Centering in Dynamic Semantics 
Daniel Hardt 
Department of Computing Sciences 
Villanova University 
Villanova, PA 19085 
hardt@vill, edu 
Abstract 
Centering theory posits a discourse 
center, a distinguished discourse en- 
tity that is the topic of a discourse. 
A simplified version of this theory 
is developed in a Dynamic Seman- 
tics framework. In dm resulting sys- 
tem, the mechanism of center sh/ft 
allows a simple, elegant analysis of 
a variety of phenomena involving 
sloppy identity in ellipsis at~d "pay- 
check pronom~s". 
1 Introduction 
Centering (Grosz et al., 1995) and Dynamic 
Semantics* both concern the sequential process- 
ing of discourses, with particular emphasis on the 
resolution of pronouns. In Dynamic Semantics, 
the semantic structure of a discourse gives rise to 
constraints on the resolution of anaphoric expres- 
sions. Centering theory claims that a discourse 
always has a single topic, or center. Constraints 
on the resolution of anaphoric expressions arise, 
in part, from the ways in which the center can 
change in a discourse. There is an important dif- 
ference in the way discourses are viewed in Cen- 
tering and in Dynamic Semantics. In Dynamic 
Semantics, a discourse is viewed as a monotonic 
increase in information, as discourse referents are 
constantly added to the domain of discourse. Cen- 
tering draws attention to a particular role that a 
discourse entity can hold; fl:om time to time, t, he 
current center will be shifted wit.h a new center. 
In this paper, I will implement a simplified version 
of the centering theory in a dynamic system, and 
of phenomena involving sloppy identity in ellipsis 
and "paycheck pronouns". 
Since Montague, a major goal of semantics has 
been to describe a compositional method for con- 
verting a syntactic representation of a sentence 
into a logical representation of the sentence mean- 
ing, and dmn to evaluate that representation with 
respect to a given context. A primary insight of 
dynamic semantics is that sentences have a sys- 
tematic relation to context in two ways: not only 
are they evaluated with respect to the current con- 
text, but they also systematically change that con- 
text. This insight has particular relevance \['or the 
apparent puzzle presented by sloppy identity and 
related phenomena. While anaphoric expressions 
are normally thought to be identical in meaning to 
dmir antecedents, they receive a different interpre- 
tation than their antecedents in these cases. Given 
the dynamic perspective, the puzzle evaporates: 
the anaphoric expression and its antecedent might 
represent exactly the same meaning, since mean- 
inn is fundamentally a potential to be evaluated 
with respect to some context. What changes is 
tile context, in the discourse intervening between 
antecedent and anaphoric expression. 
Consider the following example involving 
sloppy identity in VP ellipsis: 
(1) Tom1 loves his1 cat. John1 does too. 
\[loves hisl cat\] 
The sloppy reading results from a change in 
context, in which the value of 1 becomes John 
rather than Tom. This allows an extremely simple 
account of the "recovery mechanism" involved in 
sloppy identity; the elided VP is exactly identical 
to its antecedent. Several authors (Garden% 1991; 
Hardt, 1.994) have suggested a dynamic account 
along these lines, arguing that sloppy identity and 
related phenomena reflect the reassignment of an 
index in tile discourse context. 2 
Alternative approaches postulate complex re- 
covery mechanisms for sloppy identity, such as 
higher-order matchiug (Dalrymple et al., 1991) 
or the syntactic matching of parallel dependen- 
cies (Fiengo and May, 1994). Below, I will argue 
that tile dynamic account is more general and em- 
pirically adequate, as well as being simpler than 
alternative accotmts. 
The clynamic account raises the following prob- 
lem: since the index of the tile initial "controller" 
is reassioned, it becomes inaccessible in subse- 
519 
served tor the discourse center, and the discourse 
center will always occupy another index as well as 
0. We. will us(; the * to designate references to the 
discourse ce.nter. Thus tim above examt)le will be 
notated as follows: 
(2) '.l.'omj, loves his, (:at. John2, does too. 
\[loves his. cat\] 
In tile first senteIlce, To'm, is the value of ill- 
dex 1, and ix also the discourse center, i.e., the 
value of index 0. The pronoun his* is equivalent 
to his0, and dins refers to tile discourse center. 
In tile secon(1 sentence, John becomes the value 
of index 2, and also replaces 5Ibm as the discourse 
center and thus John becomes the value of index 0. 
This center shift gives rise to the sloppy reading. 
llowever, both 'Ibm and John remain a('eessible in 
subsequent discourse. 
The paper ix organized as follows: In Section 
Two, i present a dynamic fl'amework based on 
the system described in (Muskens, 1996), with 
extensions for the discourse center, VP ellipsis, 
and t)ayt:heck t)ronouns. Section Three (:oneerns 
an "expanded paradigm" for sloppy identity; it; 
is shown that the t)roposed approach uniformly 
accounts for a broad range of sloppy identity phe- 
nomena, including some not previously examined 
in the literature. Conclusions and plans for future. 
work are given in Section l~bur. 
2 A Dynamic Framework 
The basic dynamic framework is the dynamic logic 
system of (Muskens, 1996). This framework has, 
for the sake of simplicity, restricted l;he study 
of anaphora to pronouns that are extensionally 
identified with their antecedents :~. I will extend 
Musk(ms' system to permit anaphora involving 
VP's as well as NP's, and to allow antecedents 
to be dynamic ms well as ordinary (extensional) 
objects. 
In Muskens' system, linearized I)RT boxes are 
integrated with the type logic (Church, 1940) that 
underlies Montague Semantics. Linearize(t DI{T 
boxes are simply a more concise way of writing 
standard DIt3 ~ boxes (Kamp, 1980). Muskens 
shows that DR~I' boxes can t)e viewed as abbrevia- 
tions for expressions in ordinary type logic. Con- 
sider the following discourse.: tile; discourse: Aj 
farmer walks. H el laughed. 
This is represented by the following linearized 
I)RT box: 
\[u:t \[ farrner(ul ), walk(u1 ),taugh(u~ )\] 
3There are several researchers who have extended 
dynamic frameworks to account tbr ellipsis and re- 
lated phenomena: (Klein, 1984) is an early examt)le. 
lAsher, 1993) examines a variety of extensions to the 
I)R~.\[ ~ framework. (van Eijek and Francez, 1993) ex~ 
plorc similar issues of indexing and ellipsis in a dy~ 
namic setting. (Gardent, 1.99t) also extends a dy~ 
namie semantics system for ellipsis and anaphora. 
This is an abbrevial;ion for dm following type 
logic formula: 
Aij (i\[ut \]j Afarmer (u j) Awalks(u:, j) Alaughs(u, j)) 
In the above formula, the variables i and j ret)- 
resent inpul: and output st;ates, and tim variabh~ 
u, (akin to a discourse marker) is a flln(:tion froln 
states to individuals, lit what, folk)ws, we use 
the DltPF abbreviations without further comment,. 
The reader is referred to (Muskens, 11996) for fur-- 
ther examl)les and the details of l;tl(', system. 
We now define a siinple fragment of English, 
base(1 on the one given in (Muskens, 1996). 
-> P, 
.lotu,,,, :W(\[u,,, = 
hc,  where 
it ~ At,q \[I I'~q\] 
an(l -> ; 
walk -. Sv \[I walk(v)\] 
 :at -* Xv \[I cat(v)\] 
l(,v,, Aq 
Not:e that the t;ransladon for h,e,, refers to 
dr(ant(lte,~)). Tiffs is detined as the discourse re4> 
resentation of the antecedent of he~(see (Muskens, 
71.996, page 20)). The l;ranslation for and is the 
sequencing operator, ;. As described in (Musk(ms, 
1996), the sequencing of two boxes K,K' is an ab- 
breviation for the following type logic expression: 
\[K~; g2\]l -~ 
{<i,j> I ~k (<i,k> ~ \[t~'t~ & <k,j> e \[K2~)} 
Typically, two DI/~.\[' boxes appearing in se- 
quence can be 'm, evged inLo a single box, consisting 
of the union of the discourse markers ill the two 
boxes and the union of the conditions. This is de- 
scribed in the Mcwi'n9 Lcmma of (Muskens, 1996, 
page 8). In the representations that follow, we 
will often merge boxes without (;Oltli\[lellt to silll- 
plify representations. Ilowever, the merge of two 
boxes is not always possible if there is a reas- 
sigmnent of an index, i(; will not be possible to 
perform the merge. This will arise in the cases of 
sloppy identity exalnined below. 
The above t\]'aginent, following the Kamp/lteim 
accounts, considers only one type of anaphora, in- 
volving individuals. We will extend the fragment 
in the following ways: 
• we will add the idea of a di.scour.se center to 
the system 
• we will allow dynamic properties to be added 
to contexts, as antecedents for VP ellipsis 
• we will allow dynamic individuals to be 
added to contexts, to accoullt for "paycheck 
prollOllllS ~ 
520 
2.1 Discom'se (~('nt e,r 
W(', de, film position 0 in Om context as t;h(; I)is- 
co'u, rse (,'cntc.r'. AI; any {~iv(m l)oi\]~l; in the dis- 
course, tim (liscours(~ (;Ifl;il,y d(}sigmtt;('.(1 as Lhc dis- 
('ore:s(; ('eJlt(;r o(:(:ut)ics posil,ion 0 as well as il;s 
ot;he.r l)osil;iou. We (lesign;tl:c 1;his with a, *, as 
iu th(; following (~Xaml)l(;: 
(3) A,* farmer walks, llc* la.ughcd. 
This is r(!l)r('.se.nl;(xl ;t,q follows: 
\[,u,,,,, I,~0 : ,,,, fa,',n,;,(.,), ~alk(,,,),la,,~h@:,)\] 
In this (lis(:ours( h (:Ira enl;ity iill;rodu(:e.d /)y A I * 
farrn(:r is th(' discourse, (:(mi;('J, and thus o(:cut)i(~s 
position 0 a.s well as position \]. 
We must a(hl a,(hlitiomfl \]'uh's for indefinite (',x-- 
t)r(',ssions and n;un(',s, when t;hey add an ()t)j(~(:t l,o 
c(mtx;xt; t,hat, is (,tie dis(:oms('~ (:(~nter. 
a I', l'~@,,,,,,,,I ,,o :: ..1; l,, (,,.,);p..(..)) 
,John.n* :5 
~l.(\[u,,,u,. lu0 :~ u.,,,~,, ~--.loh@;l'(H.)) 
W(', will apply a v(',ry siinplili(',d version ot' (:en- 
tering ttmory, consisting o\[ l;h(,, following con- 
sLra.ints: 
• Every discourse Hl, t;(!l'~tIl(:e ((;xceI)l; t;he dis- 
cours(~' initial utl,(;ranc(;) musl; h~ve a (:(.'nt('.r. 
• If any t)rollOlllIS occ\]lr ill an lt|;l;('.l'~lJl(;¢;, al; 
least one. t)I'OllO1111 liltlS|; r(~f(~l' to |;h(?. (:(;ll{;(w. 
We define, two types ()f transitions fi'om ()he ul> 
l;erancc i;o I;he n(;xI;: 
I. (\](;'ll, te, l" (\]oltl, i'lt?tatio'lt: tit(', (:('.Iil;(~l; w.nmius l;\]le 
S3InO, 
2. (/c'nter Shift: tim (:(,alter (:hang('.s 
Tim ~wl:mfl (:ent('.ring theory involves ml ad(li- 
tiona.l data struci;urc, the forward-looking centers, 
and define.s fimr transition types, with a 1)re\[e.r- 
en(:e or(h;ring among tJmm. The reader is r('J'e, rred 
to (Gt'osz et al., 1995) for a fltll at:(:ount; of this. 
For our purposcs~ wc will ro.ly on t;hc mcdumisnl 
of center ,shift to iull)l('.m(ml; the \]'(;assignm(ml; i,ha|; 
we. argue, is (:ru(:ial to l;ho. dynamic a(:(:t)tlltI; o\[ 
slot)l)y id(mtity. 
2.2 VP Ellipsis 
Ne.xt, we extend tim system for VP (~llil)sis: tirsl;, 
verbs at'(; sep;uated into a base form and a,n infl(~(:-. 
lion (INFI 0. This fa(:ilit;~)J;es |;hc |;reaJ;m(ml; of VP 
ellipsis; the \[NFI, (:ategory adds the. new prop(~rl:y 
l;o (;he (:Olfl;(;xl;~ just as the (lcgcrmin(~\]' "a" a(l(ls 
~ new individual to the (:ontexl;. An ;dt(',\]nal;ive 
meaning for t\]te INFL (:ateg(>ry is giv(;n for V\])t;; 
OCCllrI'(}IIC(;S~ where a 1)rof)crl;y is acc(~sse, d from Lhe 
inlmt (:onl;(',xt. 
INFLu => A I' Ax \[1>,, I 1',~ = P\] ; P(x) 
INFI,,z :& dr(ant(INH~,,)) 
'l;h(; INFI, (:a,t('gory ra.ug('~s over verba,l inih~(: 
lions (PAST, PI~,I';S, ('¢(:.) and ;mxiliary v(,rbs ((lo, 
should, el:('..) 4 
Consid('x th(, folh)wint,, (~x~uni)l(~ of V\] ) ellip- 
sis: 
(d) a. Tom walks. ,lohn does too. 
b. Ton,,* 1)l{.l",Sz w~flk. John:~* do(~s:e too. 
The, two s(',nt(,'n(:es r('('.(dve I,ll(~ following inl;cr 
1)retal;ions: 
Total* l)l{,l",S2 walk. -~ 
i.~ - .\ x\[I .v,.\]k(~)\], w,~lk@,)\] 
,/ohn:~* does2 Vlq~;2 too. :> 
Nex(;, we join t;he, two s(',nl;(m(;(', int, ert)r(,A;at, i()\]\]s 
with the s(!qucn(:ing Ol)exator , and we apply the. 
wdue of 1)2 1;o 11:~: 
TOlIII * l'l{,t'\]S2 walk../ohn:~* (loes2 VPE2 t;oo. -5 
\[u(,, ul, Ih I u(, : : u,, u, --: Tom, 
\]% - A x\[I w~lk(:~)\], walk(u,)\]; 
\[110, 113 I,*,, -: .:~, ,,:, :-.lot.x, w~lk(.:,)\] 
Next, we will (:o\]mid(',r a,n cxmnl)h~ involving 
sloppy i(hml,ity, rib (to this, it will t)c n(',c,(~ssary 
t:o add genitiv(,. Nl)'s, such as "his (:at" to ore sys- 
I;(HII. 
his (he.,~'s,~) => 
xl'. p~ (\[,_~,. I of(u.,, H,.)\]; P\[0~,,,); \]'~(-.,) ) 
We n(',(~d two in(li(:e,s: n is the, index of h,c: this 
is ml individual deline, d in input (:ont0,xt. The iu- 
(h'.x 7n, is 1;ira index of l:he obje.(:t 1)oss(~ssed 1)y h('..,d 
this obj('x:t is adde, d to th(', Ollti)ttt context. (For 
clarit;y, we will ofl;en write h, is,cat,,,.; 1)ut the '%1 > 
licial usage" is hc.,~ %, cat.) 
Now, we. examine a simph; case of sloppy iden- 
Lily in Vl' ellipsis: 
(5) }L. rlnOlIl IOV(Lq his cat. ,Iohn does too. 
t). 'Ibm,* l)l~.f';S2 love his* (:at:~. ,lolm4* 
(10(;82 I;oo. 
Tom|* I'IH",S~ love his* (:a.t:~ -> 
\[u,), ul, \]'~, u:~ \[ Uo --ut, ul :-- Tom, 
i'~ = Ax(\[ua\] (>f(u:~, uo), 
,:a; (.,:~), lov,,(x,~:, )1), ,,f(,~:,,,~,,),(:~(,~:,), 
lov,.0,, ,,,:,)\] 
,John4* (locs:~ (1oo) :-> 
\[u(,, u4 lu4 -: u(), u4 = ,/ohn\] ; l'~,(u4) 
Next, we join tim two scnt(',n(:es I, ogeJ;h(u' an(l 
apt)ly th(, value of I)2 to u4: 
4We ignore the. semantic conlailmtion of INFI,, 
apm't f\]'ont the above.-described interaction wil;h the. 
discourse conte.xt. 
521 
Tom1* PRES2 love his* Cata (and) 
John4* doesu (too) => 
\[U0, Ul, P2, U3 \[ Uo = Ul, Ul = Tom, 
P2 = Ax\[u31 of(u3, Uo), 
cat(ua), love(x,ua)\], 
of(ua,uo),cat(ua), love(u,,ua)\] ; 
\[Uo' U4 I U4 = 110' U4 = John\] ; \[Ua I of(ua, uo), eat(ua), love(ua,u3)\] 
The antecedent for the VPE is "love his cat". 
This object (PJ is introduced into the context by 
PRES> P2 represents the property of "loving u0's 
cat", where uo is the discourse center defined in 
the input context. In the first sentence, the center 
is TOM. The second sentence shifts the center to 
JOHN. It is this change in context that gives rise 
to the sloppy reading. Thus a sloppy reading is 
made possible when there is a center shift. 
Finally, we allow the possibility that, a property 
might be the discourse center. This means we 
must add an alternative rule for INFL, so that it 
adds a property that is the discourse CEntEr: 
INFL,~* ::> 
A PAx \[Pn I P0 -=- P,~, Pn = P\]; P(x) 
2.3 Paycheck Pronouns 
The phenomenon of "paycheck pronouns",5 is il- 
lustrated by the following Example 
(6) Smith spent his paycheck. Jones saved 
it. 
The reading of interest is where the pronoun 
"it" refers to Jones' paycheck, although its an- 
tecedent ("his paycheck") refers to Smith's pay- 
check. Our account for this parallels the account 
of sloppy identity in VP ellipsis. The antecedent 
"hisi paycheck" introduces a dynamic individual: 
a relation between contexts that introduces i's 
paycheck to the output context, where the value 
of i is dEtErminEd by the input context. The fol- 
lowing rule makes it possible for NP's like "his 
paycheck" to add dynamic individuals to the con- 
text. 
his (he~'Sm) => 
P, I xm = I of(u..un)\]; 
xm(PJ 
5This term comes from Kartunnen's example: The 
man who gave his paycheck to his wife was wiser 
than the one who gave it to his mistress. Various ac- 
counts of this phenomenon have been proposed, such 
as (Cooper, 1979; Engdahl, 1986; Jacobson, 1992; 
Gardent, 1991). (Heim, 1990) proposed extending the 
Sag/Williams account of VPE to the case of paycheck 
pronouns. Gardent makes a proposal similar to the 
current account: a dynamic approach in which pay- 
check pronouns and VPE are treated uniformly. 
We use variables of the form ui to denote ordi- 
nary extensional individuals; we use variables of 
the form xi to denote dynamic individuals. There 
are two distinct effects on the output context. 
First, the dynamic individual Xm is added to con- 
text: this object addsan individual Um to a given 
context, such that Um is of un in that context. 
Second, Xm is applied to the property P2. This 
actually adds u,~ to the current context. 
Finally, we need an alternative form for pro- 
nouns thai; refer to dynamic individuals: 
hen ~ 6 where 6 = dr(ant(he,~)) 
The pronoun hen recovers xn from the current 
context. The desired reading can now be derived 
as follows: 
(7) a. Smith spent his paycheck. Jones saved 
it. 
b. Smith1* PAST2 spend his* paychecka. 
Jones4* PASTa save ita. 
We take the two sentences individually. The 
first sentEnCE introduces the dynamic individual 
xa, as follows6: 
his* paychecka. => 
AP2 \[xa I xa = IP(\[u3 \] of(ua,u0), paycheck(u3)\]; 
P(ua)) \]; 
xa(P2) 
spend his* paycheck> => 
Av \[xa I x3 = AP(\[ua \[ of(ua,uo), paycheck(ua)\]; 
P(ua)) \]; 
I spend(v,u')\]) 
spend his* paychecka. 
Av \[xa Ix3 : IP(\[u3 I of(ua,u0), paycheck(ua)\]; P(u:0) 
\]; 
\[u3 I of(ua,uo), paycheck(ua)\];\[ I spcnd(v,ua)\] 
Smith ~* PAST2 spend his* paychecka. 
\[u0,Ul,P2,xa \]u0 = ul,ul = Smith, 
xa = AP(\[ua I of(ua,uo),paycheck(ua)\]; P(ua))\]; 
\[113 I of(ua,uo), payEheck(ua),spend(ul,ua)\] 
We continue with the second sentence. 
save it3 
AQAv(Q()m'\[ I save(v,u')\])) dr(ant(ita)) 
We substitute the value of xa for dr(ant(ita)): 
save ira 
AQAv(Q(),u'\[ I save(v,u')\])) AP(\[ua I of(ua,u0),paycheck(ua)\];P (ua))\] 
We perform A reductions, resulting in: 
6To simplify the representation, we omit the values 
for VP variables P2 and Ps, since they are not relevant 
to the current example. 
522 
save ita => 
Av (\[ua \[ of(ua,u0),paye.heck(ua)\]; 
\[I save(v,,:,)\])) 
Jones4* I'AST.5 save ita. => 
\[Uo,U4,1'5,ua luo = u4,u4=Jones, of(ua,uo) , 
payeheck(ua), save(u4,ua)\] 
The coInplete discourse is rel)resented as fol- 
lows: 
Smith :t* PAST2 st)end his* t)ayche(:k:~. 
.lones4* PAST5 save ita. => 
\[u0,ul,P2,xa \] u0 = u~ ,u~ = Smith, 
X 3 -= 
M'(\[ua I (,f(ua,u0),paydmek(u:,)\];P (Ua)) 
\[ua \] of(ua,uo), payeheck(ua),st,end(u~ ,ua)\]; 
\[uo,u4,Ps,ua luo = u4,u4=aones, 
of(ua,no),l,ayeheck(u:0, save(u4,ua)\] 
The dynamic individual xa adds the paycheck 
of u0 (the discourse center) to the context. In 
the second sentence, the discourse center is ,\]o'n,c& 
Thus we get the reading in which "Jones saved 
Jones' tmyeheek", as desired. 
3 An Expanded Paradigm tbr 
Sloppy Identity 
The proposed theory permits a simple, llniforln 
treatment of sloppy identity in VPE and pay- 
check pronouns. This uniformity extends fln'ther. 
We simply permit sloppy identity for any pro- 
form, whenever the anliece(le.nl; contains a preform 
within it. This is schematicMly represented as fol- 
lows: 
Cl ... b, .... \[,,,\] ...\] ... c2... b,"\] 
(C1, C2: "controllers" of sloppy variable YP) 
Ilere, XP is the anl;ecedent for some preform 
XP', and YP is the sloppy variable~, i.e., a pro- 
form embedded within XP. A sloppy reading re- 
suits whenever there is a center shift involving 
C1 and C2. That is, the interpretation of YP 
switches from controller C1 to C2. 
Since the dynamic theory treats VP ellipsis uni- 
formly with NP proforms, XP and YP both range 
over NP and VP. This predicts four possibilities. 
All four possibilities in fact occur, as shown by the 
following examples: 
(8) Tom \[v,' loves IN,' his\] cat\]. John 
does too. 
(9) Smith spent IN;' IN,' his\] paycheck\]. 
aeries saw;d it. 
(10) I'll help you if you \[v*' want me to 
\[v*' \] \]. I'll kiss you even if you don't, r 
rThis example was provided by Marc Gawron 
(p.c.), who attributed it to Carl Pollard. 
(11) When Harry drinks, I always conceal \[NP 
my belief that he shouldn't 
\[vp \] \]. When he gambles, i can't con- 
ceal it. 
ExaInlfles (8) and (9) have already been dis- 
cussed. (8) is the familiar (:as(', in which the VP an- 
tecedent (XP) contains a sloppy pronoun (YP). 
YI' switches from C1, ~lbm, to C2, John. In ex- 
ample (9), we have at, NI' antecedent (XP) con- 
taining a sloppy pronoun (YP), and the two con- 
trollers tbr YI ) are Smith and Jones. l,',xample 
(10) involves a VP anteee(lent ('ontaining a sloppy 
VP ellit)sis; l;he VP ellipsis switches from help you 
to kiss you. Finally, example (1.1.) involves an NP 
atttece(tent (:ontaining a sh)ppy VP ellipsis, switch- 
ing froIn drinks to gambles. 
We have already seen how the sloppy reading 
is derived for (St and for (9). We now show the 
deriwttion tbr (10) (example (11) can be derived 
in a similar fashion.)8: 
1~ WILl;2* hell) youa \[if\] youa PRES4 want ntel 1;o 2. 
I I WILL.;* kiss youa \[even if\] youa DO4 NOT. :-> 
\[Ul,l)0,Pe,ua,P4 I ul = 1,1)0 -- P:~,Ua - You, 
P= = av(\[ I help(v,u:d\]), 
: av(\[ I want(v,po(u,))\]), 
help(u, ,u:~),want (ul ,helt,(Ul,Ua))\] ; 
\[1'o,1',~, \] P0 = Ps, 
1',~ -- ,\v(\[ I kiss(v,ua)\]),NOT(P4(ua))\] 
The variable P4 represents the t)roI)erty of 
"wanting ul to Po". Below, we substitute the 
value Av(\[ I want(v,Po(t h))\]) for P~, and then sub- 
stitute the wflue Av(\[ \[ help(v,u:0\]) for P0, and 
apl)ly it to ua, giving the following, result: 
It WILL2* hel t) youa \[if\] youa PRES4 want me1 to2. 
It WlM, a* kiss youa \[even if\] youa 1)O4 NOT. :* 
\[ul,Po,P2,ua,l'4 \[ul = I,Po = P2,ua = "~2)u, 
P2 = Av(\[ I help(v,ua)\]), 
= I help(u1 ,nat,want (ul ,help(u, ,ua))\] ; 
\[Po,Pa I Po = Ps, P5 = lv(\[ I kiss(v,ua)\]), 
NOT(\[ I want(ua,kiss(u,,ua))\]), 
It is the. "center shiflT involving P2 ("help you") 
and P5 ("kiss you") that inakes thedesired read- 
ing possible. That, is, "what ua doesn't want is for 
111 to kiss ua". 
The dynamic theory explains all four of these 
eases in the same way; the embedded proform in 
the antecedent (:an be sloppy, because the con- 
troller for the embedded proform can undergo a 
center shift. The eases illustrated by (10) and (11.) 
8We construct a representat, ion as if the connec- 
tives if and even if were simple conjunctions. This 
allows us to avoid the complex issues involved in rep- 
resenting such "backwards conditionals" in a dynamic 
system. 
523 
have not, to my knowledge, been discussed t)revi- 
ously in the literature. It is not clear how such ex- 
amt)les could be handled by alternative theories, 
such as (Fiengo and May, 1994) or (Dah'ymple el; 
al., 1991), since these theori(',s do not treat NP 
and VP anaphora in a uniform fashion. 
4 Conclusions and Future Work 
The dynamic perspective provides a Kamework 
for a silnple, intuitive account of sloppy identity 
and related phenomena, by explaining the inter- 
pretive facts in terms of changes in context. This 
requires contexts to change in a way that is some- 
what foreign to the dynamic perspective; a given 
position in tile (:ontext must be reassigned, or shift 
its value. To implement this, I have incorporated 
the notion of discourse center, together with the 
me('hanism of center shift, into a dynamic sys- 
rein. This makes it possible to give a novel, dy- 
nalnie account of sloppy id(mtity t)henomena, i 
have shown that this approach aeeotmts for an 
expanded paradigm of sloppy identity, going be- 
yond tile data addressed in alternative a(:counts. 
In future work, we will investigate ineorI)orating 
additional aspects of centering theory, including 
the tbrward-looking centers list, and the prefer- 
en(;e orderings on transitions. 
5 Acknowledgments 
Thanks to Claire Gardent, Aravind Joshi, Shah)m 
Lappin, Mats Rooth, Stuart Shieber, Mark Steed- 
man, and Bonnie Webber for help in develoi)- 
ing tile basic aI)proach described in this pal)er. 
Thanks to ll,einhard Muskens for helpful con> 
ments on an earlier version of this work. This 
work was partially supported by a Villanova Uni- 
versity Summer Research Grant (1995), and an 
NSF Career Grant, no. IRI-9502257. 
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