Focus and Higher-Order Unification 
Claire Gardent 
Computational Linguistics 
Universitgt des Saarlandes, 
D 66041 Saarbr/icken 
claire~coli, uni- sb. de 
Michael Kohlhase 
Computer Science 
Universitgt des Saarlandes, 
D-66041 Saarbrficken 
kohlhase(~cs, uni-sb, de 
Abstract 2 Focus theory 
Pulman has shown that Higher-Order 
Unifcation (HOU) can be used to model 
the interpretation of focus. In this pa- 
per, we extend the unification based ap- 
proach to cases which are often seen as 
a test-bed for focus theory: utterances 
with multiple focus operators and second 
occurrence expressions. We then show 
that the resulting analysis favourably 
compares with two prominent theories of 
focus (namely, Rooth's Alternative Se- 
mantics and Krifka's Structured Meano 
ings theory) in that it correctly gener- 
ates interpretations which these alter- 
native theories cannot yield. Finally, 
we discuss the formal properties of the 
approach and argue that even though 
HOU need not terminate, for the class 
of unification-problems dealt with in this 
paper, HOU avoids this shortcoming and 
is in fact computationally tractable. 
1 Introduction 
In this paper, we argue that Higher Order Uni- 
fication (HOU) provides a linguistically adequate 
tool for modeling the semantics of focus. Building 
up on (Pulman, 1995), we develop a unification- 
based analysis of focus which we show favourably 
compares with two prominent theories of focus, 
Rooth's Alternative Semantics and Krifka's Struc- 
tured Meanings theory. For data which is gener- 
ally viewed as a test-bed for focus theory (utter- 
ances with multiple focus operators and second 
occurrence expressions), we show that contrary 
to Rooth's and Krifka's theories, the HOU treat- 
ment yields a transparent analysis while avoiding 
under- and over-generation. 
Focus is a much debated notion. In this paper, we 
assume a simplifed version of Jackendoff's defini- 
tion: a focus is the semantic value of a prosodi- 
tally prominent element. We take the identifica- 
tion of prosodically prominent elements as given. 
To set the stage for this paper, we will briefly 
review the folklore, i.e. the main issues of fo- 
cus theory. It is commonly agreed that focus 
triggers the formation of an additional seman- 
tic value which we will call the Focus Seman- 
tic Value (FSV). The name and definition of 
the FSV varies from author to author: Jackend- 
off (Jackendoff, 1972) calls it the presuppositional 
set, Rooth (Rooth, 1992) the Alternative Sct and 
Krifka (Krifka, 1992) the Ground. In this paper, 
we assume a definition of the FSV which is in 
essence Rooth's Alternative set, that is, tile set 
of semantic objects obtained by making an ap- 
propriate substitution in the focus position. For 
instance, the FSV of (la) is defined as (lb), the 
set of properties of the form like-lug y where y is 
an individual (in what follows, focus is indica~ted 
using upper-case; we also follow Montague's con- 
vention that for any type % D~ is the set of objects 
of type r and wff~ is the set of wits of type r). 
(1) a. Jon only likes MARY 
b. lyc 
It is also usuMly agreed that certain linguis- 
tic elements associate with focus in that the 
meaning of the utterance containing these ele- 
ments varies depending on the choice of focus. For 
instance in (2a-b), the focus operator only asso- 
ciates with focus so that the difference in focus be- 
tween (2a) and (2b) induces a difference in mean- 
ing between the two utterances: in a world where 
aon introduced Paul to Mary and Sarah, and no 
other introduction takes place, (2a) is necessarily 
430 
false whilst (2b) is true. 
(2) a. Jon only int,vduced Paul to MARY 
b. .Ion only intr'od,tced PAUL to Mary 
To model this "association with focus" phe- 
nomenon, the semantics of associating-elements 
(e.g. focus operators, qttantifieational adverbs) is 
made contingent on the FSV which itself, wtries 
with the choice of focus. The following example 
illustrates this. Suppose that the meaning of o'nlg 
is determined by the following rule: 
\[NP only VP\] 
vP\[P ~ FSV A P(NP') -+ P = VP'\] 
where NP', VP' represent the meaning of NP 
and VP respectively, and t, kS'V stands for the fo- 
cus semantic value of the VP. As we have seen 
above, the FSV of (la) is (lb), hence by the above 
semantic for only, the semantics of (1 a) is: 
VP\[P G {Ax.l(x, ~j) I Y ~ D~} A P(j) 
-~ \[' = ~z.l(x, m)\] 
Intuitively, the only property of the form like- 
ing y that holds of Jon is the property of like ing 
Mary. 
3 The basic analysis 
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,'") 
Assuming the typed A calculus as our seman- 
tic 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, "writlen (;d, is {(;d(t'...*'~) I 
t "i < wff~,}. 
As mentioned before, this yields a focus seman- 
tic value which is in essence i{ooth's Alternative 
Set 1 . 
IThongh in fact, our definition is more syntactic 
than Rooth. In Rooth's approach, the I"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"SV can he more accu- 
rately 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 con- 
tributing 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. 
qb illustrate the workings of onr approach, we 
now run through a simple example. Consider (la). 
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: 
(;d = AyAx.l(x, y) 
And by definition (3.1), the FSV is: 
c;g = y) ly e wp;} 
Assuming the semantic of only given above, tke 
semantic representation of (la) is then: 
VP\[P c {Ae~.l(x, y) \] v ~ wife} A P(j) 
P = 
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. 
4 Linguistic applications 
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 interpreta- 
tions 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. 
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. 
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 Oeeur- 
ren(:e Restriction (l)ah'ymple et al., 1991)'s: the 
occurrence directly associated with the focus is a pri- 
mary occurrence and any solution containing a pri- 
mary occurrence is discarded as linguistically invalid. 
For instance, *n is a primary occurrence in the equa- 
tion 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 discus- 
sion of how it can be extended to {bcus, see ((?,ardent 
and Kohlhase, 1996). 
431 
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. 
4.1 Two alternative theories of focus 
Rooth's Alternative Semanti(:s 
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 rep- 
resentation). Focus is then seen as introducing a 
free variable whose value is determined by the cur- 
rent context and is filrthermore constrained to be 
an element or a subset of the FSV. For our pur- 
pose, the following characteristics are particularly 
important: 
• Given Rooth's definition of the Alternative 
Set, a focus operator associates with any tb- 
cus occurring in its scope. 
• Any NP may be subject to Quantifier Rais- 
ing. Importantly, this includes focused NPs. 
• Quantifier Raising may not apply to quanti- 
tiers occurring in a scope -island. 
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. 
Kritl~t's Structured Meanings 
Krifl(a's approach defines a rule-to-.rule seman- 
tics which assigns to any syntactic constituent, a 
meaning which can be either a k term or a struc- 
tured meaning, i.e. a tuple oF the form {Gd,/") 
where Gd is Krilka's I,'ocus Semantic Value and 1," 
is a (possibly cornl)Iex) \[bcus. 
For our purpose, an iinportant characteristic o\[' 
Krifka.'s approach is the tight syntax/semantic in- 
teraction it presupposes. In particular, the theory 
requires that a focus operator combines with a 
syntactic constituent C whose, structured se.man- 
tics C' -- (Gd, F) provides the focus (1,') this op- 
erator associates with. In other words, the right - 
adjacent sibling of a \[b(:us operator must contain 
all and only the loci this operator associates with. 
As we shMl later see, some of the data does not 
seem to square with this assumption. 
4.2 Multiple Focus Operators 
Utterances with multiple locus operators 3 are 
known pathological cases of focus theory: 
(3) a. (Jon only~ read the letters 
that 5'arah sent to PAUL1) 
b. Jon also~ onlgt read the letters 
lhat 5UE.e sent to PAUL:,. 
In the given context, the preferred reading of 
(3b) can be glossed as follows: it is also the case 
Jbr 5'U~,), that Jon only, read the lette'r~s she sent 
to PA ULI i.e. ,\]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. 
The HOU analysis 
Under the ItOU approach, (3b) is analysed as 
lbllows. First, the' meaning of onlyl read the let- 
ters that SUl'Se sent to PA UL1 is derived. 'Fo de- 
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 
sent to PA UL1 
t,. (:~(t)) = ~'.,'--,t(x,l(.%p)) 
e. (:~ = %),~.,*a~l(~,~(s,v)) 
-+ = a..,.e(.l(x, 
Analysis then proceeds further and the ground 
equation 
(,'~(.~) - az.W)\[ f' < a,X,'.,'ead(x, l(,~, Y)) 
AP(~) --> ~' - a,.,'~(,t(:~, 1(.% p))\] 
must be solved to determine the meaning of also2 
only, read the lellers that SUE.e sent to PAUL,. 
A possible solution for G 2 is 
< l(,,,, :@) 
v = v))\] 
Assuming the following semantics ff)r 
Nf-' also VP 
1,',s'v/, t'(N1") A l' ¢ Vt"\] 
we obtain the desired reading 
~P\[ P 6 kukx. onlyl wad the letters that 
to 
AP(j) A P ~ kz.z only1 read the. letters 
that Sue.2 sent to Paul1\] 
3The subscripLs indicates which operators associate 
with which focus. '\['here m'e there for clarity only, and 
have no theoreti(:al imporl,. 
4 l'br clarity, we have simplified the semantic tepre- 
sentation of (3b); nothing hinges on this. 
432 
Comparison with 17,ooth and Kritlm 
As HicnlJonc'd in section 4.1, under /,tie Alter- 
native 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\[" <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 577t£, 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\[" 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 '~ . 
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 ~s- 
nociates 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. £'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\[)- 
lions about, synt, ax, Sll<:h (:onstoitu(,nts (Io llot, exist; 
so thai, t}lo desired ini;erprel, at, ion Ci%llliOt; be g~eii- 
crated. 
4.3 Se('.OIld ()('clirrei1co, Exi)ressiolls 
We <:all se<:ond o(;curretice exl)r<~ssions (SOE) /ll;- 
l;('.r?l, llC(~S which pa.riAally or c(>nll)l(~l, cly tel)ca.i; a 
\])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 
(5c). 
(5) a. A: 
B: 
1). A: 
B: 
(% A: 
B: 
,Ion o'~dy likes MA t~Y. 
No, I)t'\]Tf','I~, <mlg likes Maw. 
,\]on only likes MA I~Y. 
Huhu, Pctc'r mull lik..~ Matq. 
Jon o~U~j likes MA t~ Y. 
,~'o 'w/ml? h'vc~ 1~l'/TVql£ out?! 
likc,~ Marg. 
An imt>ori;~ml; prol)(;rt;y of S()l'\]s is l.hat the :l:<;- 
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-" 
plii;ude and dural;ion (ci\['. (llari;els, 1995)). ()n i.t,(; 
(){;\[11.,,1' ha, ud, all l;hree l;h<'ol'ies ol" \['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\[' l;hc whole, i:ollsLilJueili, only includes />A ULi. 
Since l'lo t'ocils opcral,ors o(\]ciil in Lifts coil:d,il, liciil,, it; 
R)llows thai; gliC}l colisLil;llenL doe,'; HOt, (!xi%i,. 
t,hc question arises of whether these t;heories can 
account rot SOl+,s. 
The HOU analysis 
()u r l>l:oposal is to analyse S()Es as involving a. 
(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,. 
'l'his is mod(;led as follows, l,<;t, £',b%m 
and 7'£'em be the seni;mtic repr<~sentatiou of 
l, hc source (i.e. anl,<;c<;dcni,) and t, arget, (i.<;. 
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\[" aJl ,q()E tuust, 
t'<+st>e<:l; l.he \['ollowhlg equal;ions: 
A+. (>; / 'l ,...> ,5' f'") --+ 5;fY<;+. 
A'~. (7't' i , . . . , "l' P" ) = "/',%'era 
hll;uil;ivcly, t.hese two equal;ions re.quire thai, l;ar- 
gel, and Sotlrc(~ (;\]&llSO shai'(; & COllllllOll sclua, nl, i(;s 
ATt, the senianl, ics of l, hc dca.c('cnl.c~(I anal)hot. 
(~iv<'.n 1.his proposal, i;he a.nalysis of (Sa) involves 
i,h rec e(lUal0ions: 
/l,~(5) = Vl'\[l' c= a:xA~.l(~, n) 
/x/'(5) + s,= ),.,.l(,,,, ,,0\] 
a,,(s,) -: Vl'\[l' ~ (;<l A t'0,) + s, - A~,.l(.,, ,,,)\] 
(,',l(s,') _ A.;./(.,, ,,,.) 
Sin<:(~ ncil, liel: (Td lior \]"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><" solv<;d /.hus yMdhlg {.he vahic ~n. fi>r l;hc \['ocus 
\[".) '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. 
433 
inherit its FSV fl'om its source clause (by unifica- 
tion). In (Gardent et al., 1996), we show in more 
detail how the analysis accounts for the interac- 
tion of focus with anaphora and definiteness in the 
case of a particular instantiation of SOEs, namely 
corrections. 
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. 
In the standard case (i.e. the case where the fo- 
cus is prosodically marked), this quantification do- 
main 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 quantifica- 
tion 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. 
But now, consider the following example: 
(6) a. don only likes MARY. 
b. * No, PETER only likes Sarah. 
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 fo- 
cus 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 seman- 
tic value Ax.l(x, rn). Hence although Rooth's ap- 
proach captures some cases of SOEs, it does not 
seem to provide an adequate characterisation of 
the phenomena at hand. 
The Structured Meanings proposal distingui- 
sires between proper- and quasi-SOEs. Proper- 
SOEs involve an exact repetition of some previ- 
ous linguistic material, and are analysed as in- 
volving an anaphor which is constrained by the 
restriction that it be a segmental copy of its an- 
tecedent. 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 se- 
mantic value of its antecedent only likes MARY 
in (5a). In contrast, quasi SOEs only involve 
semantic equivalence between repeating and re- 
peated 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. 
However this theory faces a number of method- 
ological and empirical difficulties. First, it is non 
compositional because tire meaning of the deac- 
cented 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 han- 
dles separately two classes of data (i.e. quasi- and 
proper SOEs) which naturally belong together. 
Indeed, the HOU approach can be shown to pro- 
vide a uniform treatment of quasi - and proper- 
SOEs (cf. (Gardent et al., 1996)). 
5 Formal properties of the HOU 
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 prob- 
lem is only semi-decidable so that tile unification 
algorithm need not terminate for unsolvable prob- 
lems. 
Fortunately, in our case we are not interested 
in general unification, but we can use the fact 
that our formulae belong to very restricted syn- 
tactic subclasses, for which much better results 
are known. In particular, the fact that free vari- 
ables only occur on the left hand side of our equa- 
tions reduces the problem of finding solutions to 
higher-order matching, of which decidability has 
been proven for the subclass of third-order for- 
mulae (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 feasi- 
ble see (Prehofer, 1994). 
6 Conclusion 
In this paper, we have argued that Higher-Order 
Unification provides an adequate tool for com- 
puting Focus Semantic Values. To this end, we 
have considered data which is viewed as a test-- 
bed for focus theory and shown that, whilst exist- 
ing theories either under-generate, over-generate 
or are methodologically unsatist%ctory, the ttOU 
approach yields a simple and transparent analysis. 
There appear to be two main reasons for this. 
434 
l,'irst, the HOU analysis makes minimal as- 
sumptions about the role syntax is called to play 
in determining the I"SV. lit is detined on a purely 
semantic level in the sense that unification oper- 
ates on semantic representations, and relies nei- 
ther on quantifier raising, nor on a rule-to-rule 
definition of the FSV. As we have seen, this type 
of approach is a plausible way to avoid under- 
generation. 
Second, the HOU approach permits an equa- 
tional analysis which can naturally be Nrther con- 
strained by additional equations. The interest of 
such an approach was illustrated in our treatment 
of SOEs which we characterise as involving two 
phenomena: the computation of an I"SV, and the 
resolution of a &'accented anaphor. Not only did 
we show that this analysis is methodologically and 
empirically sound, we also showed that it finds a 
natural realisation in the equational framework of 
IIOU: each linguistic phenomena is characterised 
by some equation(s) and the equations may mu- 
tually constrain each other. For instance, in the 
case of SOEs, we saw that the equations character- 
ising the deaccented anaphor help determine the 
unidentified FSV of the utterance containing the 
unmarke(I focus. 
Clearly, our approach extends to cases of a(t- 
verbial quantification. For lack of space we could 
not develop the theory here; let us just point 
out that yon Fintel's criticism (von Fintel, 1995) 
of semantic approaches to tbcus, also applies to 
Krifka's Structured Meanings analysis, but not 
to the ItOU approach presented here. Von Fin- 
tel points out that in certain cases of adverbial 
quantification, a focus operator associates with 
an unmarked tbcus and dots not associate with 
a marked tbcus occurring in its scope - as should 
be clear fl'om this article, this is unproblematic for 
our analysis. 
Of course, there art still many open issues. 
First, how does the proposed analysis interact 
with quantification? Second, how does it extend 
to a dynamic semantics (e.g. Discourse Represen- 
tation Theory)'? 
7' Acknowledgments 
The work reported in this paper was flmded 
by the l)eutsche l,~orschungsgetneinschaft (I)FG) 
in Sonderforschungsbereieh S1"B-378, Project C2 
(MSA). 
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