assigning a value to Gd. On the basis of this value, we can then define the FSV, written Gd, ms follows: Definition 3.1 (\['beus Semantic Value) Let Gd be of type c~ = ~ -+ t and ~ be the number of loci (~ < k), then the fibcus Hemantic Value derivable f,'om Gd, &quot;writlen (;d, is {(;d(t'...*'~) I t &quot;i < wff~,}.</Paragraph> <Paragraph position="1"> As mentioned before, this yields a focus semantic value which is in essence i{ooth's Alternative Set 1 .</Paragraph> <Paragraph position="2"> IThongh in fact, our definition is more syntactic than Rooth. In Rooth's approach, the I&quot;SV definition is purely semantic whereas in our approach the FSV is indirectly defined by solving equations and the value thus obtained (i.e. the value of Gd) is a term, that is, a syntactic object. Hence, our I&quot;SV can he more accurately contpared to Kratzer's presuppositior~ skeletort,. This means that our approach inherits the adwmtages of Kratzer's approach (c\['. (Kratzer, 1991)). In par-FinMly, we assume as in (Pulman, \[995), that loci ~ire stored and discharged non deterministically as the need arises, thus contributing to the definition of the ground equation. li'urthermore, equations are set up at the level at which there are needed e.g. at the VP level in the case of a pre-.verbal focus operator.</Paragraph> <Paragraph position="3"> qb illustrate the workings of onr approach, we now run through a simple example. Consider (la).</Paragraph> <Paragraph position="4"> To determine the meaning of only likes MARY, the FSV of the VP nmst be known, ttence the following equation lnust be solved: : Gd(,,O By tIOU, the value of (-Id is then2:</Paragraph> <Paragraph position="6"> Assuming the semantic of only given above, tke semantic representation of (la) is then:</Paragraph> <Paragraph position="8"> In short, we obtain a reading similar to that of l-tooth, the difference being in the way the FSV is determinecl: by ItOU in our approach, by means of a semantic definition in Rooth's.</Paragraph> </Section> <Section position="4" start_page="430" end_page="433" type="metho"> <SectionTitle> 4 Linguistic applications </SectionTitle> <Paragraph position="0"> In this section, we show that the ItOU approach f~wourably compares with I~.ooth's and Krifka's anMysis in that it correctly generates interpretations which these two theories fail to yield. As we shall see, the main reason for this is that the |\[OU approach makes minimal assumptions about the role syntax plays in determining the FSV. In particular, it relies neither on the use of Quantifier l{aising, nor on the assumption of a rule to rule definition of the FSV. In this way, it avoids some of the pitfalls these theories encounter.</Paragraph> <Paragraph position="1"> ti(:ular, it adequately captures the interaction of focus with VP ellipsis as illustrated by Kratzer's notorious ex~ttnI)le: I ordy wer~t to TANGLE'WOO1) because you did.</Paragraph> <Paragraph position="2"> 2 Unification yields another possible value of C'd, namely A yXx.l(x,m). In what follows, we assume a restriction similar to the DSP's Primary Oeeurren(:e Restriction (l)ah'ymple et al., 1991)'s: the occurrence directly associated with the focus is a primary occurrence and any solution containing a primary occurrence is discarded as linguistically invalid. For instance, *n is a primary occurrence in the equation Xx.l(x,,n) = Gd(m) so thai; the solution Gd = AUA.9:.I(x, 7n) is invalid. For a formal treatment of l)SP's Primary Occurrence Restriction and a discussion of how it can be extended to {bcus, see ((?,ardent and Kohlhase, 1996).</Paragraph> <Paragraph position="3"> We begin by a brief summary of l~.ooth's and Krifka's theories and stress the properties relevant for the present discussion. We then confront the three theories with the data.</Paragraph> <Section position="1" start_page="431" end_page="431" type="sub_section"> <SectionTitle> 4.1 Two alternative theories of focus Rooth's Alternative Semanti(:s </SectionTitle> <Paragraph position="0"> In l~,ooth's approach, the FSV is detined by re(:ursion on the truth conditional structure which is itself derived from LF (i.e. Logical Form, the Government and Binding level of semantic representation). Focus is then seen as introducing a free variable whose value is determined by the current context and is filrthermore constrained to be an element or a subset of the FSV. For our purpose, the following characteristics are particularly important: * Given Rooth's definition of the Alternative Set, a focus operator associates with any tbcus occurring in its scope.</Paragraph> <Paragraph position="1"> * Any NP may be subject to Quantifier Raising. Importantly, this includes focused NPs.</Paragraph> <Paragraph position="2"> * Quantifier Raising may not apply to quantitiers occurring in a scope -island.</Paragraph> <Paragraph position="3"> Note that Rooth's approach criticaJly relies on quantifier raising as a means of moving a focused NP out of the scope, of a focus operator. However this only applies if the focus NP is not eml:)edded in a scope island.</Paragraph> <Paragraph position="4"> Krifl(a's approach defines a rule-to-.rule semantics which assigns to any syntactic constituent, a meaning which can be either a k term or a structured meaning, i.e. a tuple oF the form {Gd,/&quot;) where Gd is Krilka's I,'ocus Semantic Value and 1,&quot; is a (possibly cornl)Iex) \[bcus.</Paragraph> <Paragraph position="5"> For our purpose, an iinportant characteristic o\[' Krifka.'s approach is the tight syntax/semantic interaction it presupposes. In particular, the theory requires that a focus operator combines with a syntactic constituent C whose, structured se.mantics C' -- (Gd, F) provides the focus (1,') this operator associates with. In other words, the right adjacent sibling of a \[b(:us operator must contain all and only the loci this operator associates with. As we shMl later see, some of the data does not seem to square with this assumption.</Paragraph> </Section> <Section position="2" start_page="431" end_page="432" type="sub_section"> <SectionTitle> 4.2 Multiple Focus Operators </SectionTitle> <Paragraph position="0"> Utterances with multiple locus operators 3 are known pathological cases of focus theory: (3) a. (Jon only~ read the letters that 5'arah sent to PAUL1) b. Jon also~ onlgt read the letters lhat 5UE.e sent to PAUL:,.</Paragraph> <Paragraph position="1"> In the given context, the preferred reading of (3b) can be glossed as follows: it is also the case Jbr 5'U~,), that Jon only, read the lette'r~s she sent to PA ULI i.e. ,\]on didn't read the letters shc.~ sent to c.g. Peter. In othc'r words, the preferred reading is that also.2 associates with b'Ul'2~ and onlyj with PAUL:I.</Paragraph> <Paragraph position="2"> The HOU analysis Under the ItOU approach, (3b) is analysed as lbllows. First, the' meaning of onlyl read the letters that SUl'Se sent to PA UL1 is derived. 'Fo de- null termine the FSV of the VP, the ground equation (4b) must be solved for which (de) is a solution. Applying the semantics of only given in section 2, the se,r~antics of (4a) is then as give,, in (4d) 4. (4) a. only, wad the letters that 5'U1'2.2</Paragraph> <Paragraph position="4"> Analysis then proceeds further and the ground</Paragraph> <Paragraph position="6"> must be solved to determine the meaning of also2 only, read the lellers that SUE.e sent to PAUL,.</Paragraph> <Paragraph position="7"> A possible solution for G 2 is with which focus. '\['here m'e there for clarity only, and have no theoreti(:al imporl,.</Paragraph> <Paragraph position="8"> 4 l'br clarity, we have simplified the semantic tepresentation of (3b); nothing hinges on this. Comparison with 17,ooth and Kritlm As HicnlJonc'd in section 4.1, under /,tie Alternative Semanl;ies al>l)roach, a \[\>cus o\[)cral;or Iie(:essariiy associal,es with any f'ocus OCCllrrilig; in its scope. Igu'thermore in (310, t, hc SCOl)e o\[&quot; <mlgl is the whole Vl ) read Uw letters /,hal, 51/15'~ .sesd, I,o lb'l ULI. \[l<;li(:( 1, if no qua nl, ilier raising; o(:(:urs, oltlyl associates with bot, h 577tPS, an(I I)AIJLI. '\['\[it/S ill or(let to g(;nerat;<; l;h<; desired reading, oq'U/( 2 Hltlsl; 13(; lllovod out, of {lie scoI)o, o\[&quot; ol/ly i . l\[owevcr, since the NI > ihc letters l, hat ,5'UI';:, sold, to PAULt is a scope island, quanl;ilier raising in in'/I)ossible, l lence, the desir('d l:eading (:a, illiOt 1)<; ge.n<;rat, ed '~ .</Paragraph> <Paragraph position="9"> Itc('Ml t;\[iat; in l,he ,ql;l:u<:t,urc(l Me;u\]ings al) f)roacli, the righl, sibling of a fo<:us Ol)(Wa.Loi: liitlSl, <onl;ain all and only th(; \['ocus i,his opera t, or ~snociates with (of. se<:l,ion 4.1). llence, t,o ~eilor ate i;hc d<'sir<'d roadmg in (3t)), l;h<'re n-iusl; exist a synf,~u:t, ic <:onsf, il, ucrll; whi<:h is righl, adja<:ent, \[;o onlyl and whicii conl;ains l>A 17i, l but not. PS'UI'2:>/;; similarly, l;h('re ill/iS{ exist a synD.~ctic ('onst.il;uenl; which is right, ad.jaccni; t;o also and w\[ti(:h (!OliLa.il\]s 5'Uh'.~ but nol; ILill/Li. (Jivcn sl;an(\[ard it,SSillil\[)lions about, synt, ax, Sll<:h (:onstoitu(,nts (Io llot, exist; so thai, t}lo desired ini;erprel, at, ion Ci%llliOt; be g~eiicrated. null</Paragraph> </Section> <Section position="3" start_page="432" end_page="433" type="sub_section"> <SectionTitle> 4.3 Se('.OIld ()('clirrei1co, Exi)ressiolls </SectionTitle> <Paragraph position="0"> We <:all se<:ond o(;curretice exl)r<~ssions (SOE) /ll;l;('.r?l, llC(~S which pa.riAally or c(>nll)l(~l, cly tel)ca.i; a \])reviollS Ill, l,(;r~tllCe. Typical c as<;s of S()1% arc: ~(:,,.,.<;<:tioils (r,~), ~<:ho s<.,,~.<~,l<:<,s (aid ,~,l~u w,.i~,,~.s l>eai;cd tria.l;cria, I is d(:a(:c(;nl;(;d, thai, is, it, is char a,cl;<~risec/ by a,ll illll)orl, a, lll; r(;(lll(:t;R)n ill pit, oh, ;'till-&quot; plii;ude and dural;ion (ci\['. (llari;els, 1995)). ()n i.t,(; (){;\[11.,,1' ha, ud, all l;hree l;h<'ol'ies ol&quot; \['OCliS consi(l<~i:ed hero arc basc'<l Oil l,hc &ssiiiHi)l.ioli l;haJ, focus is t>rosodically umrk(:d &lid thus, id<'nt, iIial>le. I lcn<:<h '~This l)oint, is in(l(!pendenl.iy iiot.c.<l in (l{ool, h, 1,0:)',*). ~'l'his is a ,dmplifical;ion: t;hc colist;il.ucni, linty in \['ad, cont;~-'~in ,S'U\['\]2 bill; this \['OC/lSed N I ) siiould ,:tlrea,(ly have bccn bolilld }>y so\[no focus ot)cr.~tLor s(', l;}l;'d, i;hc t'()CliS o\[' l;hc whole, i:ollsLilJueili, only includes />A ULi.</Paragraph> <Paragraph position="1"> Since l'lo t'ocils opcral,ors o(\]ciil in Lifts coil:d,il, liciil,, it; R)llows thai; gliC}l colisLil;llenL doe,'; HOt, (!xi%i,.</Paragraph> <Paragraph position="2"> t,hc question arises of whether these t;heories can account rot SOl+,s.</Paragraph> <Paragraph position="3"> The HOU analysis ()u r l>l:oposal is to analyse S()Es as involving a.</Paragraph> <Paragraph position="4"> (lca.c<:enl,ed anat)hor whMl consists of the r0p(;ated ,uai;erial, and is subject: to the <:oridil;ion t, hal, its senialitic rcl>resent;ai;ion must unify wit, h t;h<~ s<;mantle reprcsent;a.l, ion of it, s ant;ecedenl,.</Paragraph> <Paragraph position="5"> 'l'his is mod(;led as follows, l,<;t, PS',b%m and 7'PS'em be the seni;mtic repr<~sentatiou of l, hc source (i.e. anl,<;c<;dcni,) and t, arget, (i.<;.</Paragraph> <Paragraph position="6"> ana l)\[t()ric ) claus(; rcslmct, ively , and ~l'l)l...Tl )'~, ,b'l 't . ..,b'l )'~ be the l, arg<;1, and source para.lhfl c'lenieni;s', l.\[len t,\[ic inl,erFirct;a,l.ion o\[&quot; aJl ,q()E tuust, t'<+st>e<:l; l.he \['ollowhlg equal;ions: A+. (>; / 'l ,...> ,5' f'&quot;) --+ 5;fY<;+.</Paragraph> <Paragraph position="7"> A'~. (7't' i , . . . , &quot;l' P&quot; ) = &quot;/',%'era hll;uil;ivcly, t.hese two equal;ions re.quire thai, l;argel, and Sotlrc(~ (;\]&llSO shai'(; & COllllllOll sclua, nl, i(;s ATt, the senianl, ics of l, hc dca.c('cnl.c~(I anal)hot.</Paragraph> <Paragraph position="8"> (~iv<'.n 1.his proposal, i;he a.nalysis of (Sa) involves</Paragraph> <Paragraph position="10"> Sin<:(~ ncil, liel: (Td lior \]&quot;o(:tt8 ~-tre, initially givon, the third e<llla, l, ioli ;tl)ovc is uni,yl)C(t a.ll(I (:~tliiiol; I)e solved t)y I luel,'s algorit;hnP. \[n I, ha.t, sit, ual, ion, we <:au <fit, her ilsslilliO. SOIIIC delaying inechanisnl or some exl.erision o\[ I luet,'n algorithm t;hag can cope with I>yi)('. va.riabl<;s (el'. (I)oughcrl:y, t9!)3; I lusl;adl>, 1.9,9 I)). l{.enohiliion of the tirsi; <?(tuat;ion yMds i>hc followiug sohltion: A,, -- &,jW'\[l' ~_ {X:,.l(,,, :,DI9 ~ ,,,Z/;\] A l'(~). + s' :_: .x.,.l(.:, ,,,)\] I~>y at)l)lying An 1;o p, l.ll<; le\['l~ hand side of/.he second C(lUa.l;ion is l,hen (h;l,<'iqriinc(\[ so l;hat I.hc se<:on(t equal, ion t)<'x:o\[nes Vl,\[1, ~ ~:,p,.,./(.;, :,D A l,(;,) .> t':: ..',.;./(.,, ,,0\] :: Vl'\[l' ~ (ida S'O,)--> ;' :-: .\..,.l~k,,(,,. ,,,)\] an(I tim value of (/d is i(Icni;ificd as I)eiug (;d :- X~SXa:./(z, :q) (Nol,e fui:l;her, l;llrd; t;he l;hird equal;ion (:a.\[i llOW I><&quot; solv<;d /.hus yMdhlg {.he vahic ~n. fi>r l;hc \['ocus \[&quot;.) 'l'hal; i,q, l;lle llOI; appro;mh 1;o S()I,;.<; ali<>ws tin I,o <:orre(:i, ly (:al)t;illJe t_hilJ; \['a(:l, l,\]l;-tt H.II ~()\[!; (Hill ~As in (l)ah'ynii>lc (,i., a\]., I991), wc i,ake l,hc hlcni, i\[ica.{i(;il O\[ l)arallc\[ (':l(!lncnl;s as given R)r t,he i\[iolit~.:\[ll,. '~t'\]v(!lt I,h()llgh Lhis is liOl; cxpli<:il.ly si,al:ed, I)ul iii,n.l/',<'; mialysis (\]>llllIl~lll, 19,95, I)&ge (~) \['&CCS ,:'l, ,';iRlil;ir l)rol)l<!iu.</Paragraph> <Paragraph position="11"> inherit its FSV fl'om its source clause (by unification). In (Gardent et al., 1996), we show in more detail how the analysis accounts for the interaction of focus with anaphora and definiteness in the case of a particular instantiation of SOEs, namely corrections.</Paragraph> <Paragraph position="12"> Comparison with Rooth and Krifka Under the Alternative Semantics approach, SOEs are captured as follows. It is assumed that the quantification domain of focus operators is a variable whose value is contextually determined.</Paragraph> <Paragraph position="13"> In the standard case (i.e. the case where the focus is prosodically marked), this quantification domain of focus operators is usually identified with tire FSV of the VP. However, in the SOE cases, the assumption is that the quantification domain of focus operators is identified with the FSV of the source clause. Thus in (5a), the quantification of only in the second clause is identified with the FSV of the preceding utterance i.e. the set of properties of the ~brm like ing somebody.</Paragraph> <Paragraph position="14"> But now, consider the following example: (6) a. don only likes MARY.</Paragraph> <Paragraph position="15"> b. * No, PETER only likes Sarah.</Paragraph> <Paragraph position="16"> Clearly, this dialog is ill-formed in that (6b) is no appropriate correction for (6a). However, under the Alternative Semantics approach, it will not be ruled out since the FSV of (6a) provides an appropriate quantification domain for the focus operator in (6b): as required by the semantic of pre verbal only, it is a set of properties whose elements can be identified with the VP semantic value Ax.l(x, rn). Hence although Rooth's approach captures some cases of SOEs, it does not seem to provide an adequate characterisation of the phenomena at hand.</Paragraph> <Paragraph position="17"> The Structured Meanings proposal distinguisires between proper- and quasi-SOEs. Proper-SOEs involve an exact repetition of some previous linguistic material, and are analysed as involving an anaphor which is constrained by the restriction that it be a segmental copy of its antecedent. For instance, the semantics of only likes Mary in (5b) is not determined by the semantics of its parts but is instead identified with the semantic value of its antecedent only likes MARY in (5a). In contrast, quasi SOEs only involve semantic equivalence between repeating and repeated material (for instance, in a quasi-SOE a repeated element may be pronominalised). Krifka claims that quasi-SOEs have prosodically marked loci and thus do not raise any specific difficulty.</Paragraph> <Paragraph position="18"> However this theory faces a number of methodological and empirical difficulties. First, it is non compositional because tire meaning of the deaccented material in proper-SOEs is solely defined by the meaning of its antecedent (rather than the meaning of its parts). Second, the prosodic data is rather unclear: the assumption that quasi-SOE contains a prosodically marked focus is a moot point (cf. (Bartels, 1995)) and if it proves to be false, the analysis fails to account for quasi-SOEs. Third, it is counterintuitive in that it handles separately two classes of data (i.e. quasi- and proper SOEs) which naturally belong together.</Paragraph> <Paragraph position="19"> Indeed, the HOU approach can be shown to provide a uniform treatment of quasi - and proper-SOEs (cf. (Gardent et al., 1996)).</Paragraph> </Section> </Section> <Section position="5" start_page="433" end_page="433" type="metho"> <SectionTitle> 5 Formal properties of the HOU </SectionTitle> <Paragraph position="0"> approach The unification problem can be stated as follows: Given two terms of a logic M and N, is there a substitution, or, of terms for variables that will makethe two terms identical (i.e. ~r(M) = (r(N))? It is well-known that for Higher-Order Logic (e.g. the typed A calculus) the space of solutions can be infinite and furthermore, the HOU problem is only semi-decidable so that tile unification algorithm need not terminate for unsolvable problems. null Fortunately, in our case we are not interested in general unification, but we can use the fact that our formulae belong to very restricted syntactic subclasses, for which much better results are known. In particular, the fact that free variables only occur on the left hand side of our equations reduces the problem of finding solutions to higher-order matching, of which decidability has been proven for the subclass of third-order formulae (Dowek, 1992) and is conjectured for tile general case. This class, (intuitively allowing only nesting flmctions as arguments up to depth two) covers all of our examples in this paper. For a discussion of other subclasses of formulae, where higher-order unification is computationally feasible see (Prehofer, 1994).</Paragraph> </Section> class="xml-element"></Paper>