AN INTERPRETATION OF NEGATION 
IN FEATURE STRUCTURE DESCRIPTIONS 
Anuj Dawar 
Department of Computer 
and Information Science 
University of Pennsylvania 
Philadelphia, PA 19104 
K. Vijay-Shanker 
Department of Computer 
and Information Science 
University of Delaware 
Newark, DE 19716 
Feature structures are informational elements that have been used in several linguistic theories and in 
computational systems for natural language processing. A logical calculus has been developed and used as a 
description language for feature structures. In the present work, a framework in three-valued logic is 
suggested for defining the semantics of a feature structure description language, allowing for a more 
complete set of logical operators. In particular, an interpretation of the negation and implication operators is 
examined within this framework. We extend this approach to interpret descriptions that involve existence (or 
nonexistence) of values for attributes. A definition of augmented feature structures is proposed, and one 
particular interpretation of the description language with a negation operator is described. A sound and 
complete proof system is presented for the logic thus obtained and its computational aspects studied. 
1 INTRODUCTION 
A number of linguistic theories and computational ap- 
proaches to parsing natural language have employed the 
notion of associating informational elements, consisting of 
features and their values, with phrases. Such elements, 
called feature structures, have been used in linguistic 
theories, such as Generalized Phrase Structure Grammar 
(GPSG; Gadzar et al. 1985) and Lexical Functional Gram- 
mar (Kaplan and Bresnan 1983), and in computational 
formalisms, such as Functional Unification Grammar (Kay 
1979) and PATR-II (Shieber 1984). 
Rounds and Kasper introduced a logical formalism to 
describe feature structures with disjunctive specification 
(Kasper 1987; Kasper and Rounds 1986; Rounds and 
Kasper 1986). The language is a form of modal proposi- 
tional logic. To define the semantics of this language, 
feature structures are formally defined as acyclic finite 
automata. The detailed definition is given in Section 2. A 
fundamental property of the semantics is that it is mono- 
tonic in the sense that the set of automata satisfying a given 
formula is upward-closed under the operation of subsump- 
tion. This is important, because we consider a formula to be 
only a partial description of a feature structure. This 
property is precisely formulated in Section 2. 
Several researchers have expressed a need for extending 
this logic to include the operators of negation and implica- 
tion. These two are related in that, in most logical systems, 
it is possible to use one to define the other (in the presence 
of a disjunction operator). In this paper, we shall concen- 
trate on the problem of extending the logic to include 
negation, while also showing that it yields a satisfactory 
interpretation of implication. 
Karttunen (1984), for instance, provides examples of 
feature structures in which a negation operator might be 
useful. For instance, the most natural way to represent the 
number and person attributes of a verb such as sleep would 
be to say that it is not third person singular, rather than 
expressing it as a disjunction of the other possibilities. We 
express this agreement constraint by the following formula: 
agreement : -1 (person : third A number : singular) (1) 
Pereira (1987) provides the following example formula 
that expresses the semantic constraint that the subject and 
object of a clause cannot be coreferential unless the object 
is a reflexive pronoun: 
Computational Linguistics Volume 16, Number 1, March 1990 11 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
obj : type : reflexive k/-7 (subj : ref ~ obj : ref) (2) 
This constraint can, in fact, be represented just as naturally 
as the following implication: 
(subj : ref ~ obj : ref) ~ obj : type : reflexive (3) 
Similarly, the feature co-occurrence constraints in GPSG 
(Gadzar et el. 1985) include implications of the form 
3 l ~ q~ (where q~ is some description). While a formula of 
the form 3 l is not part of the Rounds-Kasper logic, we 
intend it here as asserting the existence of a feature l in a 
structure. This would normally be expressed in the Rounds- 
Kasper formalism by the formula I:NIL. As we see later, 
formulae of the kind we have in this example, i.e. in which 
an existential appears negated, require special treatment 
and will motivate an extension to the feature structure 
formalism. 
Various interpretations have been suggested that define 
a semantics for these operators (see Section 3), but none 
has gained universal acceptance. Pereira (1987) set forth 
certain properties that any such interpretation should sat- 
isfy. We suggested that three-valued logic provides us with 
a framework appropriate for defining the semantics of a 
feature description logic (which we will call FDL) that 
includes a negation operator (Dawar and Vijay-Shanker 
1989). We also showed that the three-valued framework 
(based on Kleene's three-valued logic; Kleene 1952) is 
powerful enough to express most of the existing definitions 
of negation and implication. It is therefore possible to 
compare these different approaches. We also presented one 
particular three-valued interpretation for FDL, motivated 
by the approach to negation given by Karttunen (1984), 
that meets the conditions stated by Pereira. 
In the present work, we give an exposition of these 
results, and we also examine another three-valued interpre- 
tation for FDL, obtained by using a modified notion of the 
feature structures that serve as models. This new interpre- 
tation, while preserving the desirable properties of the 
previous one, also provides a satisfactory semantics for the 
problematic case, mentioned above, of formulae with a 
negated existential. 
In Section 2 we present an exposition of the Rounds- 
Kasper logic. In Section 3 we examine some existing ap- 
proaches to defining the semantics of negation, and we also 
present the framework of three-valued logic within which 
we define our own interpretation. In Section 4 we exhibit 
the modified notion of feature structures as models for 
FDL, and we give the semantics of FDL in terms of these 
modified feature structures. Finally, in Section 5, we present 
a proof system for the language and esta61ish some compu- 
tational results. 
2 ROUNDS-KASPER LOGIC 
In this section, we take a look at the calculus developed by 
Rounds and Kasper to describe feature structures. The 
symbols in the language are taken from two primitive 
domains: 
1. Atoms ( A ), and 
2. Labels ( L ). 
The set of well-formed formulae (W), is given by: 
NIl 
TOP 
a where a ~ A 
l:q~ wherel~Land~ W 
~b/\ ~k where ~b, ~b ~ W 
q5 V ~b where 4~, ~b ~ W 
Pl '= P2 wherepl, P2 ~ L* 
To define the semantics of this language, feature struc- 
tures are defined as acyclic finite automata. These are 
formally defined as follows: 
Delinition 1. An acyelie finite automation is a 7-tuple 
A = (Q,~,F,&qo, F,?O, where: 
1. Q is a nonempty finite set (of states), 
2. z is a countable set (the alphabet), 
3. F is a countable set (the output alphabet), 
4. 6 : Q x ~ ---, Q is a finite partial function (the 
transition function), 
5. qo ~ Q (the initial state), 
6. F C Q (the set of final states), 
7. X : F --~ r is a total function (the output function), 1 
8. the directed graph (Q, E) is acyclic, where pEq iff 
for some l ~ ~, 6(p, l) = q, 
9. for every q G Q, there exists a directed path from q0 
to q in (Q, E), and 
10. for every q C F, 6(q, l) is not defined for any 1. 
We can define a partial ordering of information on 
acyclic finite automata. This partial ordering is given by 
the subsumption relation, defined as follows: 
Definition 2. Given two acyclic finite automata, A = 
( QA, ~"A, I'A, 6A, qOA, FA, ~kA ) and B = ( Qs, ~,s, rs, 6s, qos, 
Fs, ),s), we say that A subsumes B (A E B) iff there is a 
homomorphism from A to B, i.e. there is a mapping h : 
QA ~ Qn such that: 
1. h(6A( q, l) = tro(h(q), l), 
2. tB(h(q)) = 1 A (q) for all q E FA, and 
3. h(qoA) = qoB 
Unification, which is the primary information-combining 
operation on feature structures, can now be simply defined 
as the operation of finding a least upper-bound (if any 
upper-bound exists) under the above ordering. 2 
We can now give the semantics of a formula over the set 
of labels L and the set of atoms A. The domain over which 
this is done is the set of acyclic finite automata A ----- Q, L, 
A, 6, q0, F, X). The satisfies relation (~) is defined as 
follows: 
Definition 3. An acyclic finite automation A = (Q, L, A, 
6, q0, F, X) satisfies (~) a formula in the following cases: 
12 Cnmputational Linguistics Volume 16, Number 1, March 1990 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
A ~ NIL always 
A ~ TOP never 
A~ a iffQ=F={qo}and~(qo)=a 
A ~ l:4) iff A / l ~ 4) 
A~ 4)/~p iffA~ 4)andA~ 
A ~ 4) V ~b iffA ~ 4)orA ~ ~p 
A ~ PI ~- P2 iff(i(q0, Pl) = 5(qo, P2) 
In the above, 6 is extended in the standard way to members 
of Z*, i.e. (5(q, E) = q and 5(q, wl) = ~(6(q, w),l) and A/I is 
the automaton obtained from A by making (5(q 0, l) the 
initial state and eliminating all unreachable states. 
A fundamental property of the semantics given above is 
that the set of automata satisfying a given formula is 
upward-closed under the operation of subsumption. The 
property is stated in the following theorem (Rounds and 
Kasper 1986): 
Theorem 1. A E_ B if and only if for every formula, 4), if 
A~ 4) thenB~ 4). 
Rounds and Kasper also showed that the satisfiability 
problem for their logic is NP-complete. 
3 PREVIOUS APPROACHES TO NEGATION 
In this section we examine the problem of adding a nega- 
tion operator to the language described in the previous 
section. We do this by presenting various approaches to 
defining the semantics of the extended language. We look 
at these approaches in terms of both their linguistic appro- 
priateness and their computational properties. We will also 
show that the framework of three-valued logic that we 
present can be used as a basis for comparison of the 
different approaches. 
3.1 CLASSICAL NEGATION 
By classical negation, we mean an interpretation in which 
an automaton A satisfies a formula --7 4) if and only if it does 
not satisfy 4). Johnson (1987) defined an Attribute Value 
Logic (AVL), similar to the Rounds-Kasper Logic, that 
included a classical form of negation. Smolka (1988) pre- 
sented a classical semantics for negation in a Rounds- 
Kasper-like framework. While such approaches are appro- 
priate under one view of feature structures, they are not 
satisfactory from the viewpoint of feature structures seen 
as partial descriptions. This is because the crucial property 
of monotonicity is lost, as can be seen from the following 
example: 
Example 1. 
A = \[person:second\] 
\[person:second\] 
B = \[number : singularJ 
4) = -q(person : second/~ number : singular) 
As can easily be seen, by the classical semantics, A ~ 4) 
and A E B, but B ~ 4). 
Kasper (1988a) discusses an interpretation of negation 
and implication in an implementation of Functional Unifi- 
cation Grammar that is extended to include conditionals. 
Kasper's semantics is classical, but his unification proce- 
dure uses notions similar to those of three-valued logic. 3 
Kasper also localized the effects of negation by disallowing 
path expressions within the scope of a negation. This 
restriction may not be linguistically warranted as can be 
seen from Pereira's formula example in Section 1. 
3.2 INTUITIONISTIC LOGIC 
Moshier and Rounds (1987) described an extension of the 
Rounds-Kasper logic, including an implication operator 
and hence, by extension, negation. The semantics is based 
on intuitionistic techniques. The notion of satisfying is 
replaced by one of forcing. Given a set of automata K, a 
formula 4), and A such that A ~ K, A forces in K ~4) (A F K 
---~) if and only if for all B E K such that A E_ B, B does not 
force 4) in K. Thus, to show that a formula, 4), is satisfiable, 
we have to find a set K and an automaton A such that A 
forces in K 4). 
Moshier and Rounds also gave a complete proof system 
for their logic, and showed that the satisfiability problem, 
while decidable, was PSPACE-complete, thus making it 
even more intractable than the original Rounds-Kasper 
logic. Furthermore, Langholm (1989) has shown that not 
all formulae in the Moshier-Rounds logic can have heredi- 
tarily finite sets of minimal models. These computational 
problems, along with questions about the linguistic appro- 
priateness of its semantics, render the linguistic value of the 
intuitionistic approach questionable. 
3.3 THREE-VALUED LOGIC 
Here we take a look at how three-valued logic can be used 
to define the semantics of FDL. We also take a look at one 
particular interpretation of FDL that uses the automata of 
Section 2 as models. This interpretation is essentially the 
same one we presented earlier (Dawar and Vijay-Shanker 
1989). This is an interpretation of negation that is intu- 
itively appealing, formally simple, and computationally rto 
harder than the original Rounds-Kasper logic. The pri- 
mary intention here (as in our earlier paper) is, however, to 
explore the use of three-valued logic in defining the seman- 
tics of FDL with negation. To this end, we will examine 
other interpretations also within the three-valued frame- 
work. Then, in the next section, we motivate a modified 
notion of automata models and redefine our interpretation 
with respect to it. 
3.3.1 THE THREE-VALUED FRAMEWORK 
With each formula we associate the set (Tset) of automata 
that satisfy the formula, a set (Fset) of automata that 
contradict it, and a set (Uset) of automata that neither 
satisfy nor contradict it. 4 The Uset contains all automata 
that are not in either of the other two sets. Different 
interpretations of negation are obtained by varying defini- 
tions of what constitutes "contradiction." The reason for 
Computational Linguistics Volume 16, Number 1, March 1990 113 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
having some automata that neither satisfy nor contradict a 
formula is as follows: an automaton is to be viewed as a 
partial information structure. Given a description (for- 
mula), 4~, a feature structure A may not carry enough 
information to suggest that it satisfies or falsifies 4~. How- 
ever, it may be possible to extend A to either satisfy or 
falsify ~b. For example, we will place \[person : third\] in the 
Uset of ~b = (number : singular/k person : third). Of 
course, this feature structure can be extended to falsify or 
satisfy 4~ as in: 
and 
pnumber : singular\] 
erson : third J 
number : plural\] 
erson : third J 
We will define the Tset and the Fset so that they are 
upward-closed with respect to subsumption for all formu- 
lae. Thus, we avoid the problem of nonmonotonicity associ- 
ated with the classical interpretation of negation. In our 
logic, negation is defined so that an automaton A satisfies 
--~ if and only if it contradicts q~. 
Formally, the semantics is defined by a partial interpre- 
tation function, I. If WFF is the set of well-formed formu- 
lae of FDL, and A the set of acyclic finite automata, 5 the 
interpretation I is a partial function: 
I : WFF x A ~{True, False} 
I(4~, A) is True iff A satisfied ~b. It is False if A contra- 
dicts ~b 6 and is undefined otherwise. Thus, the following 
hold: 
Tset(~) = {AIt(~, A) -- True} and 
Fset(ck) = {AIt(~, A) = False} 
3.3.2 A THREE-VALUED INTERPRETATION 
We now look at one such interpretation function that uses 
the strong Kleene truth definition for conjunction and 
disjunction. 
Definition 4. The partial interpretation function I is 
defined as follows: 
1. I(NIL, A) = True for all A; 
2. I(TOP, A) = False for all A; 
3. I(a, A) = True 
ifA is atomic and ~'(q0) = a 
I(a, A) = False 
if A is atomic and X(qo) = b 
for some b, b #: a (see Note 2.) 
l(a, A) is undefined otherwise; 
4. I(l : ~p, A) = I(dp, All) if All is defined. 
(see Note 3.) 
I(l : ep, A) is undefined otherwise; 
5. I(dp I A qb2, A) = True 
if I(q~ I, A) = True and I(qb 2, A) = True 
I(~b t A q~2, A) = False 
if I(~b l, A) = False or I(~b2, A) = False 
l(~b I A 4~2, A) is undefined otherwise; 
6. /(~b I V ~b2, A) = True 
if I(~b x, A)= True orI(ep 2, A) = True 
/(~1 V ~2' A) = False 
if I(~bl, A) = False and I(~b2, A) = False 
/(q~l V ~b2, A) is undefined otherwise; 
7. I(--CA A) = Trueif I(ep, A) = False 
I(--@, A) = False ifI(~b, A) = True 
I(-~, A) is undefined otherwise; 
8. I(p I ~- p2, A) = True 
if 6(%, Pl) and 6(qo, P2) are defined 
and 6(%, Pl) = 6(%, P2) 
I(Pl ~ P2, ,4) = False 
if Alp I and Alp 2 are both defined 
and are not unifiable 
I(Pl ~ P2, A) is undefined otherwise (see Note 4.). 
where, 
q~, 4h, ~2 E WFF 
A = <Q,L,A, 6, qo, F,X>CA 
a,b@A 
l~L 
.Pl, P2 ~ L* 
NOTES 
1. We have not included an implication operator in the 
formal language, since we find that defining implication 
in terms of negation and disjunction (i.e. 4~ =~ ~k---- 
--~ v if) yields a semantics for implication that corre- 
sponds exactly to our intuitive understanding of implica- 
tion. 
2. As one would expect, an atomic formula is satisfied by 
the corresponding atomic feature structure. On the 
other hand, only atomic feature structures are defined as 
contradicting an atomic formula. An interpretation of 
negation that defines a complex feature structure as 
contradiciting a (and hence satisfying ---a) is also possi- 
ble. Our definition was motivated by the linguistic inten- 
tion of the negation operator as given by Karttunen 
(1984), where, for instance, we require that an automa- 
ton satisfying the formula case : --ndative have an atomic 
value for the case feature. However, we now feel that 
this problem would best be dealt with in a multi-sorted 
logic and hence, in the interpretation we present in the 
next section, we have adopted the other alternative 
mentioned here. 
3. In definition 4 above, we state that: I(1 : ep, A) = I(¢b, A~ 
l) ifA/l is defined. When A/l is defined, I(~, A/l) may 
still be True, False, or undefined. In any of these cases, 
I(1: cp, A) = l(dp, All). 7 I(l : dp, A) is not defined if A/l 
is not defined (as illustrated by the example given earlier 
where ~b ---- (person : third A number : singular). Not 
only is this condition required to preserve upward clo- 
sure, it is also linguistically motivated. 
14 Computational Linguistics Volume 16, Number 1, March 1990 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
In the next section, we will make the distinction in 
feature structures between not being defined versus 
cannot be defined. In this section, we will say that 
I(l : 4~, A) is not defined if A is not defined for l and 
I(l : 4~, A) = false if A cannot be defined for I. 
4. We have chosen to state that the set of automata that 
are incompatible with the formula Pl ~- P2 is not the set 
of automata for which 6(q0, P0 and 6(q0, P2) are defined 
and ~(qo, Pl) :~ 6(q0, P2), since such an automaton could 
subsume one in which 6(q0, Pl) = 6(qo, P2). Thus, we 
would lose the property of upward closure under sub- 
sumption. However, an automaton, A, in which/~(qo, Pl) 
and 6(qo, P2) are defined, and AlP l is not unifiable 8 with 
Alp 2 cannot subsume one in which tS(qo, Pl) = 6(q0, P2). 
The monotonicity property for the above interpretation 
can be stated as follows: 
Theorem 2. Tset(qb) is upward-closed under the subsump- 
tion relation for all formulae ~b. 
Proof. The proof is by induction on the structure of 4~ and 
can be found in Dawar 1988.\[7 
We now take a look at some examples mentioned earlier 
and see how they are interpreted in the logic just defined. 
The first example expressed the agreement attribute of the 
verb sleep by the following formula: 
agreement : 7(person : third/k number : singular) (4) 
This formula is satisfied by any structure that has an 
agreement feature which, in turn, either has a person 
feature with a value other than third, or a number feature 
with a value other than singular. Thus, for instance, the 
first two structures satisfy the given formula, whereas the 
third structure is undefined with respect to the formula. 
cat : NP : second\]\] agreement : \[person 
\[agreement \[person: third \]\] 
: \[number : plural J\] 
cat : NP \] 
agreement : \[person : third\] 
On the other hand, for a structure to contradict formula 
(4), it must have an agreement feature defined for both 
person and number with values third and singular respec- 
tively. 
Turning to another example mentioned earlier, the for- 
mula: 
obj : type : reflexive V --l( subj : ref ~. obj : ref ) (5) 
is satisfied by the first two of the following structures, but is 
contradicted by the third (here co-index boxes are used to 
indicate co-reference of path-equivalence). 
lob j: \[type : reflexive\]\] 
\[ref: \[\] 
obj : l \[type : reflexive 
subj : \[ref : \['iq \] 
\[ref:lq3 11 °"-,: b,,,e :,,on,-e.x,vd / 
subj : \[ref : \[\] \] J 
3.3.3 OTHER THREE-VALUED INTERPRETATIONS 
OF NEGATION 
We briefly examine here how the three-valued framework 
may be used to provide interpretations other than the one 
presented above. 
The classical interpretation of negation can, of course, be 
expressed by making I a total function such that wherever 
I(qL A) was previously undefined, it is now defined to be 
False. 
Moshier and Rounds consider a version in which forcing 
is always done with respect to the set of all automata, i.e. 
K*. This means that the set of feature structures that 
satisfy --~ is the largest upward-closed set of feature struc- 
tures that do not satisfy ~b (i.e. the set of feature structures 
incompatible with q~). We can capture this in the three- 
valued framework described above by modifying the defini- 
tion of I in the following cases: 
• I(a, A) = True 
if A is atomic and X(q0 ) ---- a 
I(a, A) = False otherwise 
• I(1 : 4~, A) = True 
if A/l is defined and I(~b, A/l) = True 
I ( l : 4~, A) = False 
if A/I is defined and 
VB(A/I E_ B =~ I(cb, B) = False) 
I(1 : qb, A) is undefined otherwise. 
• I(¢1 A ¢2, A) = True 
ifI(q~l, A)= True andI(qb 2, A)= True 
1(¢ 1 A 02, A) = False 
if 
VB(A E_ B ~ I(c~1, B) ~ True or l(~b2, B) ~ True) 
I($1/k 4~2, A) is defined otherwise; 
• I(¢ l W ¢2, A) = True 
ifI(4h, A) = Trueor I($2, A) = True 
I(4a I V ¢b2, A) = False 
if 
VB(A E B =~ I(c~ l, B) ~ TrueandI(4~2, B) ~ True) 
I(~b I /k 4~2, A) is undefined otherwise; 
• I(p I ~-p2, A)= True 
if 6(qo, Pl) and b(qo, P2) are defined 
and 6(q o, Pl) = 5(qo, P2) 
Computational Linguistics Volume 16, Number 1, March 1990 15 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
I(pl "~ P2, A) = False 
if Alp 1 and AlP 2 are both defined 
and are not unifiable or if A is atomic 
l(Pi ~" Pc, A) is undefined otherwise. 
As mentioned earlier, our approach was motivated by 
Karttunen's implementation as described in Karttunen 
1984. In the unification algorithm given, negative con- 
straints are attached to feature structures or automata 
(which themselves do not have any negative values). When 
the feature structure is extended to have enough informa- 
tion to determine whether it satisfies or falsifies the for- 
mula, then the constraints may be dropped. We feel that 
our definition of the Uset captures the notion of associating 
constraints with automata that do not have sufficient infor- 
mation to determine whether they satisfy or contradict a 
given formula. 
As discussed in Section 3.1, Kasper (1988a) used the 
operations of negation and implication in extending Func- 
tional Unification Grammar. Though the semantics defined 
for these operators is a classical one, for the purposes of the 
algorithm Kasper identified three classes of automata asso- 
ciated with any formula: those that satisfy it, those that are 
incompatible with it, and those that are merely compatible 
with it. We can observe that these are closely related to our 
Tset, Fset, and Uset respectively. For instance, Kasper 
states that an automaton A satisfies a formula f : v if it is 
defined forfwith value v; it is incompatible with f: v if it is 
defined forfwith value x(x ~ v) and it is merely compati- 
ble with f: v if it is not defined forf. In three-valued logic, 
we incorporate these notions into the formal semantics, 
thus providing a formal basis for the unification procedure 
given by Kasper. Our logic also gives a more uniform 
treatment to the negation operator, since we have removed 
the restriction that disallowed path equivalences in the 
scope of a negation. 
4 INTERPRETING FDL WITH AUGMENTED 
FEATURE STRUCTURES 
We have seen examples (Section 1) of formulae that assert 
the existence of certain features. While 31 is not a formula 
in the Rounds-Kasper syntax, we can regard it as syntactic 
sugar for the formula I:NIL, which is indeed satisfied 
exactly by those automata that have a feature l. 
However, the formula -d:NIL is not satisfiable in the 
logic we have defined. This is because any automaton that 
does not have a feature labeled l subsumes one that does. 
We have, however, seen examples of formulae where 3 l 
occurs in the scope of a negation (for instance, Kasper 
\[1988b\] uses the formula 3Mood-type--, Rank : Clause). 
We certainly intend that such formulae be satisfiable. 
Since feature structures are partial information struc- 
tures, if they are not defined for an attribute l, it could be 
due to lack of information about the value for the attribute 
l. On the other hand, here we wish to capture the fact that if 
a feature structure A satisfies the description --73 l, then not 
only is A not defined for l, but it is also the case that it 
cannot be defined for l. That is, it is erroneous to extend A 
to state a value for the attribute I. 
The problem stems from the fact that in the formula 
-7--1l, we are trying to capture the information that a 
feature structure not only does not have a value for the 
feature l, but cannot be extended to have a value for l; i.e. 
we have the information that, in the current context, the 
information structure that we are building is not going to 
acquire a value for the feature l at any future time. This 
kind of "negative" information is not expressible in auto- 
mata models as we have defined them. As they stand, they 
can only capture "positive" information. To include the 
negative information we need, we will define an augmented 
notion of feature structures and redefine our interpretation 
function accordingly. 
To use the analogy with finite state automata, note that 
in a deterministic fsa we often consider states that do not 
have outgoing arcs defined on certain labels as having those 
arcs leading to an "error" state. Since we view fsas as 
complete structures, this distinction between arcs that are 
not defined and those that cannot be defined is unimpor- 
tant. However, when we view our automata models as 
partial information structures, we must distinguish be- 
tween the case in which a feature is simply not defined 
(leaving open the possibility that it may be defined in some 
extension) and the case in which we know that a certain 
feature cannot be defined. 
In what follows, we capture the information of certain 
labels leading to "error" states without explicitly defining 
such states, but by attaching to each state in the structure a 
finite set of labels. This set contains those labels that cannot 
be defined from that state. We already have an elementary 
form of this notion in our restriction on final states, when 
we specify that they cannot have any outgoing arcs. We are 
effectively saying that no label can be defined from these 
states. We formalize all these notions below. 
4.1 AUGMENTED FEATURE STRUCTURES 
In this section, we give definitions relating to our aug- 
mented notion of f-structures. As we stated above, the 
augmentation consists of attaching to each nonfinal node in 
the f-structure graph a finite set of labels. These labels are 
exactly those for which we know that no outgoing arcs can 
be defined from that node. The set is finite since we require 
that our information structure at any point be finite. We 
formally define our extended notion of f-structure as 
follows: 9 
Definition 5. An acyclic finite automaton is an 8-tuple 
,4 == (Q,~,F, 6, qo, F,X,S), where: 
1. Q is a nonempty finite set (of states), 
2. 2; is a countable set (the alphabet), 
3. F is a countable set (the output alphabet), 
4. 6:Q x Z ~ Q is a finite partial function (the 
transition function), 
5. qo E Q (the initial state), 
16 Computational Linguistics Volume 16, Number 1, March 1990 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
6. F C Q (the set of final states), 
7. X : F ~ r is a total function (the output function), 
8. S:Q\F--* P'~"~Y~) is a function from the nonfinal 
states to finite subsets of Z, 
9. the directed graph (Q, E) is acyclic, where pEq iff 
for some I ~ Z, 6(p, l) = q, 
10. for every q ~ Q, there exists a directed path from q0 
toqin (Q,E), 
11. for every q ~ F, 6(q, l) is not defined for any l, and 
12. whenever I ~ S(q), 6(q, l) is not defined. 
We can now define the subsumption ordering on these 
structures as follows: 
Definition 6. Given two f-structures, A = ( Q.1, ~.1, rA, 61 , 
q0.1, FA, ~.1, S.1) and B = (Qn, ~n, F~, 6n, qon, Fn, ~n, Sn), 
we say that A subsumes B (A~B) iff there is 
a homomorphism from A to B, i.e. there is a mapping 
h: Q.1 --* Qs such that: 
1. h(qo.1) = qon, 
2. h(6A(q, l)) = 6 n(h(q), 1), 
3. Xn(h(q)) = ~ (q) for all q ~ F~, and 
4. SA(q) C Ss(h(q)) for all q ~ Q.1\F.1. 
This definition of subsumption ensures that, for any 
automaton A, if I ~ S.1(6(q o, p)) then, for any automaton 
subsumed by A, the pathp is defined, but the path pl cannot 
be defined. 
4.2 THE LANGUAGE 
We now give the interpretation of FDL in terms of f- 
structure models as we have just defined them. The syntax 
of the language is the same as before. 
We first give the following auxiliary definitions: 
Definition 7. An f-structure A = (Q, 2;, I', 6, qo, F, ~, S) 
is: 
• atomic if and only if Q = F = {q0 }, 
• null if and only if Q = {q0} and F = ~ and 
• complex otherwise. 
We can now define the revised semantics: 
Definition 8. The (revised) partial interpretation function 
I is defined as follows: 
1. I(NIL, A) = True for all A; 
2. I( TOP, A) = False for all A; 
3. I(a, A) = True ifA is atomic and k(q0) = a 
I(a, A ) = False if A is atomic with k (q0) :~ a or if A 
is complex 
l(a, A ) is undefined otherwise; 
4. I(l:dp, A) = I(4~, A/l) if A/l is defined. 
I(l: 4~, A ) = False if l C S( qo ) or if A is atomic 
I( l : 4~, A ) is undefined otherwise; 
5. l(ckl /~ 42, A) = True 
if 1(q~1, A ) = True and l(q~2, A ) = True 
I(4~ A (a2, A ) = False 
if l(4h, A ) = False or I(4~2, A ) = False 
I(~b I /~ ~b2, A ) is undefined otherwise; 
6. I(q~l V q~2, A) = True 
if I(~bl, A) = True or I(q~2, A ) = True 
I(q~l V 4~2, A ) = False 
if I(4h, A) = False and I(q~2, A) = False 
I(~b~ V 4~2, A ) is undefined otherwise; 
7. I(--~, A ) = True if I(4,, A ) = False 
I(-74~, A ) = False if I(q~, A ) = True 
I(---~, A ) is undefined otherwise; 
8. I(p I ~ P2, A) = True 
if 6(qo, Px ) and ~(q0, P2) are defined 
and 6(q o, Pl ) = 6(qo, P2) 
I(Pl ~" P2, A ) = False 
if Alp x and A/p 2 are both defined and are not 
unifiable or 
Pl = wlx and 1 ~ S(6(q o, w)), or 
P2 = wlx and I ~ S(6(q o, w)) 
I(p~ ~- P2, A ) is undefined otherwise (see Note 4). 
where, 
~b, q~l, ~2 ~ WFF 
A = (Q,L,A, 6, qo, F,X ) ~A 
a, bEA 
I~L 
Pl, P2, w, x E L* 
We are now in a position to prove the following monoto- 
nicity property for our logic. We express it in terms of the 
knowledge (or information) ordering --<k on the truth values 
{_1_, True, False } defined by _1_ <k True, _1_ k False, True ~t k 
False and False ~t k True. In the following, I(~b, A) = _t_ is 
used for I(4>, A ) undefined. 
Theorem 3.1 A E B if and only if for every formula, q~, 
I(q~, A ) -<k I(~b, B). 
Proof. ~ Suppose for every formula, 4~, I(q~, A) --<k 
I(~b, B). Every pathp defined in A must also be defined in 
B, since I(p :NIL, A ) = True and hence I(p :NIL, B) = 
True. Since for every state qi in A, there is a path pf such 
that qi = 6A(qOA, Pi) we can define a map h such that 
h(qi) = 6B(qon, Pi ). To see that this map is indeed 
functional, note that, if there is a q E QA such that q = 
6A(q0A, Pl) = 6A(q0.1, P2) for distinct Pl and P2, then 
I(Pl ~" P2, A) = True. Thus l(p 1 ~ P2, B) = True and 
6B(qos, Pl) and 6n(qo s, P2) do indeed describe the same 
state. 
One can immediately see that this map satisfies proper- 
ties 1 and 2 of being a homomorphism given above in the 
definition of subsumption. To verify the other two condi- 
tions, note that if hA(qi) = a for some qi E A, then, 
I(pt:a, A) = True. Hence I(pi:a , B) = True and 
XB(6n(q0n, Pi)) = a. Thus condition 3 is satisfied. The 
argument for condition 4 is similar. We have, therefore, 
established that h is a homomorphism and hence that A 
v-- B. =~ The consequent is trivially true with I(4~, A) = 
_1_, so we will only consider the case when it is either True 
Computational Linguistics Volume 16, Number 1, March 1990 17 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
or False. The proof is by induction on the structure of the 
formula. 
Basis: 
NIL 
Trivial, since I(NIL, A ) = True, for all A. 
TOP 
Trivial, since I( TOP, A ) = False, for all A. 
a 
Note that ifA is atomic and A C B, then A = B. 1° 
Thus, if I(a, A) = True, then A = B and we are done. 
If I(a, A ) = False, either A is atomic and the argument 
is the same as before, or A is complex. But then, since 
A E B, B is also complex and I(a, B) = False. 
Pl ~-P2 
If I(p I ~ P2, h ) = True then there is a q E QA such 
that q = 6 A (qoa, Pl ) = 6a(qoA, P2)" Let h be a homo- 
morphism witnessing A E_ B. Then, by the definition of a 
homomorphism, h(q) = 6n(qon, Pl ) = 6B(qon, P2) and 
therefore, I(p I ~- P2, B) = True. 
In the case in which l(p I ~ P2 , A ) = False, we have 
two possibilities. Either Alp I and Alp 2 are both defined 
and not unifiable, in which case, clearly by the definition 
of subsumption, the same will be true of B, or pl = wlx 
(choosing Px without loss of generality), for some label l 
and some paths (possibly empty) w and x such that I E 
Sa(6a(qo A, w)). But then, as we pointed out earlier, this 
would mean that the path wl and hence the path Pl can- 
not be defined in B either. Thus, in either case, l(pl ~- 
P2, B) = False. 
Induction Step: 
1:4, 
Since A r"- B, if All is defined, so is B/I and All E B/1. 
But then, by induction hypothesis, I(ep, All) <-k I \[(4~, 
B/l) and therefore I(1 : c~, A) <-k I\] (l:q~, B). 
If All is not defined and I(l:¢k, A) = False, one of two 
possible cases applies: either A is atomic, in which case 
A = B or l E Sa(qoA), in which case I E Sn(qo B) by 
the definition of subsumption, and we are done. 
If I(4~/~ ~b, A ) = True, then I(4~, A ) = True and 
I(~b, A) = True. But then, by induction hypothesis, 
I(4~, B) = True and I(~b, B) = True. Thus 
I(~b /~ ~k, B) = True. 
Similarly, if I(4~ A ~b, A ) = False, I(cb, A ) = False or 
I(~b, A ) = False. Hence, by induction hypothesis, 
I(~, B) = False or I(ff, A ) = False, and therefore 
I(q~ /~ ¢z, B) = False. 
The argument is similar to the one in the previous case. --q, 
Since I(--~, A ) = True if and only if I(cb, A ) = False 
and vice versa, clearly I(--~, A ) __<k I(--~, B), since 
I(~b, A) _<k I(4~, B).~ 
The following simple corollary corresponds to the mono- 
tonicity result we established for our original three-valued 
semantics. 
Corollary. For all 4~, Tset(4~) is an upward-closed set. 
As we mentioned earlier in this section, Langholm (1989) 
describes negatively extended feature structures in a fash- 
ion very similar to what is described above. The interpreta- 
tion he chooses for the description language is, however, 
intuRionistic in character. We believe that the modifica- 
tions that we suggested to our interpretation (in Section 
3.3.3) to capture the special case of intuitionistic logic in 
which forcing is always done with respect to K*, when 
applied to our new interpretation yield exactly the interpre- 
tation described by Langholm. 
5 PROOF SYSTEM 
In this section, we give a proof system for the logic de- 
scribed above that is essentially an adaptation of the tab- 
leau proof system described by Moshier and Rounds (1987) 
for their intuitionistic interpretation of the feature logic. 
The proof system works, not with individual formulae, 
but with sets of labeled signed formulae. The Moshier- 
Rounds tableau proof method worked with sets of sets of 
labeled signed formulae. However, this extra level of com- 
plexity !is not needed here. 
We first introduce the notion of a labeled signed formula: 
Definition 9. A labeled signed formula is a triplet 
(w,X, q~), where w ~ L *, X ~ {True, False} and q~ 
WFF. (w, True, dp) will be written as wT4~, and 
( w, False, 4a ) as wF4~. 
We can now define the notion of an f-structure satisfy- 
ing a labeled signed formulae: 
Definition 10. An f-structure, A, satisfies a labeled signed 
formula ~ (written A ~ ~) in the following cases: 
A t = wTep if and only ifA/w is defined and I(4~,A/w) = 
True 
A t = wFcb if and only ifA/w is defined and I(4~,A/w) = 
False 
Definition 11. A set c of labeled signed formulae is closed 
if and only if at least one of the following holds: 
• wT4~, wF4~ ~ c, 
• wTa, wTb C c, 
• wTa, wxTc~ ~ e, 
• wTa, wTI:NIL ~ c, 
• wT(p ~ px) ~ c, 
• wFNIL ~ c, or 
• wTTOP ~ c 
forsome/~ L,w,p ~ L*,x ~ L ÷,a,b E A and~b 
WFF. 
Lemma 1. Any closed set of labeled signed formulae is 
unsatisfiable. 
Proof. Immediate from the definition of a closed set. \[\] 
Definition 12. A set of labeled signed formulae, c, is 
downward-saturated if and only if c is not closed, and 
18 Computational Linguistics Volume 16, Number 1, March 1990 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
wT-'adp ~ c 
wF~(~ ~ c 
wT(l:4~) ~ c 
wF(l:dp) ~ c 
wlT(p~ ~ P2) ~ c 
wlF(p~ ~ P2) ~ c 
wT(px ~- P2) ~ c 
wT(p I ~. p2) ~ c 
wT(p~ ~. P2 ), 
wT(P2 ~ P3) ~ c 
=~wFq~ ~ c 
=wT4~ ~ c 
=~wlT4a ~ c and 
wTI:NIL ~ c 
=~wlF~ ~ corwFl:NIL ~ c 
~wT(lpx ~ lp2 ) ~ c 
~wF(lp~ ~- lp2 ) ~ c 
~wTp~ :NIL ~ c and 
wTp2:NIL ~ c 
=~wT(p2 ~" Pl) ~ c 
=wT(p~ ~'P3) ~ c 
wT(pl ~ P2), wpl Tdp ~ c~wp2Td p ~ c 
wT(pl ~" P2), wp1F~ E c~wp2F(b ~ C 
wF(pl ~ P2), wPl Ta E c=>wp2Fa ~ c 
wT(4~ /~ ~b) ~ c =>wTdp C c and wT~b E c 
wF(ep /~ ~) ~ c ~wFdp ~ c or wF~b ~ c 
wT(~ W ~k) ~ c =~wT(~ ~ c or wT~k ~ c 
wF(c~ V ~b) ~ c =~wF~ ~ c and wF~k ~ c 
Lemlna 2. If a finite set of labeled formulae, c, is down- 
ward-saturated, it is satisfiable. 
Proof. Consider the automaton, A ---- (Q, L, A,/~, q0, F, 
~, S ), constructed from c as follows: 
1. For every path w for which there is a formula ~b such 
that wT4~ ~ c or wF4~ ~ c, include a state qw in Q, 
with/~ defining a path from qo to qw labeled w. 
2. For every pair of paths p~ = wx~ and PE ---- wx2 such 
that wT(x~ ~- x2) ~ c, let q,~ and q,2 be the same 
state. 
3. For every formula wTa ~ c, include q~ in F and let 
h(qw) = a. 
4. For every formula wFa ~ c, if there is no label I such 
that there is a state q~t ~ Q, then include qw in F and 
let h(q~) ---- b for any atomic value b such that b does 
not occur in any formula in c. ~1 
5. For every formula wFl: NIL ~ c, include l in S(qw) 
6. For every formula wF(p~ ~- P2) ~ c, if states q~p, and 
qwp~ are defined and neither of them is in F, then add 
new states q~ and q2 to Q and F, and for some label 
1 that does not occur in any formula of c, define 
~(qwp,, 1) = q~ and ~(q~, 1) = q~ with h(q~) = a and 
~,(q2) = b where a and b are distinct atomic values. If 
exactly one of the two states (say, q~p, ) is not in F, 
add just one new state q to Q and let 6(q~,, 1) = q for 
a label I that does not occur in c. 
If, however, one of the paths (say, p~) does not have 
the associated state (qwp,) defined, let p be the longest 
prefix ofp~ such that qw~ does exist, and let p~ = plx. 
Include I in S(q~). 
Claim 1 : 
The above construction of an automaton is well de- 
fined. 
We need to verify that the above definition yields an 
automaton that meets our definition of an acyclic finite 
automaton without any conflicts. The possible conflicts 
that could arise would be that: X does not define a 
function; the graph of the automaton had a cycle; for 
some q ~ F and some label 1, 6(q, 1) is defined; or, for 
some state q and some label l, 1 ~ S(q) and ~(q, 1) is 
defined. However, in each of these cases, it is easy to see 
that were it to arise in the construction given above, the 
original set c would in fact be closed, contradicting the 
hypothesis that it is downward-saturated. 
Claim 2: 
The automaton so constructed satisfies all the labeled 
signed formulae in c. 
We establish this claim by induction over the structure 
of the formulae in c. For the base cases (namely labeled 
signed formulae of the forms: wXa, wXNIL, wFI:NIL, 
and wX(p~ ~ P2)) it follows immediately from the con- 
struction that they are satisfied by A. For the other cases 
(wX(q~ V ~b), wX(4J A ~b), and wX~ ~b), their sub- 
formulae are also in c since it is downward-saturated. 
But by the induction hypothesis, these sub-formulae are 
satisfied by A. That completes the result.\[\] 
The entailment relation (k) on sets of labeled signed 
formulae is defined as follows: 
Definition 13. Let c and d be two sets of labeled signed 
formulae. Then c I- d if and only if c 4= d and one of the 
following holds: 
1. wT~4~ E candd = c U {wFdp) 
2. wE-14~ ~ c and d ---- c U {wT~b} 
3. wTl:dp E candd ---- c U {wlTdp, wTI:NIL} 
4. wFl:c~ E candd = c U {wlFdp} 
5. wFl:cb E candd = c U {wFI:NIL} 
6. wlT(pl ~" P2) E candd = c U {wT(lpl ~ lp2)} 
7. wlF(p I = P2) E candd = c U {wF(lpl ~- lp2)} 
8. wT(pl ~-P2) E c and d = cU{wTpl:NIL, wTp2: 
NIL} 
9. wT(p 1 ~" Pz) ~ candd ----- c U {wT(p2 ~'Pl)} 
10. wT(p I ~-P2), wT(p2~'P3) E c and d = c U 
{wr(p~ ~ p3)} 
11. wT(pl ~-p2),wplTdp E candd = c U {wp2Tdp} 
12. wT(p I ~P2), wplF4~ ~ candd ---- c U {wp2F~b} 
13. wF(p I ~. p2), wplTa E candd ---- c U {wp2Fa} 
14. wT(cb A ~b) ~ candd = c U {wTep, wT~b} 
15. wF(4~ A ~b) E c and d -- c U {wF~} 
16. wF(~ A 4,) E candd = c U {wF~b} 
17. wT(ep V ~b) E candd = c U {wT4~} 
18. wT(dp V ~b) ~ candd = c U {wT~b} 
19. wE(oh V ~b) E c and d = c U {wF¢, wF~b} 
We denote by k* the reflexive and transitive closure of this 
entailment relation. 
Theorem 4. (Soundness) If c t-* d for sets of labeled 
signed formulae c and d, and d is downward-saturated, 
then c is satisfiable. 
Proof. This follows immediately from Lemma 2 and the 
fact that c I-* d implies c Cd.\[\] 
Lemma 3. For any set of labeled signed formulae c, there 
are only finitely many sets of labeled signed formulae d 
such that c I-* d. 
Computational Linguistics Volume 16, Number 1, March 1990 19 
Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 
Proof. To prove this, we inductively define the notion of 
length of a formula, as follows: 
In(a) = ln(NIL ) = In(TOP) = 1 
ln(pl "~ P2) = length(pl ) + length(p2) 
ln(-~) = ln((a) + 1 
ln(¢ W ¢) = ln(¢ /~ ¢) = ln(ck) + ln(~) + 1 
where length denotes string length. 
Also, define the length of a labeled signed formula 
wX¢ as ln(¢) + length(w). Let ~ be any labeled signed 
formula such that 4> ~ c but 4, C d for some d such that 
c I-* d. Observe that the length of • is bounded by the 
length of the longest formula in c and that • does not 
contain any symbols that do not occur in c. The result 
follows.E\] 
Lemma 4. For any set of labeled signed formulae c, if 
there is no set of labeled signed formulae d such that c I- 
d, then c is either closed or downward-saturated. 
Proof. Clearly, if c is closed under all the entailment 
rules listed above, then it satisfies all the implications 
listed in the definition of downward saturation. Hence, if 
it is not downward-saturated, it must be closed.E\] 
Lemma 5. For any satisfiable set of labeled signed formu- 
lae c that is not downward-saturated, there is a satisfi- 
able set of labeled signed formulae d such that c l- d. 
Proof. Since c is satisfiable, it is not closed. Since it is not 
downward-saturated, by hypothesis, there must be a d 
such that c I- d. However, it is clear from the definition of 
entailment that if all such d are unsatisfiable, then so is 
c.I--\] 
Theorem 5. (Completeness) For any satisfiable set of 
labeled signed formulae, c, there is a downward-satu- 
rated set of labeled signed formulae d such that c F-* d. 
Proof. By Lemma 3, there must be a d such that c I-* d 
and for no d' d i- d'. All such d are either closed or 
downward-saturated by Lemma 4. However, not all of 
them can be closed since then by Lemma 5, c would be 
unsatisfiable. Hence, at least one of them is downward- 
saturated. E\] 
Theorem 6. (NP-Completeness) The satisfiability prob- 
lem for the logic we have defined is NP-Complete. 
Proof. It follows from the proof of Lemma 3 that the 
length of any derivation c I-* d is bounded by n 2, where n 
is the sum of the lengths of the formulae in c. Since this 
bound is polynomial, the problem is in NP. It is NP-hard 
because the satisfiability problem for the Rounds- 
Kasper logic, which is a special case, is NP-hard. E\] 
6. CONCLUSIONS 
A logical formalism with a complete set of logical operators 
has come to be accepted as a means of describing feature 
structures. While the intended semantics of most of these 
operators is well understood, the negation and implication 
operators have raised some problems, leading to a variety of 
approaches in their interpretation. 
In Dawar and Vijay-Shanker 1989 and the present work, 
we introduced the framework of three-valued logic as a 
means of defining the semantics of a feature structure 
description language with negation. This framework per- 
mits us to say that a formula such as ~1:4~ does not have a 
truth value defined in a feature structure that does not have 
a feature l. This enables us to define an interpretation that, 
unlike the classical approach to negation, is monotonic, as a 
log\]\[c describing partial structures should be. 
We presented one particular interpretation of FDL within 
this three-valued framework and compared it with other 
apl:,roaches to defining the semantics of negation. We 
showed that several different such approaches could be cast 
in the three-valued framework. In particular, we showed 
that the special case of the Moshier-Rounds intuitionistic 
approach, in which forcing is always considered with re- 
spect to K* could be captured in our framework. 
One motivation cited by Moshier and Rounds for consid- 
ering forcing sets other than K* was so that formulae of the 
form -~I:NIL could be considered satisfiable. The same 
reason led us to examine an augmented notion of feature 
structure models for FDL that yields an interpretation that 
is conceptually simple, motivated by the preservation of 
monotonicity, and is computationally no harder than the 
original Rounds-Kasper logic. We also showed that our 
interpretation meets the conditions set out by Pereira (1987) 
for a satisfactory interpretation of negation. 
NOTES 
In the original Rounds-Kasper formulation, the output function is not 
required to be total. This is because every terminal node in the 
transition graph is considered to be a final state. However, since the 
notion of finality of a state is not crucial to the formalism, we have 
chosen this equivalent alternative for presentation. 
2. Strictly speaking, we should be taking the least upper bound in the 
ordering on equivalence classes of automata under isomorphism. 
3. See Section 3.3.3. 
4. A similar notion was used by Kasper (1988a), who introduces the 
notion of compatibility. We shall compare this approach with ours in 
greater detail in Section 3.3.3. 
5. In this paper we will not consider cyclic feature structures. 
6. And therefore it satisfies the formula ---~. 
7. Equality here is strong equality (i.e. if I(~, (Aft) is undefined then so 
is I(l:~, A)). 
8. Two automata are not unifiable if and only if they do not have a least 
upper bound. 
9. Langholm (1989) has defined a similar notion of negatively extended 
feature structures. We will take up a comparison of his approach with 
ours later in this section. 
10. Up to isomorphism. 
11. We are implicitly assuming that the sets of atoms and labels are both 
infinite. If this is not the case, the definition of closure of a set of 
labeled signed formulae and this construction can be suitably modi- 
fied. 

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