A TREATMENT OF NEGATIVE DESCRIPTIONS OF 
TYPED FEATURE STRUCTURES 
KIYOSHI KOGURE 
NTT Basic P~esearch Laboratories 
9-11, Midori-cho 3-chome, Musashino-shi, Tokyo, 180 Japan 
kogure@atom.ntt.jp 
Abstract 
A formal treatment of typed feature structures 
(TFSs) is developed to augment TFSs, so that neg- 
ative descriptions of them can be treated. Negative 
descriptions of TFSs can make linguistic descriptions 
compact and thus easy to understand. Negative de- 
scriptions can be classified into three primitive nega- 
tive descriptions: (1) negations of type symbols, (2) 
negations of feature existences, and (3) negations of 
feature-address value agreements. The formalization 
proposed in this paper is based on A'it-Kaci's com- 
plex terms. The first description is treated by extend- 
ing type symbol lattices to include complement type 
symbols. The second and third are treated by aug- 
meriting term structures with structures representing 
these negations. Algorithrrts for augmented-TFS uni- 
fication have been developed using graph unification, 
and programs using these algorithms have been writ- 
ten in Conmaon Lisp. 
1 Introduction 
In unification-based or information:based linguistic 
frameworks, the most important objects are struc- 
tures called 'feature structures' (FSs), which are used 
to describe linguistic objects and phenomena. A fea- 
ture structure is either atomic or complex: an atomic 
FS is denoted by an atomic symbol; a complex FS 
consists of a set of feature-value pairs each of which 
describes an aspect of an object. Partial information 
on aJ~ object is merged by applying the unification 
operation to FSs. 
ILeseareh on unification-based linguistic theories 
has been accompanied by research on FSs themselves. 
Several extensions on FSs or on feature descriptions 
and formal treatments of the extensions have been 
proposed. 
Disjunctive and negative descriptions on FSs help 
make the linguistic descriptions simple, compact, and 
thus easy to understand. For disjunctive feature de- 
acrq)tions, Kay\[14\] introduces them into FUG (lqlnc- 
tmnal Unification Grammar) and gives the procedu- 
ral semantics. Karttunen\[ll\] also proposes proce- 
dural treatments of disjunctions in conjunction with 
relatively simple negations. ILounds and Ka.sper\[19, 
13\] propose a logic-based formalism--feature logic- 
which uses automata to model FSs and can treat dis- 
Junctive feature descriptions, and they obtain impor- 
tant results. 
For negative descriptions of PSs, one of the most. 
fundamental properties of FSs, the partiality of in- 
formation they carry, makes its insufficient to adopt 
relatively simple treatments. Classical interpretation 
of negation, for example, does not, allow evaluation 
of negations to be freely interleaved with unification. 
Moshier and Rounds\[17\] propose a formal framework 
which treats negative feature descriptions on the b`a~is 
of intuitionistic logic. Ilowever, their |bHnalism has 
trouble treating double negations. Dawar\[5\] l)rOl)OSeS 
a formal treatment b~ed on three-valued logic. 
In order to treat feature domains of complex FSs 
and to treat taxonomic hierarchies of symbolic tim 
ture values, type (or sort) hierarchies have been in- 
troduced, allowing definition of typed (or sorted) 
feature-structures (TFSs). A TFS consists of a type 
symbol from a lattice and a set of rearm:e-value pairs. 
A TFS can be seen as a generalized concept of both 
atomic and cornplex FSs. Pollard and Sag/18\] iatt'o- 
duce sorts into IIPSG (Ilead-drivcn Phr~Lse Strllcttn'e 
Grammar) and use sorted FSs to describe linguistic 
objects. 
Ait-Kaci\[1\] proposes an Mgebraie fratnewot'k using 
the C-types and ~-types, one of promising lbt'maliza- 
tions of TFSs, based on lattice theory. This lbrmal- 
ization was originally ainmd at formalizing and in- 
tegrating various kinds of knowledge representat.ioiT 
frameworks m AI. In this approach, types are defined 
,as equivalence clmsses of complex term structures. A 
subsumption relation is defined on these term struc-. 
tures. The join and meet operations on thenT cor- 
respond to tile generalization and uniilcation Ol)era- 
tions on TFSs, respectively. This approach essentially 
adopLs 'type-as-set' seulantics. Subtype relationships 
on type correspond to subsnmption relationships on 
denotations of types. Based on this framework, an 
extension to Prolog, LOGIN\[2\], has becn developed. 
Smolka\[20\] proposes a feature logic with subsorts. 
In this approach, negative descriptkms can be decom- 
poscd into three kinds of prinfitivc negations, namely, 
negations of sorts or complement sorts which denote 
tile complements of sets that positive counterlmrl.s lie- 
note, negations of feature existences, and negations 
of feature-address agreement or feature-address dis 
agreement. Slnolka extends t~aturc descriptions but 
a feature-structure interpretation of an extended de 
scription does not include negat.iw~ information and 
corresponds to a simple TI"S. 
Some TIeS based m~tural language processing sys- 
tems have been developed\[7, 24, 12, 15, 8, 22\]. Car- 
imnter and Pollard\[4\] propose an interlhce to buikl 
type lattices. 
Formalizations of extended FSs and of extettd('d 
feature-descriptions, described above, arc classilicd 
into two classes: (1) extensions of FSs themselves, 
and (2) extensions not of FSs themselves hut of 
Dature-descriptions. Previous attempts to introduce 
type hierarchies fall into the former clzLss while pre 
vious treatments of disjunctive and neg~diw~ &'scrip- 
tions mainly fall into the latter. 
ACRES DE COIJNG,92, NAMES, 23-28 AOt33" 1992 3 8 0 Pl~oc. OF COL1NG-92, NANTES, AUG. 23-28, 1992 
This paper proposes an extension to Ait-Kaci's ~/,- 
type that incorporates three kinds of the primitive 
negative descriptions described below into the q:-type. 
Ai't-Kaei's t-type formalization uses term structures. 
In this paper, both these type structures and the tyl)e 
symbol lattice on which term strnctures are delined 
are e×tcuded to treat negative descril)tions. Nega 
tions of type symbols are treated by extending type 
symbol lattices, aud negations of feature cxistmmes 
attd feature-address disagreements are treated by ex- 
tending term structures. This extension can be seen 
as intuitionistie. The extension is classified into class 
(1) abow'.. 
Based on this paper's formalization, unilieation al- 
gorithms have been developed usiug graph unification 
techniques\[23, 16\]. Programs based on these alger 
rithms have been implemented in Common Lisp. 
2 Requirements of Negative 
Descriptions of TFSs 
In describing linguistic information using (tyl)ed) fea- 
ture structures, negative descriptions make the de-. 
scription compact, intuitive, and hence easy to under- 
stand. For example, we want to deserihe the gram- 
rnaI, ical agreement for an English verb, say "eat", nat- 
urally a.s follows. 
......... , I. r ,,e,. ..... :'"h\] sg (1) 
This description specifies compactly and directly that 
it is not the case that the person attribute is third 
and that the number attribute is singular. If we 
could not use such complex negative descriptions, we 
would write it using disjunctive descriptions with sim- 
ple complement types as follows. 
sy,(ag,'eeme,d ag,{l ...... ,, ~3rd\] \]'\[ 
syii(agreeme;',l all, tinumbe'r msg\] I J" (2) 
or { 
sy,(.g,~,ae,,t ,,g~\[ve~ .... 1st\]\] 1 
sy,ftag,'eemenl aglIp ....... 2nd\]\]} (3) 
sy,( a.qreeme,d ag, f,,umber pl\]\] J 
In this case, (1) is e*Lsier to understand than (2) or 
(3). 
In the above ease., we can describe the informa- 
tion because the complex negative descriptions C~tll 
be transformed into the disjmlction of simple negative 
descriptions {with ml almost same inteuded mean- 
ing) and because both person and number features 
take their values from {lst, 2nrl, 3rd} and {st, pl}. 
However, it is not always the case that such transfor- 
mations are possible and that feature takes its value 
from a finite set. 
Let us consider more. complicated cases using dif- 
t 1 ference lists expressed using featm'e structures. The 
empty list of categories is represented as follows. 
x~ H) 
In the above example, the tag symbol, X1 shows that 
features in and out must take the same value. 
tin HPSG and JPSG (Japanese Ptlrase Structure 
Grammar), a difference list is very convenient \['or express- 
ing subcat and slash feature values. 
llow can oniy nomemptiness be expressed? This 
is impossible using complement type symbols or dis 
junctions becmlsc we can consider the set of MI finite 
length lists whose elements can bc taken froltl inlinitc 
sets. l)ireet or indirect extension of feature struetures 
is required. 
So far, we have discussed the requirement of nega- 
tive descriptions of type symbols and of l;eature-value 
agreeumnts from the viewpoint of capability of de- 
scribing linguistic inR)rmation. There are other ad 
vantages of allowing negative descriptions. Consider, 
for exannlde , debttgging processes of gramJt,atical de- 
scriptlous by parsing sample sentences. We may ob 
taiu unexpected results Sllch ll.~ il ~l'FS with an tlnex 
peeled type symbol, a TFS with an unexpected lea 
tare value agreement and so on. \[1/ such sittlations, 
negatiw~ descriptions can be usefld tools R)r delecting 
their re~mons. 
To t/l;tke linguistic descriptions compact and thus 
ea.uy to understand, to treat natural language efll- 
clently, and to detect error reasons rapidly, it is neces- 
sary to develo 1) formalizations and nu'.thods of treat- 
ing negative descriptions. 
a Formal Treatment of Negative 
Descriptions of TFSs 
As stated earlier, a typed t~:at, ure structure (TI"S) 
cousists Of ~t tYl)e syulbol alld a set of feal, tlre-vs.ble 
pairs. Thus, descriptions of TFSs are chLssitied into 
descriptions of TFSs having: 
(1) a certain type symbol (or having a subtype syn,- 
hol of a certain type symbol), 
(2) a feature, and 
(3) two feature-address vahtes that agree. 
A TFS can be described by using conjunct, ions and 
disjunctions of such kinds of descriptions. A eonjmle- 
tiw* and disjunctive TFS can be formalized as Nit- 
Kaei's t-type and ~-type, respectively. That is, a 
t-type, which has a complex term structure called a 
g, term a.s its syntax, represents a conjunction of such 
kinds of descriptkms or at col0unctiw~ typed feaLltrl! 
structure, and an e-type is a maximal set of ¢ types 
representing the disjunction of them. 
Negative counterparts of these descriptions are 
ebLssified into deseriptions of TFSs: 
(1') not having a certain tyl)c symbol (or having a 
type symbol which is not subsunmd by a certain 
type symhol), 
(2') not having a certain feature, and 
(3') having two thature-addrcss values that do not 
agree. 
By ineorporatiug strllettlres represelll, illg stlch lll!g- 
ative descriptions into a O term, a'FFS with the net 
ative descriptions can be formalized. Such a lerm is 
called an allglnented t-term and a type with an allg- 
mented ~/, term ~m its syntax is called an allgllu!nted 
O-type. From augmented g:-t.erms, an augmented 
terul eilll be COllStl'lleted ill the S~Lllle Illallll#!l" tlHlt fill 
(-terlu is eonstrlleted frolu ¢-t, errns. 
Next, augmented C-terms and C-types are defined. 
Terln structures are first allglueuted with strtlctllres 
representing inhibited features and disagreement of 
feature address values. Then, type symbol htttiees 
are extended to inch,de complement type symbols as 
suggested in \[1\]. 
AcrEs DE COLING-92, NAN'rgs. 23-28 AO£~r 1992 3 8 1 1)l~oc. OF COLING-92, NANTES, AUG. 23-28, 1992 
3.1 Typed Feature Structures as 
Augmented C-Types 
In order to define complex term structures, a signa- 
ture is used to specify their vocabulary. It serves as 
the interface between their syntax and semantics. A 
signature is formally defined as follows. 
Definition 1 A signature is a quadruple (7-,<_T 
,2-, V) consisting of: 
1. a set 7- of type symbols containing T and _L, 
2. a partial order _<7- on 7" such that 
(a) ± is the least and T is the greatest element, 
and 
(b) every pair of type symbols a, b E 7- have a 
least upper bound or join, which is denoted 
by a VT" b and a greatest lower bound or 
meet, which is denoted by a AT b, 
3. a set .T" of feature symbols, and 
4. a set I\] of tag symbols 
where 7-, 2- and l? are pairwise disjoint. 
A simple 'type-as-set' semantics is adopted for 
these objects. That is, a type symbol in 7- denotes 
a set of objects in an interpretation. Here, 7- and 
.1_ denote the sets called the universe, written as U, 
and the empty set 0, respectively. Another element 
a denotes a nonempty subset of U, written as \[a\]. 
The partial order <~- denotes the subsumption rela- 
tion between these sets; for any type symbols a, b, 
and c, 
1. a <~ b if and only if Is| c lb\], 
2. a Y:r b = c if and only if \[a\] O \[b\] = \[el, and 
3. a AT- b = c if and only if \[a\]N \[b\] = \[c\]. 
A feature symbol denotes a function from a subset 
of U to U. A feature path is a finite string of feature 
symbols and denotes the function obtained by tile 
composition of the functions that tile feature symbols 
denote. 
A term is defined from a signature. First, a term 
domain is defined as a skeleton built from feature 
symbols. 
Definition 2 A term domain A on 2- is a set of finite 
strings of feature symbols in 2" (inclnding the empty 
string ~) such that 
1. Aisprefix-elosed: Yp, q(52-*,ifp.q(s A, then 
p (5 A; and 
2. A is finitely branching: if p (5 A, then {f (5 
2"1 p .f (5 A} is finite 
where '.' is the string concatenation operator. 
An element of a term domain is called a feature 
address or a feature path. By definition, the empty 
string e must belong to all term domains and is called 
the root address. A term domain is represented by 
a rooted directed graph within which each arc has a 
feature symbol as its label. 
A suhdomain of a term domain, corresponding to 
a subgraph, is defined ms follows. 
Definition 3 Given a term domain A and a feature 
address p t5 A, the subdomain of A at p is defined to 
be the term domain Alp := {p' I P' P* (5 A}. The set 
of all subdomains of A is denoted by Subdom(A). 
Next, flesh is put on the term structure's skele- 
ton as defined as a term domain by assigning several 
kinds of objects to each feature address. Ait-Kaci's 
term structure, the basis of the C-type, is defined by 
assigning a type symbol and a tag symbol to each 
feature address as follows. 
Definition 4 A term is a triple (A, r, v) where A is 
a term domain on .T, r is a type symbol function fi'om 
2-* to T such that r(f* - A) = {T}, and v is a tag 
symbol 5ruction front A to Y. 
Given a tag symbol fimction v, Addr. denotes the 
function from a tag symboJ to tile set of addresses: 
Addro(X) :-- {pGAIv(p)=X}. (5) 
In order to treat negations of feature existences attd 
feature-address value disagreement, the term struc- 
ture defined above is augmented by assigning addi- 
tional objects, a set of inhibited features and a set of 
disagreement tag symbols, to each feature addrcss. 
Definition 5 An augmented term is a quintuple 
(A,r,o,¢,X) where A is a term domain on 5 v, r 
is a type symbol timer(on from ~'* to T such that 
r(2-* - A) = {T}, v is a tag symbol function front 
A to V, ¢ is an inhibited feature filnction front 5 r* 
to 2 ~ such that ¢(p) is finite for any p (5 A and 
~(~'* - A) = {0}, and X is a disagreement tag sym- 
bol function from J'* to 2 v such that X(P) is finite 
for any p (5 A and X(f'* - A) _- {0}, 2 
The inhibited feature fimction ¢ specifies which fea- 
tures cannot exist at a given address. There is thus 
inconsistency if there is an address p in A such that 
¢(p)n{fe2-lp.f(sA} # O. (6) 
The disagreement tag symbol fimction X specifies, 
for a given address, substructures with which its ar- 
gument disagrees. There is thus inconsistency if there 
is an address p in A such that 
,(p) e x(1,). (7) 
The disagreement address function Disagr., x frmn 
A to 2 ~:', based on v and X, takes an address as its 
argument, and gives the set of addresses with Milch 
the argument address must disagree, called the dis- 
agreement address set and defined as: 
Disagrv,x(P) := U Addr.(X), (8) 
Xex(v) 
Augmented terms are hereafter referred to simply 
as terms unless stated otherwise. 
Definition 6 Given a term ~ : (A,r,v,¢,X) and a 
feature address p in A, the subterm of/at the address 
p is the term tip = (A/p,r/p,v/p,~b/p,x/p) where 
rip :Jr* ~ T, v/p : Alp ~ V, ¢/p :2-" ~ 2 F, and 
X/P : .T" ~ 2 v are defined by 
(r/p)(q) := 7-(p-q), (9a) 
(v/p)(q) := v(p.q), (91) 
(¢/p)(q) := ¢(p.q), (9r) 
(X/P)(q) := X(P'q). (9(1) 
For a term t = (A, r, v, ¢, X), a type symbol a (sim- 
ilarly, a tag symbol or a term t') is said to occnr in t 
if there is a feature address p in ,X such that r(p) = a 
(similarly, v(p) = X or X (5 X(P), or lip = t'). 
A term t = (A r, v, ¢, X) is said to be regular if the 
set of all subterms of t, Subterm(t) := {t/p \] p (5 
A}, is finite, tlereafter, we will consider mdy regular 
terms. Ill a regular term, only finite numbers of type 
symbols and tag symbols occur. 
2For any set S, 2 s denotes the set of subsets of S. 
ACRES DE COLING-92, NANTES, 23-28 AOt3T 1992 3 8 2 PRO(=. OF COLING-92, NANTES, AUG. 23-28, 1992 
~e,apty 
Xl:{}:{} :dlist 
: \['"o,,, x2:{j...,~:{}:r \]x2 
Xa:{}:{}:dlist 
\[ X4:{}:{X6}:list \] 
= in \[first X5: {} : {} : T \] 
out X6 : {} : {X4} : list 
Figure 1: Examples of Augmented Terms ill Matrix 
Notation 
lem~ty tnonempty 
Xl:{} :{}:dlist X3:{}:{) :dlist 
,. (~o.t .Clio,,, x4:{}:{x6}.(_ .... _%. 
X2:{fi,'st}:{} list I x6:{}:{x4} 
ll.t li"~t I ii.,, 
xs:{}:{} 
T 
Figure 2: Examples of Augmented 'l~rms ill Directed 
Graph Notation 
In a term, any two feature addresses bearing tile 
same symbol are said to corefer. Thus, tile corefer- 
enee relation g of a terln is a relation defined on A ,as 
the kernel of the tag flnlctiou v; i.e, ~ := Ker(v) = 
v -I o v. IIere, g is an equivalence relation and a ~- 
class is called a corefereuee class. 
Definition 7 A terln t is referentially consistent if 
the same subtern* occurs at all feature addresses in a 
coreference class. 
If a term is referentially consistent, then by defini- 
tion, for any Ph p:Z E A, if v(pl) = v(p2) then, for all 
p such that Pt ' P C A, it follows that P2 ' P (5 A and 
v(pl " p) = v(p~ . p). Therefore, if a term is referen- 
tially consistent, g is a right-invariant eqnivalence or 
right-eongrueuee on A. That is, for any Pl, P2 E A, 
if Pt*¢P2 then (Pl ' P)~:(P2 ' P) for any p such that 
Pl .pEA. 
Definition 8 A well-formed term (wft) is a 
referentially-consistent regnlar term. The set of all 
well-formed terms is denoted by 14,'.TtrT. 
A term can be represented in matrix notation. Ex- 
amples of terms are showu in Figllre 1. In this figure, 
T, dlist and list are type symbols, in, out and .first 
are feature symbols, and X1, X2, ... are tag sym- 
bols. A matrix represents a set of feature-value pairs 
preceded by a tag symbol, followed by a set of iuhib- 
ited features and followed by a set of disagreement tag 
symbols. In the term te,,vlv, its snbterms at in and at 
out corefer while t,~o,,,,,vty is a term ill which its sub- 
terms at in aud at out should not corefer. The term 
te.m£1y should not have the feature address in ..first 
Willie tnonempty II&S that address, 
A term can also be represented by directed graphs 
(DGs). t~,,~,t~ anti t ........ ply in Figure 1 are shown as 
DGs in Figure 2. 
The set WY5 r of well-formed terms includes many 
terms that llave tile same type syml)ol function, tile 
same coreferenee relations, the same inhibited feature 
function, and the same disagreelnent address fllllC- 
lion but different tag symbol fiUlCtions. These terms 
have the same infornlation and can describe the same 
liugttistic object or tile same linguistic phenomena. 
These ternls construct equivalence classes by reualll- 
lug tag symbols in a certain manner. 
Delinltion 9 Two terlns tl = (Al,rl,Vl,¢q,?(1} 
and t~ = (A2, r2, V~, ~2, X-~) are altlhabetical variants 
of each other if and only if 
1. Al = A2, 
2. Ker(vl) = Ker(v2), 
3. rl = r2, 
4. ¢1 = ¢2, and 
5. Disagr~,,x ` = Disagr~,x ,. 
This is written as 11 ~t~. 
According to 'type-as-set' semantics, tile symbols 
T aud ± denote, respectively, tile le&st informative 
type tile whole universe U aud the overdefined or 
incousistel,cy type.--the empty set 0. Therefore, a 
term containing ± should be interpreted as inconsis 
tent. Such an inconsistency is called a type inconsis- 
tency. 'Ib treat such inconsistency, a relation 1~1 on 
W.~'T is llefiued as follows. 
Definition 10 For ally two terms tl, t=, G \]4,'.T'T, 
tl gl t2 if and mdy if.£ occurs in both tl and i 2. 
There are other kinds of inconsistency as mentioned 
earlier. If a term contains an address p such that 
¢){P)fq {f ~: J:'lp'f (~ A} i¢ 0, it is inconsistent 
because it means that there are features that should 
uot exist at. the address. Such an inconsistency is 
called a feature inconsistency. 
Ill addition, if a terln contains an address p such 
that v(p) E X(P), it is inconsistent because it means 
that tile subterm at p does not agree with itself. Such 
an inconsistency is called a tag illconsisteucy. 
llence, the three kinds of inconsistency are treated 
integratedly by a relation .~ on )4,'S'T delincd as fol- 
lows. 
Definition 11 For any two terms it, Z2 C W.T'T, 
tl U 12 if and ouly if each of them contaius at legist 
one address p such that 
t. r(p) : ±, 
2. ¢(p)n{f e Jlp.f e A} ¢ O, or 3..(p) e x(v). 
Clearly, if J~ occurs in a terln, it also occurs in all 
ternls in its ¢~-class. This is also trne for feature incon- 
sistency and tag inconsistency, lh.'nce, the relations (~ 
and -U are such that their union ~ becomes an equiv- 
alence relation. Thus, we call detincd the augnlented 
t-types as follows. 
Definition 12 An augmented &-tyl)e (or ~b-tyl)e for 
short) It\] is an element of tile quotient set, q~ := 
Syutactic structures of augmented g,-tyl)es will I)e 
(:ailed augmented ~p-ternls. An augmented typed- 
feature-structure Call t)e formalized as an anglllented 
t-type. 
The set of type symbols 7- has the partial order ~7- 
which denotes a subsumption relation between the 
set denoted by type symbols. The partial ordering 
on 7 can lie extended to augnmuted g~-terms and t- 
types. Tile sul)smnption orders on )&.T"T and on 
are ilefined t~s follows. 
Acrgs DE COLING-92, NANTES, 23-28 AO~' 1992 3 8 3 PROC. OF COLING-92, NAm'ES, AUG. 23-28, 1992 
Definition 13 Let tl = (AI, rt, vl,~bl, Xt) and t2 = (A2,r~,v2,C~,X2) 
be WFTs. il is said to be sub- 
sumed by t2, written tt _< i2, if and only if either 
tt ~J_or 
1. A~ __ At, 
2. Ker(v~) C_ Ker(vl), 
3. vp e Y', n(p) _<r r~(v), 4. Vp E 2-*, #:(p) _c ~t(p), 
~nd 
5. Yp (5 5 r*, Disagr~,x,(p) C_ Disagro~,x ~ (p). 
The subsumption order on • are defined by \[/1\] _< \[t2\] 
if tl _< t2 is well-defined. 
Lattice operations on • can be defined to be com- 
patible with the above subsumption order relation as 
follows. 
Theorem 1 If (7";_<7") is a lattice, then so is ~. 
Proof. This theorem can he proved in a very simi- 
lar manner to the counterpart for A'/t-Kaci's 0-terms. 
Therefore, instead of providing the proof in detail, 
only the definitions of the least upper bounds-- 
or joins--and greatest lower bounds~r meets--are 
provided below. Let t t : (ml,7"l,Pl,g\])l,XI) and 
t~ = (A~,r~,v2,ck2,X2) be WFTs. 
First, the join of t~ and t2, ta = tl V t2 = 
(Aa, ra, Va, ~ba, Xa), is defined as follows: 
Aa = Alna= (10a) 
va : Aa ---* ~1 such that 
Ker(va) = ~x Ntis, (lOb) 
and Vp E .T* 
rs(p) = rx(p) Vz T~(p), (10c) 
~ba(p) =-- (pl(p)N~b2(p), and (lOd) 
XS(P) = {us(q) I q E (Disagro,,xt(p) 
NDisagro~,x~ (p))}. (10e) 
Next, the meet of t, and t2, t4 = t, A t~ = 
(A4, r4, v4, ~b4, X4), is defined as follows: 
A 4 = At*\], (lla) 
v4 : A~ ~ I; such that 
Ker(v4) = r\[*l, (llb) 
and Vp G 9 r" 
r4(p) : VT{7"i(q)\]P~pq, i : 1,2},(lie) 
U{~i(q) lpnpq, i= 1,2}, (lid) ~(v) = 
and 
x4p) = 
where 
A\['\] = 
Al,,l = 
g\[.l = 
U{v4(q) lqaqr, 
r C (Disagrv~,×~(p) 
oDisagro~.×~(P)) }~1 le) 
co 
U At"l, 
n=0 { 
A1 UA~ for n = O, 
A \['-q U 
{p E 9 r  I ptct'lq, q E Ai"-q} 
for n > I, 
x\["\], 
n=0 
tempty V tnonempty 
X/:{} :{}:dlist 
= \[in XS:{}:(}:llst \] out 
X9 {} {} list 
~etnp\[y A ~Tlollelllpl~ 
X10 : {} : {} : dlist 
I Xll : {first} : {Xll} : list 
= it, \[first X12:{} : {} :T \] 
out Xll 
Figure 3: Examples of Join and Meet. of Augmented 
tb-Terms 
o /£2 ) ' 
for n = 0, 
i¢\['d = ~\[,- l\] O 
{(p~ • p,p~. v) I pl~t"-'lv2), 
for n> 1 
attd r~,uA~ is the rellexive extension of ~i from Ai 
to A1UA2 for i= 1, 2. 
The conditions (lla-lle) define a meet, that col- 
lapses to J- whenever conditions (lie--lie) l)roducc 
some address p such that type inconsistency, feature 
inconsistency, or tag inconsistency occurs at p. 
The V is a join operation and A is a meet operation 
which are compatible with the subsumptiou order de- 
fined in Definition 13. \[\] 
Examples of join and meet operations on aug- 
mented e-terms are shown in Figure 3. The join and 
meet operations on augmented ~-types correspond 
to the generalization and unification operations on 
TFSs. 
A'it-Kaei defines an ~-type as a maximal set of ~b- 
types. It is also possible to defir, e an augmented ~- 
type as a maximal set of augmented ~b-types in the 
same manner, making disjunctive and negative de- 
scriptions possible. 
3.2 Type Symbol Lattice Extension to 
Include Complement Type Symbols 
q_¥eating a negative desGil)tion of a given type syln- 
bol, say a, requires a type symbol I) such that b has 
only information that unification of it with a yiekls in- 
consistency, or such that aVT h = -V and aAT b = ±. 
Such a symbol is called a complement type symbol of 
a and written as a ~. If a given type symbol lattice 
(7-; _<7") is a Boolean lattice, that is, a comI)lcmented 3 
distributive lattice, we do not need to do anything. 
Otherwise, we nmst extend the lattice to include the 
cmnplements of the type symbols contained in the 
given lattice. 
For a finite type symbol lattice T, for example, 
a Boolean lattice T ~ can he constructed a.s follows. 
Let ..4 := {at ..... aN} be the set of atolns of 7-, 
that is, type symbols which cover j_.4 If there are 
,ton-atomic type symbols which cover only one sym- 
bol, for each such symbol a, a new atom is added 
aA lattice is called complemented if its all elements 
have complements.t3\] 
~a is said to cover b if b <7 a attd b <7 c <7- a 
implies e = b. 
Ac'IXS DE COLING-92, NANTES, 23-28 ^otrr 1992 3 8 4 Paoc. OF COLING-92, NANTES. AUG. 23-28. 1992 
tsymbol: node structure {a type symbol} 
arcs: ~a set of arc structures) __ 
~a set of feature symbols) 
es: ~a set of rtode structures) anoaes: fo~a~: ~a .odo s.nc*nro/ I NZL 
arc structure 
\[ #atn,~! I (a feat .... ymbol} 
\[ vM .... I {a node structure} 
Figure 4: Data Structures 
Fnnetion Unify(nodel, node~) 
begin 
node1 := Dereference( node l ); 
node~ := Det,e\]erence( node2 ); 
if node1 = node2 then 
return(node1); 
qodel .forward := node~; 
node2.tsymbol := nodel.tsymbol AT node2.tsymbol; 
if node2.tsymbol = J_ then 
return(J_) 
node2.ifeatures := nodel.i\]eatures LI node~.J\]eatures; 
if node2.ifeaturesr'l 
{arc.feature I arc • nodel .arcs LJ node2.arcs} 
# 0 then 
return(.L ); 
aodee.dnodes := node1 .dnodes O node2.dmMes; if {node1, node2} {7 node2.dnodes # ~ theai 
return(.L ); 
arcpairs := Shared-Arc-Pairs(node1, node~); 
for (arc1, arc2) in arcpairs do 
begin 
value := Unify( arcl .value, arce.value); 
if vMue = .1. then 
return(l); 
end; 
arcs :: Complement~Arcs(node1, node'2); 
rlodcS2.aFcs := arcs LJ llode~.arcs; 
return(node*); 
end 
Figure 5: A Destrnctive Graph Unification Function 
so that a covers all additional type symbol. The ex- 
tended lattice "T ~ is tile set of subsets of A with set 
inclusion ordering. An element {al}iet E "T' denotes 
Uie/\[al\]. The join and mcct operations on T' are 
the set-nniou and set-intersection operations, respec- 
tively. The complement of an element {ai}ie/ in T' 
is the set-complement of it with respect to .4, that is, 
{~ • .4 l a ¢ {ad,e~}. 
4 Implementation of Augmented TFS 
Unification 
The unification operation for augmented 1/,-terms or 
augmented TFSs has been implemented using graph 
unification techniques. A term structure is repre- 
sented as a directed graph by assigning a graph node 
to each x-class as in Figure 2. The unification oper- 
ation for such DGs corresponds to a graph merging 
operation. This takes two DGs and merges ~-cla.sses 
of the same feature-aAdress into a n-class. 
In a destructive graph unification method, which is 
very simple, suci~ a graph is represented by tile data 
structures in Figure 4. A node structure consists of 
live fields: lsymbol for a type symbol, arcs for a set 
of feature-vafile pairs, ifeatures for a set of inhibited 
features, dnodes for a set of disagreement nodes 
i.e., disagreement K-classes, and forward. The field 
for'warY1 is used for the Union-Find algoritfim\[9\] to 
calculate unions of K-classes in tile salne nlanner ,'Lq 
lluet's algorithm\[10}. By traversing two DGs' nodes 
with the same feature-address sinmltaneously, calcu- 
lating the union of their x-classes, and copying arcs, 
their unification can be calculated as in Figure 5. 
The function Unify takes two input nodes and puts 
them in a K-class by letting one input be tim forward 
field values. The flmction then examines three kinds 
of inconsistency; namely, type inconsistcncy, fea- 
ture inconsistency, and tag inconsistency. Tim fimc- 
tion finally treats arcs in order to make tile result 
graph right-cougruent. For treating arcs, tile function 
Unify assumes two fimctions, Shared_Arc_Pairs and 
Complement_Arcs. The function Shared_Arc_Pairs 
takes two nodes as its inpnts aud gives a set of 
arc pairs each consisting of both inputs' arcs with a 
shared feature. The flmctiou Complement_Arcs also 
takes two nodes and gives a set of arcs whose features 
exist in the first node but not in the second. 
An inhibited feature fimetion is implemented using 
tile tfeatnres field of nodes. When unification of two 
nodes results in a node witfi an arc witfi a feature in 
i features, it yields J- because of feature inconsistency. 
A disagreement tag symbol fnnetion is implemented 
using dnodes. Unification of two nodes which have 
each other in their dnodes yields 3. because of tag 
inconsistency, q_'hese computations require negligible 
additional computation. 
qb simplify the exphmation, the destructive version 
of graph unification is used above. Other versions 
based ou more efficient graph unillcation methods 
such ;~s Wroblewski's and Kogure's method\[23, 16\] 
have also been developed. 1,'urthermore, it is easy 
to modify other graph unification methods\[21, 6\] to 
allow augmented TFSs. 
,5 Conclusion 
\]'his paper has proposed an augmentatiotl of fea- 
ture structures {FSs) which introduces negative in- 
formation into FSs ill unification-based tbrmalisms. 
Unification-based linguistic formalisnm nse l".qs to de- 
scribe linguistic objects and phenotneua, l~ecanse lin- 
guistic information (:an |)e described compactly using 
disjunctive and uegatiw: descriptions, FSs and feao 
ture descriptions are required to treat such (lescrip- 
trans, in this paper, FSs have been augnlent.ed, using 
a promising method of fornudizat.iou, Ait-l(aci's $~ 
type, to allow three kinds of negatiw~ descriptions of 
them to be treated. 
In a formalizalion of typed feature structures, neg- 
ative descriptions can be decomposed rata three kinds 
of negations: negations of type sytnbols, negations of 
feature existences, aud llegations of feature-address 
value agreements. It. is shown thai the second and 
third kinds Call be treated by ailglncIItlllg tlrl'nl stlill% 
Lures to include structures representing such kinds of 
descriptions. Subsnmption relations on augmented 
terms are defined. It. is also shown that the first kind 
call be treated by exteuditlg type symbol lattices t() 
include complement type synd)ols. 
The proposed formalization cau provide efficient al- 
AcrEs DE COLING-92, NANTES, 23-28 AOl3"r 1992 3 8 $ PROC. OF COLING-92, Nhr, n'Es. AUG. 23-28. 1992 
gorithms for generalization and unification operations 
as well as treat primitive negations. The formaliza- 
tion can be integrated with logic-based frameworks 
such as \[20\] which can treat wider ranges of descrip- 
tions but which do not have such efficient algorithms 
for these operations. Logic-based frameworks can be 
used to obtain the data structures for this paper's 
formalization. 
Unification algorithms for augmented terms or aug- 
mented TFSs have been developed using graph uni- 
fication techniques. Unification programs based on 
these algorithms have been developed in Common 
Lisp. 
The augmentation of TFSs makes linguistic de- 
scriptions compact and easy to understand. In an 
HPSG-based grammar, for example, non-emptiness 
of a subcat or slash feature value can be easily de- 
scribed by nsing feature-address value disagreement. 
Moreover, negative descriptions make debugging pro- 
eessss of grammatical descriptions easier. 
Acknowledgments 
This research was performed in the VCAT project of 
the Takeuchi Research Group in NTT Basic Research 
Laboratories. The author would like to thank Ikuo 
Takeuehi, Akira Shimazu, Shozo Naito, Masahito 
Kawamori, Mikio Nakano, and other colleagues of the 
group for their encouragement and thought-provoking 
discussions. 
Ac-rEs DE COLING-92, NANTEs, 23-28 ^o~'r 1992 3 8 6 Paoc. oF COLING-92, NANTES, AUG, 23-28, 1992 

References 

\[1\] Hassan Ait-Kaei. An algebraic semantics approach 
to the effective resolution of type equations. Journal 
of Theoretical Computer Science, 45:293-351, 1986. 

\[2\] Hassan Ait-Kaci and Roger Nasr. Latin: a logic pro- 
gramming language with built-in inheritance. Jour- 
nal of Logic Programming, 3:185-215, 1986. 

\[3\] Garrett Birkhoff, Lattice Theory. Americau Mathe- 
matical Society, Providence, Rhode Island, USA, 3rd 
edition, 1967. 

\[4\] Bob Carpenter and Carl Pollard. htclusion, disjoint- 
hess and choice: the logic of linguistic classification. 
In Proceedings o\] the 29th Annual Meeting of the As- 
sociation for Computational Linguistics, pages 9-16, 
ACL, University of California, Berkeley, California, 
USA, 1991. 

\[5\] Annuj Dawar and K. Vijay-Shanker. A three-valued 
interpretation of negation in feature structure de- 
scriptions. In Proceedings of the 271h Annual Meet- 
ing of Association for Computational Linguistics, 
pages 18-24, ACL, Vancouver, British Columbia, 
Canada, 1989. 

\[6\] Martin Emele. Unification with lazy non-rednndant 
copying. In Proceedings o\] the ~9th Annual Meet- 
ing of the Association \]or Computational Linguistics, 
pages 325-330, ACL, University of California, Berke- 
ley, California, USA, 1991. 

\[7\] Martin Emele and Rdmi Zajac. RETIF: A Rewrit- 
ing System \]or Typed Feature Structures. Technical 
Report TR-I-0071, ATR, Kyoto, Japan, 1989. 

\[8\] Martin Emele and Rdmi Zajac. Typed unification 
grammars. In Proceedings of the 13th International 
Conference on Computational Linguistics, Vol. 3, 
pages 293-298, 1990. 

\[9\] J. E. ltopcroft and R. M. Karp. An Algorithm for 
Testing the Equivalence of Finite Automata. Tech- 
nical Report TR-71-114, Dept. of Computer Science, 
Cornell University, lthaca, New York, USA, 1971. 

\[10\] G~rard Huet. Rdsolution d'Equations dans des Lan- 
gages d'Order 1, 2, ..., w. PhD thesis, Universit6 de 
Paris VH, France, 1976. 

\[11\] Lauri Katttunen. Features and values. In Proceedings 
of the lOIh International Conference on Computa- 
tional Linguistics, pages 28-33, Stanford, California, 
USA, 1984. 

\[12\] Robert T. Kasper. Unification and classification: an 
experiment in information-hazed parsing. In Proceed- 
ings of the International Workshop on Parsing Tech- 
nologies, pages 1 7, Pittsbnrgh, Pennsylvania, USA, 
1989. 

\[13\] Robert T. Kasper and William C. Rounds. A logi- 
cal semantics for feature structure. In Proceedings of 
the 241h Annual Meeting o\] the Association for Com- 
putational Linguistics, ACL, New York, New York, 
USA, 1986. 

\[14\] Martin Kay. Parsiug in functional unitication gram- 
mar. In D. R. Dowty, editor, Natural Language Pars- 
in9, chapter 7, pages 251-278, Cambridge University 
Press, 1985. 

\[15\] Kiyoshi Kogure. Parsing Japanese spoken sentences 
based on HPSG. In Proceedings of the International 
Workshop on Parsing Technologies, pages 132-141, 
Pittsburgh, Pennsylvania, USA, 1989. 

\[16\] Kiyoshi Kogure. Strategic lazy incremental copy 
graph unification. In Proceedings of the 131h Inter- 
national Conference on Computational Linguistics, 
Vol. 2, pages 223-228, 1990. 

\[17\] M. Drew Moshier and William C. Rounds. A logic 
for partially specified data structures. In Proceedings 
of the ldth ACM Symposium on Principles of Pro- 
gramming Language, pages 156 167, Munich, West 
Germany, 1987. 

\[18\] Carl Pollard and Ivan Sag. An Information.Based 
Syntax and Semantics--Volume 1: bhndamentals. 
CSLI Lecture Notes Number 13, CSLI, 1987. 

\[19\] William C. Rounds and Robert T. Kasper. A com- 
plete logical calculus for record structures represent- 
ing linguistic information. In Proceedings of Sympo- 
sium on Logic in Computer Science, IEEE Computer 
Society, 1986. 

\[20\] Gert Smolka. A Feature Logic with Subsorts. Tech- 
nical Report LILAC Report 33, IBM Deutschland, 
7000 Stuttgart 80, West Germany, 1988. 

\[21\] Hideto Tomabechi. Quasi-destructive graph unifi- 
cation. In Proceedings of the 291h Annnal Meet- 
ing of the Association for Computational Linguistics, 
pages 315-322, ACL, University of California, Berke- 
ley, California, USA, 1991. 

\[22\] Yoshihiro Ueda and Kiyoshi Kogure. Generation 
for dialogue translation using typed feature struc- 
tnre unification. In Proceedings of the 13th h~ter. 
national Conference on Computational Linguistics, 
Vol. 1, pages 64-66, 1990. 

\[23\] David A. Wroblewski. Nondestructive graph unifi- 
cation. In Proceedings of the 6th National Confer- 
ence on Artificial Intelligence, pages 582-587, AAAI, 
Seattle, Washington, USA, 1987. 

\[24\] R6mi Zajac. A transfer model using a typed fea- 
ture structure rewriting system with inheritance. In 
Proceedings of the PTth Annual Meeting of Associa- 
tion for Computational Linguistics, pages 1-6, ACL, 
Vancouver, British Columbia, Canada, 1989. 
