T I I E C O R R E C T A N D E F F I C I E N T IMI)LI~',MI,N.[A" '~ T I O N OF' A P P I [ ( ) P R I A T E N E S S S P E C I F I ( : A T I O N S F O R T Y P F ; D F l g A T U R I " S T R U C T U R ] ~ , S
Dale GeMemalm and Paul John I(ing*
Seminar t'iir Sprachwissenschaft, Uniw',rsitS~t Tiibingen t
A B S T R A C T
in this pa,per, we argue tha, t t y p e inferenc- ing i n c o r r e c t l y i m p l e m e n t s a.pl)rolwiate- ness specifica.tions for t y p e d [ea.ture struc- tures, p r o m o t e a combina.tion of l;ype res- o l u t i o n a n d unfilling a,s a. correct a.nd ef'~ ticient M t e r n a t i v e , a n d consider the ex- pressive limits of this a.lterna.tive approa.ch. !['hroughout, we use feature cooccurence restrictions as i l l u s t r a t i o n and linguistic m o t i v a t i o n .
1
I N T R O D U C T I O N
U n i f i c a t i o n l b r m M i s m s ma.y be either un- t y p e d (DCC~s, PATR-II, 1,F(;) or t y p e d
(npsG). A m~L,ior
reason for a d d i n g types to ~ forma,lism is to express restrictions on fea.ture cooccurences a.s in (;l's(:: [5] in o r d e r to rule o u t nonexista.nt tyl)es of objects. For e x a m p l e , t h e r e a.re no verbs which have t h e [km.ture +R. T h e simplest way to express such restrictions is by mea.ns of a.n a.ppropria.teness pa.r- t i m flmction Approp: Type × Feat ~ Type. W i t h such a.n a.pl)rol)riatleness specifica.- tion lrla.tly Sllch restrictioi,s m a y be ex- pressed, t h o u g h no restrictions i n v o l v i n g r e e n t r a n c i e s ma.y be expressed.In this pal)er, we will first in §2 survey t h e r a n g e of t y p e eonstra.ints tha.t ma.y be expressed with j u s t a. t y p e hiera.rchy and *']'he resea.rch pl'eS(!lllL('d ill |,his; paper was pay tia.lly sponsored hy '[kfilprojekt B4 "(;onsl.rahH.s on Grammar fl~r Efficient Ck:neration" of the Soi,der forschungsbereich 340 of the Deutsche ["orschungs- g e m e i n s c h a , ft. "VVe would also like to thank 'l'hilo GStz for helph,l comments ou thc ideas present.ed here. All mistakes a.rc of collrsc our OWll.
IKI. Wilhehnstr. 113, |)-721174Tfilfi,,ge,, (ler- ma.ny, {rig,King} g'~sfs.n phil.uni-I uebingen.de.
a.n N)propria.teness specification. T h e n in ~3, we discuss how such t y p e c o n s | f r o n t s linty be mainta.ined u n d e r u n i f i c a t i o n as exemplilied in t h e na.tura.1 l a n g u a g e D~rs- i n g / g e n e r a t i o n system '.l'ro]l [7]. 1 Unlike previous s y s t e m s such as ALl,:, Troll does n o t e m p l o y a.ny type infereneing, inste~M, a, l i m i t e d a m o u n t of n a m e d d i s j u n c t i o n ([1 1], [12], [6])is i n t r o d u c e d to record t y p e resol u tion possibilities. T h e a.lnount of dis- j u n c t i o n is a.lso kept small by the t e c h n i q u e
of unlilli,g
described in [9]. T h i s s t r a t e g y a.ctua.lly ma.inta.ins apl)ropri~tteness condi- tions in some ca.ses in which a. t y p e in- ferencing stra.tegy would fa.il, l)'inMly, in §4, we discuss t h e possibilities for gener- a lizillg this a.pl)roa.ch to ha.ndle a bro~Mer r~tnge of constra.ints, i n c l u d i n g c o n s t r a i n t s inw)lving r e e n t r a n cies.2
A P P R O P R I A T E N E S S
F O R , M A L I S M S
As discussed iu G e r d e m a n n ,~ King [8], one ca.n view a.pl}rol)ria.teness CO[lditions as (lelining GPSG style fea,1;tl re cooccurence re- strict:ions ( F C R s ) . In [8], we divided F C R s into
co,j,,ctive
a n ddi.q,,~ctive
ct~sses. A c o n j u n c t i v e FCI/. is a constra.int of t h e fol- lowing fornl :i[' a.n o b j e c t is of ;~ cert;fin kind t h e n ill deserves certa.in fea.tures w i t h wdues of cert~till kinds
An FCI~ stat:ing tha,2: a. verb m u s t h~we v and N t'eatures with values A- a n d - re- spectively is a.ll e x a m p l e of a. c o n j u n c t i v e FCI{. A d i s j u n c t i v e I"CI{. is of the form:
if an object is of a. cel'taiu kiud t h e n it deserves cerl;a.in [ca,1;tll'C~s w i t h vMues of certa.hi kinds,
or it deserves cerl.ahi (pei'ha.liS o t h e r ) fea.1;u res \vil, h viiiues of terra.in (perlla.ps o t h e r ) kinds,
o r . . .
(31: it i:leserw.;s i:erl;a.in (lmrhal)S o t h e r ) fea,1;llres wil.h Vi, l.[ll(~S o[ certain (perha.ps o t h e r ) khi,<ls
I
lo:: exa~]nple, the f o l l o w i n g F(',|/. sl.a.t.iug tha,t inverCed verbs lilt|S1, lie a.uxili;tries is d i s j u n c t i v e : a verb Ilitisl; ha.re the ['(~il.l.tll'(~s
I N V a n d A U X w i t h va.l/ies d a.Iid I, - a.iitl i L-, o r - ;Mid - respectivel.y.
B o t h o| these |el'illS or l,'(',lls iiHly I)(! expressed in a. foi'llla.iiSlli euiployhi<~ fiiiil.e lia,rtia.[ order (Type, E) o| types tllldel' sub- 8illnptioli> a, finite sel. Feat of
ro;./.t;tll.(~s,
and an a.pprol)ria.teness parl, ial rliilcl.ionA p p r o p : T y p e X Feat -~ Type. [uluitively; the l, y p e s fornla.lize I;lie n o t i o n ol"
kinds +,j"
objecl, t g: t,' ill' ca.oh oil|eel, of tyl>e t' i~<i Mso of l;Ylle L, il, ll(] Approp(l, f ) = l I ill' (!;i('[I o b j e c t oF t y p e t deserves [eaA.urt~ f wil.]i :i.Vi./.]lle o r t y p e ft. ~@'e call S/IC]I it. [Ol'tll;li- i S l l l i-i, ii ;I,])l)l"Opl']al, olio,~/ fOl'lllil]i~;lll.
(',iLl'-
peliLel",s A I , F , a n d ( , e r d e l i i a . i ;ill(| (i(~t,z'sT r o l l are ex:-t.niples o| illilllenienl.a.Lions o| a,pF, ro]) ria, Loliess |or illa.[iSlil,s.
l low an a.i)ln'oprhi.teness [orniaJisnl en- co<les a c o n j u n c t i v e I:(',R is ob\.'i<>us~ bll(. llOW it encodes a d i s j u i i c t i v e I"(',1{ is less so. A l i exa.niple i|]usl;ral;es best how it. is done. ~Ul)pOS0 t h a t F( ',1{ [i sl.al.es l.hal, ob- .iecls (if type t deserw! [(!a.[./ll'(!S f 'and .q,
I)oth w i t h boolea.I/ wdues a.ll(I ['lll'l,[lel'lllOF(~ t h a t the va.hies of f aild g iil/lSl al~r(!e, [>
is the disjunct]w! I"(111.
if a,u object is o[ type l
then it deserw:s f with va.lue -I- and q with w d u e +,
or it deserw.~s f with va.lue -
a.nd
9
with value -To 0ncode [3> first iul,rodLiCe sul/l.yltes , t ~
##
;+l.ll([
l"
of I(1 E I/, 1.
), O11(! SUl)tyl)e ['()l' ea,ch d i s j u u c t iu the cousequenl, of'p. T h e n e n c o d e the ]'ea.tli['e/wthl~.~ <'on(!il.illliS in l, he [irst disjunct ILy putthlg Approp(t', ./) :: ~-a,nd Approp(//~ q) - + , and encode the I'ea- t u r e / v a l u e c o n d i t i o n s in the second dis- juu(:t by p u t t i n g A p p r o p ( t ' , f ) = -. and
A p p r o p ( t ' , g ) = . .'2
This a p p r o a , c h I n a , k e s t w o i n l l ) o r t ; a , lll,
closed-world type assumptious a, bouL (.he t y p e s tli~d; Slll)SlllIle 11o ogher t y p e s (hellCe-
forth
species),
l:irst, thep;i.rtition condi-
tiOII
s t a t e s tha.t for each t y p e t, if a.n ob- ject is (31' t y p e t theu the o b j e c t is of ex- ax-I.ly o11(2 species subsulned by t. Second, theall-or-nothing cclndition
sta, tes t h a t 1'(31' each species ,q a.itd fea.ture f , e i t h e r every el" IIOol>,iecl, or
species s deserves feature .#c.3 A l l a.l)ltroltriM,eliess [orli+ia.lisill sllc]l a.s ALl:, ([2], [3])ti,;t.l. does not uieet both c.ou- ditions llla.y llOt; ]lroper[y el|cOde a, disjull('- five l"(:l/. For exalnple, consider disjunc- tive I"CI{. p. An a.I)prl;)pria.l, elleSS [ornia.l--iSlli I/lily l/O( p r o p e r l y e n c o d e 1,hi~t t / a.lld t " i'el)rt,selil, MI a.lid o i l l y t h e d i s j u n c l , s ill t h e COll.qeqll(Hlt o r [i w i L h o u t t h e i)a.rl,ition COll- d ] t i o n . <till a.llln'ol)riill.eness [orlila.liSlll l l i a , y IIOl. l l r O l ) e r l y e n c o d e t h e [t~ii.l.llle/vii.hle (:(lll- <liiriOii: deinanded liy em'h disjuncl, hi the
COli.~t!qllelil. o | p w i l h o u l , t h e a.i[-Ol'-liot;hilig c(m(til.ion.
As indicat.ed a.bove, AI, I.; is iLIi exa.tlli)le o| it. f(n'liialiSlU I.ha.l, does it(it ineel; llol;h o| 1.hese closed world aS,glllnlil,iOli.g. In AI+E :-/. ['eli.l.tlr(~ st.i'llCtlile i.<4 won typed ifl' for ea.ch arc iit the te:+d.ure sI.l'tlCl;tlr0, if' 1,he SOtll'('(~ node is labelled wil.h type /., the targel; node is lallelled w i t h 1;ype l / a.lld the il.i'c is IMlelled w i t h [ea.tlll'(~ f 1,lien Approp(/.> .f) [ l/. Furl.her|note> a ['eal, urt~ s t r u t ( t i r e is
>l'lds exanll)h: I:(JR is, for eXlmsil.ory l)nrl)oses,
quilt simph'. "l'hc prolileni o[ c.xpr('.sshig F(Jl/'s, however, is a l'Cal Iiuguisl.ic i)rol)lcin. As noted I)y Copcstakc. ct al. [4], it. was inipossihlc I.o c.xpress
CV('II Ihc .~ilii[)]oM. forilis o[ l"(JRs in l.hc.ii7 c×tciidcd VCISiOII (it' AI.E.
'['hc basic principle of expressing l"Clls also ex l e n d s Io I"(',[(s i u v o l v i u g l o n g e r p a l h s . F o r e x a m - ple, to (:llSllt't: thai. for the type l, I.he path ( f g )
lakes a vahie subsuuied I)y .% one nlust tirst hll, ro ducc the chaiu A p p r o p ( / , f ) = .,, A p p r o p ( ' a , g ) = .~.
Silch ilil.crlllCdialc I.'~'l)lts COllid ll(! hll.rodllced a.<-; part o[ a (onilli[al.iou sl.age..
4 Nob: I.hal. Ihesc cl,>s<,d world assulnplions art'
w e l l - t y p a b l e iff the feature s t r u c t u r e sub- sumes a well-typed feature structure, in ALl.:, t y p e infereneing is employed to en- sure t h a t all feature structures are well- t y p a b l e - - i n fact, all feature structures are well t y p e d . Unfortunately, well-typability is not sufficient to ensure t h a t disjunctive F C R s are satisfied. Consider, For exam- pie, our encoding of the disjunctive F C R p and suppose t h a t 99 is the fe, a t u r e s t r u c t u r e
t [ f : + , 9 : - ] . 90 is well-typed, and hence trivially well-typable. Unfortunately, 99 vb elates the encoded disjunctive F C R p. The only way one could i n t e r p r e t ~ as well- formed
By c o n t r a s t , the Troll system described in this p a p e r has an etfeetive algorithm f<>r deciding well-formedness, which is based on the idea of efficiently representing dis- junctive possibilities within the feature struetu.re. Call a well-typed feature struc- ture in which all nodes are labelled with species a r e s o l v e d f e a t u r e s t r u c t u r e and call a set of resolved feature structures t h a t have the same underlying graph ( t h a t is, they differ only in their node labellings)
a d i s j u n c t i v e r e s o l v e d f e a t u r e s t r u c t u r e .
We write f S , ~ v f 8 and 'D~.)c$ for the
collections of feature s t r u c t u r e s , resolved feature s t r u c t u r e s and disjunctive resolved feature s t r u c t u r e s respectively. Say t h a t
F ' 6 " l ~ f $ is a r e s o l v a a t of F C f,"? ill'
F and .F' have the same underlying graph
and F subsumes 1 ''l. Let taype resolution be
the t o t a l flmction ~ : f 5 " --+ D g f S such t h a t 7~(1,') is the set of all resolvants of l i'.
Guided by the llartition and all-or-
nothing c o M i t i o n s , King [13] has fOl'inti- lated a semantics of feature structures and developed a notion of a satisfiable feature s t r u c t u r e such t h a t l'7 C .T$ is satisfial~le
if[' 7~(F) 7 ~ (7). C, erdemann ,% King [8] have
also shown t h a t a feature strtlcture l]leets all encoded F C R s ifl" the feature s t r u c t u r e is satisfiable. The Troll system, which is based on this idea, effectively inqflements t y p e resolution.
W h y does t y p e resohttion succeed where. t y p e inferencing fails? Consider again the encoding of p and the feature s t r u c t u r e
9~. Loosely speaking, the a p p r o p r i a t e -
ness sl)eeifieations for t y p e t encode the
part of p that sta, tes that an o b j e c t of
tyl)e t deserves features f and g, b o t h
with boolean vahles. However, the ap-
propriateness specifications for the speci- ate sul)types t' and t" of type t encode the part of p t h a t s t a t e s t h a t these val- lies lnust agree. Well-typability only con-
siders species if forced to. In the case
of ~, well-typability can be estahlished by consklering t y p e t alone, without the l)artition condition forcing one to find a well-typed species subsumed hy t. Conse- quently, well-tyl)ahility overlooks the p a r t o f f l exehisively encoded by the ai)propri- ateness specifications for t' and t". T y p e resolution, on the other hand, always con-
siders species. Thus, t y p e resolving 9o
cannot overlook the p a r t of p exclusively encoded by tile a p p r o p r i a t e n e s s specifica- tions for t' and t ' .
3
M A I N T A I N I N G
A P P R O P R I A T E N E S
S
C O N D I T I O N S
l[ow may these D ~ . T S be used ill an inl-
plenmntation? A very i m p o r t a n t prop-
erty of the class of " D T ~ f S is t h a t they are closed under unification, i.e., i f / " and F' 6 D ~ f 8 then F U F' 6 D ' P v f S . 4
Given this prol)erty , it would in princi-
ple lie possible to list the disjunctive re- solved feature sl;ructures h/ an iinplemen- tal;ion withonl; any additional type infer- 01]¢hig proc0dnre to ma.hltahi satisfialfil- ity. It would, of course~ tier be very of[i- cieut 1.o work with such large disjunctions of featiil'e strtlctilres. These disjunetiorls of fea.ture structnres, however, have a sin-
gular l)rol)erty: all of the disjuncts have
the same shape. The disjuncts differ only
in the types labeling i;he nodes. This prop-
4In fa.ct., it ~:~rl~ I)~ SI~OW ~ that if t" a.nd 1'" 6 f S then "R ( F) tJ 1"(1"') = "R ( F tO F'). Uni/ication
of sets of fca.ture structures is defined here ill the
standard way: S t2 ,S" = {1"[ I"' 6 S and l"" G S"
(!rty a.llows a. disjultctivo fesolv(,d featur(, structti re to I)e r(;l)rosetd,(~d more et[icieutly a,s ~t sitlgle untyl)(~d l'eatur(' st.l'll(:l.llfe plus a, sel; of d(;pondlmt node la.h(~liugs, which ca.n be f u r t h e r
(;oml)a,(:t(~d using mi, Nie(l
dis. j u n c t i o n a.s in (',(~rdemann [(i], I)i'~['re t(:Fo]' exanH)le , SUl)l)OS(~ \v(~ I,,yl)(~ r(~solvc the [ea, l, urc st, r u c t u r e
t[,f ; bool,fl; bool] us-
ing o u r e n c o d i n g of p. ()he can (rosily see tha.t this fea.tur(~ strut:fur(, has only two I'es o l w l , n t s , w h i c h ca, n I)e colla.ps(~d i u t o o n e fea,1;ure strlll:ttlro w i t h llallV2d d]sjunci.ion a,s shown below:
II'll;1}
f : k , : :>["'"' ]
f : (I t - )0:t-
LU: J
,u: (I t
)
We now ha,vo a, [;(mSolml)ly COml)a(:l l'q)- resentaJ;ion hi which t.ho
l"(il{,
ha.s lie(Hi tl';tllsl;I,t(~([ iul,o a. Ila, ill(!(I ([iS.]llll(:l.ioli. Ih,wO,V(H'> (Hie should note tha, t f i l l s d i s . i u n ( : l;ion is o n l y l)l'eSeUl; b(~(:aats(~ the ['oaJ, tli'O~i .f a,]l(l g ha>l)l)en 1:o I)o Fir(~s(HIt. Tilt!S(! I(,a tures w o u l d .eed l;o Im l)res(mt i l w(~ wtwe enforchl<ej (Jaxpcnl,(H"s [:7] lcil, al w(ql i.yl)iug r(xluiroti]oilt ~ whhth ,qa,y's 1.1ial [(!al:ilr('s I. lial a,l:e a.llowed ilillSt 1)o pres,.ml., lllil. Iol.a[ well I.yping is, hi fax:t> incoinl)a.lib]e ;villi l y p e resolul, ioli~ since I;hore lilil$' w(ql I)o all inli llit;(~ seL of tota, lly w(,ll iyl)od I'esolvalil.s of ;1 l'(;a, Lllr(J st]'llcttir('~, For (~xa.llipi(~, a.ll illi(lei'.- Sl)ocifiod list stl'u('tlir(' couhl be iT(~S()/v0(I 1.o ;~ list of l e n g t h (L a. list of h:ngl.h 1, el.c,
,qhlce I, ota.I well I,yliin g is liOt i'(!quir(!([, we lm~y i~s well a.ctiwqy un[il[ r0(lulid;lnt ['0a, tlires, 5 ill t h i s (!Xalli[)l(!> i[ t, li('
f
ail(l (7 fo.a, tllrOS ;~l'e r e l i i o v o d , w e a,lO l e l l wil, h l h ( , simple, d i s j u n c t i o n {if,/'~}: which is (!quiv- a,lent, to l;]le o r ( l i l l a J ' y l,Yl)(' l.(; T h u s , iu lliis ca, so>]lO
(lisjtulcl, ion a.t all ix rc!(llliro(l10 (!11"
force the I " C I L A l l th',tt is requirc(I is tim~qntuil, ively, [eat, ui'cs arc rodundaui it I l w i r val llCS art'. eul,h'cl 5 predictaldc fl'oui ihc a p p r o l u i a i c .ross Sl>eCificatim,..%'c GStz [1)], (',cr,lemam, [7] k,r
;I. IIlOl;('. [HXX:iHC forUllllalioii.
°[n t h i s casc, il. would also have b(:ml l)~>,~iblc
to unlill Lhc oi'i<eiuai teal, life Sll'tl<ltllc I , . I . i e I*' solviug. /Snforl, unai,e, ly, llmvcvcr, this i~; l.>i ;ihvay~.
the (:asc, as C;lll |)(! S(!t'II in the [ollowiug (!Xalll])lC:
t{j:
+] :> { C / : + ] } ~ ~'.asSUml)tion tha.t t will only be ext(mded I)y unil'ying il with a.lmther (t;Oml)a.ct(~d) m(mll)(!r o['
"l)']?.Jr,_c,.
This, h.w(wer, wa.s a. simple ca.se iu which a.I1 of the n a m e d dis.jun(:tion could ho removed. It would not lmve I)('en i)os sihle to relnov(' tim fea.tur('s f ~tll(I g if thest~ 17,atu['es had I)oen involved iu re(m- tranci(+s of i[' tlt(,se lim.tures ha.d ha.d t:om- i)h+x va.lu('s, lu gt+tlera.I, howover, our eXl)e-
ri(!ll(:(~ ha,s I)(~(ql t h a t , eV(;l! wil, li v e r y (:()tit
pl('x type hi(~ra, rchi(~s a.nd |'(m, tur(; SLI'UC-
l, lll'eS [()1" l i P S ( i , v e r y i'ow n a m e d (lisjunc- lions a, re introdu('e(l. 7 q'hus~ uuilica.1;ion is e;(merally uo more (~xp(msive tha.n unifica.- li,:)H w i t h u n l y l m d l(mt.ur(~ sl.fu(:l.ur('s.
4
C O N C L U S I O N S
\ % havc~ sh,:Y, vu in this i~al),:~r tha.t the kind of consl raints ,:~Xl)r.t~ssihlo Ity api)Vol)rh~,l;o- r.~ss c~mdit.ions call he imlflemc'.nted iN a i.'actical .,.D, sle]n e,ul)loyinK typ,M featu r,:'~ st.ru(:t.uf(,s and utdlica.Lion a.s I.he I:,ritna.ry Ol)(U'a,t, ic:,n on t(>;t,l, ur<+ ,'-;t, ruct, ure~. I l u t what. Of IIlOl'(' COIII[)I(~N l;yp(~ CC'IIH|,F.~LilI|,,q it~v'.)l',.' h~y; r(~enl;ram:ies': [ntro(IL~ciug reeJH.ra.ncies i l l l . ,::<rest ralid.s allows E.' the F,O~sihillty of d(~liNiu/,, recursivc l.yl),:~s ~ such a.s the (leli n i t k m of a p p e n d in [I]. (;lea['ly the re ~olv;-~nl.~, o[ such a. recursiv(~ l.yl)(', could Not I)(~ l,reCOmlfiled a.s r,.~quiI'oxl in Troll.
Oue m i g h t , uew'rtholoss, considm' a l-
[OWil]l[ f('(Hl(, f a, ll('y- ('OIls t f a hI| S oll l l o l l -
recursiv(qy defiltcd l.ypcs. A ])ro/)leul still arises; nantcly, il lhe l'eSo[va.itts of a Frail, till't1 .qll'tlCI411"(~ ill(:ludcd sonic with a pa.rticu lar r(~onll'all(:y a.nd s()Tn(~ \viLh(',ul, then the (:,.)mliti()ll iliad, a.II resc)lva.uts ha.v(~ th,:~ same shal)(~ w o u l d m ) l o n ~ e [ ' hold. ()ue v.,ottkl l.her(q'or,.~ no(~(l i.o eml)loy a moue COml)l(,x
vorsion .r ,a.med (lis.it, f,t:tio,
(Ill], [12],
tit)I). It. ig (i,.L(~sti,.malfl(~ w h ( ' t h e f such a.d d i t i o n a l (:()mpl(~xit.y w o u l d I)e justified to' O u r CXl)ericl~(:c is derived l , ' i m a r i l y flora test- i.I" Ihc 'l'loll s y s t e m (m a tat, her lar<e,e e, ramul;G for (',(!l>lll;lll imfiial vcrh I>lHases, which was wiit-
t('n I)y I'hhard Ilillrichs a . d T s u m : k o Na, kazawa
handle this limited class of reentrancy-
constraints.
It seems then, that the class of con-
straints that can be expressed by appro-
priateness conditions corresponds closely
to the class of constraints that can be effi-
ciently preeompiled. We take this as a jus-
tification for appropriateness formalisms
in general. It makes sense to ~d)straet out
the efficiently processable constraints and
then allow another mechalfiSm, such as at-
tachments of definite clauses, to express
more complex constraints.
R e f e r e n c e s
[1] l[asmn Ait-Kaci.
A New Model of
Computation Based on a Calculus of
Type Subsum, ption.
Phi) thesis, Uni-
versity of Pennsylvania, 1985.
[2] Bob Carpenter.
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l;~ature ,5'tructurcs.
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1992.
[3] Bob Carpenter. AI,E
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[4] Ann Copestake, Antonio Saniilippo,
Ted Briscoe, and Valeria De Paiwl..
The ACQUILEX IA{B: An introduc-
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Parsing and (;c~.-
eration of UniJieation
Gram, roots.
PhD thesis, University of Illinois,
11991. Published as Beckman Insti-
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Re-
port
CS-91-06.
[8] l)ale Cerdem~nn. Trolh Type resolu-
tion system, user's guide, [994. M~ut-
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by Dale Gerdemann & Thilo G6tz.
[9] Dale Gerdemann and Paul John
King. Typed feature structures for
expressing and computationally im-
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[13] .h)hn T. Ma.xwell III and I{onahl M
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