A THREE-VALUED INTERPRETATION OF NEGATION IN 
FEATURE STRUCTURE DESCRIPTIONS 
Anuj Dawar 
Dept. of Comp. and Info. Science 
University of Pennsylvania 
Philadelphia, PA 19104 
K. Vijay-Shanker 
Dept. of Comp. and Info. Science 
University of Delaware 
Newark, DE 19718 
April 20, 1989 
ABSTRACT 
Feature structures are informational elements that have 
been used in several linguistic theories and in computa- 
tional systems for natural-language processing. A logi- 
caJ 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, the semantics of the negation and implication 
operators are examined. Various proposed interpretations 
of negation and implication are compared within the sug- 
gested framework. One particular interpretation of the 
description language with a negation operator is described 
and its computational aspects studied. 
1 Introduction and Background 
A number of linguistic theories and computational ap- 
proaches to parsing natural language have employed the 
notion of associating informational dements, called feature 
structures, consisting of features and their values, with 
phrases. Rounds and Kasper \[KR86, RK86\] developed a 
logical calculus that serves as a description language for 
these structures. 
Several researchers have expressed a need for extending 
this logic to include the operators of negation and impli- 
cation. Various interpretations have been suggested that 
define a semantics for these operators (see Section 1.2), but 
none has gained universal acceptance. In \[Per87\], Pereira 
set forth certain properties that any such interpretation 
should satisfy. 
In this paper we present an extended logical calculus, 
with a semantics in three-valued logic (based on Kleene's 
three-valued logic \[Kh52\]), that includes an interpretation 
of negation motivated by the approach given by Kart- 
tunen \[Kar84\]. We show that our logic meets the condi- 
tions stated by Pereira. We also show that the three-valued 
framework is powerful enough to express most of the pro- 
posed definitions of negation and implication. It therefore 
makes it possible to compare these different approaches. 
1.1 Rounds-Kasper Logic 
In \[Kas87\] and \[RK86\], Rounds and Kasper introduced a 
logical formalism to describe feature structures with dis- 
junctive specification. The language is a form of modal 
propositional logic (with modal operator ":'). 
In order to define the semantics of this language, fea- 
ture structures are formally defined in terms of acyelic 
finite automata. These are finite-state automata whose 
transition graphs are acyclic. The formal definition may 
be found in \[RK86\]. 
A fundamental property of the semantics is that the set 
of automata satisfying a given formula is upward-closed 
under the operation of subsumption. This is important, 
because we consider a formula to be only a partial descrip- 
tion of a feature structure. The property is stated in the 
following theorem IRK86\]: 
Theorem 1.1 A C 8 if and only i/for every formula, ~, 
if A ~ ~ then B ~ cb. 
1.2 The Problem of Adding Negation 
Several researchers in the area have suggested that the 
logic described above should be extended to include nega- 
tion and implication. 
Karttunen \[Kar84\] provides examples of feature struc- 
tures where a negation operator might be useful. For in- 
stance, the most natural way to represent the number and 
person attributes of a verb such as sleep would be to say 
18 
that it is not third person singular rather than expressing 
it as a disjunction of the other tive possibilities. Kaxttunen 
also suggests an implementation technique to handle neg- 
ative information. 
Johnson \[Joh87\], defined an Attribute Value Logic 
(AVL), similar to the Rounds-Kasper Logic, that included 
a classical form of negation. Kasper \[Kas88\] discusses an 
interpretation of negation and implication in an implemen- 
tation of Functional Unification Grammar \[Kay79\] that in- 
cludes conditionals. Kasper's semantics is classical, but 
his unification procedure uses notions similar to those of 
three-valued logic a . 
One aspect of the classical approach is that the prop- 
erty of upward-closure under subsumption is lost. Thus 
the evaluation of negation may not be freely interleaved 
with unification 2 
In \[Kas88\], Kasper localized the effects of negation 
by disallowing path expressions within the scope of a 
negation. This restriction may not be linguistically war- 
ranted as can be seen by the following example from 
Pereira \[Per87\] which expresses the semantic constraint 
that the subject and object of a clause cannot be coref- 
erential unless the object is a reflexive pronoun: 
oh3 : type : reflezive V -~(subj : ref ~ obj : re f) 
Moshier and Rounds \[MR87\] proposed an intuitionistic 
interpretation of negation that preserves upward-closure. 
They replace the notion of saris/action with one of model- 
theoretic/arcing an described in Fitting \[Fit69\]. They also 
provide a complete proof system for their logic. The satis- 
liability problem for this logic was shown to be PSPACE- 
complete. 
1.3 Outline of this Paper 
In the following section we will present our proposed solu- 
tion in a three-valued framework, for defining the seman- 
tics of feature structure descriptions including negation 3. 
This solution is a formalization of the notion of negation 
in Karttunen \[Kar84\]. In Section 3 we will show that 
the framework of three-valued logic is flexible enough to 
express most of the different interpretations of negation 
mentioned above. In Section 4 we will show that the satis- 
fiability problem for the logic we propose is NP-complete. 
lsee Section 3.4 
2see Pereira \[Per87\] p.1006 
3 We shall concentrate only on the problem of extending the logic to include the negation operator, and later in Section 3.4 discuss 
Implication. 
2 Feature Structure Descriptions with 
Negation 
We will now present our extended version of the Rounds- 
Kasper logic including negation. We do this by giving 
the semantics of the logic in a three-valued setting. This 
provides an interpretation of negation that is intuitively 
appealing, formally simple and computationally no harder 
than the original Rounds-Kasper logic. 
With each formula we associate the set (Tset) of au- 
tomata that satis/y the formula, a set (Fset) of automata 
that contradict it and a set (Uset) of automata which nei- 
ther satisfy nor contradict it 4. Different interpretations of 
negation are obtained by varying definitions of what con- 
stitutes "contradiction." In the semantics we will define, 
we choose a definition in which contradiction is equivalent 
to essential incompatibility 5. We will define the Tset and 
the Fset so that they are upward-closed with respect to 
subsumption for all formulae. Thus, we avoid the prob- 
lems associated with the classical interpretation of nega- 
tion. In our logic, negation is defined so that an automaton 
,4 satisfies -,~b if and only if it contra£1icts ~. 
2.1 The Syntax 
The symbols in the descriptive language, other than the 
connectives :, v, A,-, and ~ are taken from two primitive 
domains: Atoms (A}, and Labels (L). 
The set of well formed formulae (W), is given by: NIL; 
TOP; a; 1 : @; ~ A ~b; @ V ~b; "-~ and pl ~- P2, where a E A; 
1E L; ~,~ E W and Pa,P2 E L'. 
2.2 The Semantics 
Formally, the semantics is defined over the domain of par- 
tial functions from acycLic finite automata ~ to boolean val- 
ues. 
Definition 2.1 An acyclic finite automaton is a 7-tuple 
A =< Q, E, r, 6, qo, F, A >, where: 
1. Q is a non-empty finite set (of states), 
~. E is a countable set (the alphabet), 
4A similar notion was used by Kasper \[Kas88\], who introduces 
the notion of compatibility. We shall comps.re this approach with ou~ in greeter detail in Section 3.4. 
Sln general, a feature structure is incompatible with a formula i£ 
the information it contains is inconsistent with that in the formula. 
We will distinguish two kinds of incompatibility. A feature struc- 
ture is essentiall~/incompatible with a formula if the information in 
it contradicts the information in the formula. It is trivially incom- 
patible with the formula if the inconsistency is due to an excess of mformtstion within the formula itself. 
Sin this paper we will not consider cyclic feature structures 
19 
3. r is a countable set (the output alphabet), 
4. 6 : Q × E -" Q is a finite partial/unction (the tran- 
sition function), 
5. qo ~ Q (the initial state), 
6. F C Q (the set of final states), 
7. A : F "-* r is a total function (the output function), 
8. the directed graph (Q, E) is acyclic, where pEq iff 
.for some 1 6 Z, 6(p, l) = q, 
9..for every q ~ Q, there exists a directed path from qo 
to q in ( Q, E), and 
10. for every q ~ F, 6(q, I) is not defined for any I. 
A formula ~ over the set of labels L and the set of 
atoms A is chaxacterized by a partial function: 
~r, : {'41"4 =< Q, L, A, 6, q0, F, A >} "7" {True, False} 
~#,('4) is True iff "4 satisfies ~b. It is False if'4 contra- 
dicts ~b r and is undefined otherwise. The formal definition 
is given below. 
Definition 2.2 For any formula ¢~, the partial func- 
tion .~'¢ over the set of acyclic finite automata, "4 =< 
Q, L, A, 6, qo, F, A >, is defined as follows: 
1. if ~ = NIL then 
~('4) = True for all "4; 
~. if ~ = TOP then 
~(,4) = False for all .4; 
3. if O m a for some a ~ A then 
~(.,4) = True 
if .4 is atomic and A(q0) = a 
:7:(.4) = False 
if "4 is atomic and A(qo) = b 
for some b, b # a (see Note ~.) 
~'~( "4 ) is undefined otherwise; 
4. if @ f l : @t for some l ~ L and @x 6 W then 
~r ~ ( "4 ) __ ~r ~, ( "4 / l ) if.Aft is defined. 
(see Note 3.) 
:F,('4) is undefined otherwise; 
rand therefore it satisfies the formula "-4, 
5. if ~ = ~a A ~2 for some ~bi , ~2 E W then 
.~'+('4) = True 
if~r+,('4) = True and jr('4)= True 
y+('4) = False 
if ~r~,('4) = False or ~'~('4) - False 
~('4) is undefined otherwise ; 
6. 
7+('4) 
7~('4) 
Y+('4) 
V ~b2 for some ~,,~2 6 W then 
• .~ True 
if.~'~, ('4) = True or 9r¢2('4) = True 
= False 
if ~x('4 ) = False and F~2('4 ) = False 
is undefined otherwise ; 
7. if ~b -- "~1 for some ~h E W then 
:~('4) = True if Y:~, ('4) = False 
~r,#('4) = False if gr~x ('4) = True 
~('4) is undefined otherwise ; 
8. if¢= m 
~+('4) 
~( "4) 
7+('4) 
~ I~ for some pa,p2 E L" then 
= True 
if 6(qo,p,) and 6(qo,p2) are defined 
and 6(q0, pl) ---- 6(qo,p2) 
= False 
if "4/pa and "4/P2 are both defined 
and are not unifiable 
is undefined otherwise (see Note 4.). 
Notes: 
I. We have not included an implication operator in 
the formal language, since we find that defining im- 
pllcation in terms of negation and disjunction (i.e 
~b =~ ~b ~ -~@ V ~b) yields a semantics for implica- 
tion that corresponds exactly to our intuitive un- 
derstanding of implication. 
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. Though 
a complex feature structure is clearly incompatible 
with an atomic formula we do not view it as being 
essentially incompatible with it. An interpretation 
of negation that defines a complex feature structure 
as contradicting a (and hence satisfying -,a) is also 
possible. However, our definition is motivated by 
the linguistic intention of the negation operator as 
given by Karttunen \[Kar84\]. Thus, for instance, we 
require that an automaton satisfying the formula 
case : ".dative have an atomic value for the case 
feature. 
3. In J. above, we state that: ~'~('4) = jr', ('4/1) if.Aft 
is defined. When "4/l is defined, ~t ('4/I) may still 
20 
4. 
be True, False or undefined. In any of these cases, 
~#(.A) -- ~I(.A/I) s. ~r~(.A) is not defined if .All 
is not defined. Not only is this condition required 
to preserve upward-closure, it is also linguistically 
motivated. 
Here again, we could have said that a formula of the 
form I : ~bz is contradicted by any atomic feature 
structure, but we have chosen not to do so for the 
reasons outlined in the previous note. 
We have chosen to state that the set of automata 
that are incompatible with the formula pz ~ p2 is not 
the set of automata for which 6(qo,pl) and 6(qo,p~) 
axe defined and 8(q0,pz) ~ 6(q0,p2), since such an 
automaton could subsume one in which 6(qo,px) = 
6(q0,p~). Thus, we would lose the property of 
upward-closure under subsumption. However, an 
automaton, .4, in which 6(q0,pl) and 8(qo,p2) are 
defined and .A/p1 is not unifiable 9 with ~4/p2 can- 
not subsume one in which 6(q0,pa) = 6(q0,p2). 
2.2.1 Upward-Closure 
As has been stated before, the set of automata that satisfy 
a given formula in the logic defined above is upward-closed 
under subsumption. This property is formally stated be- 
low. 
Theorem 2.1 Given a formula ~b and two acyclie finite 
automata .4 and IJ, if ~(.A) is defined and .4 C B then 
y.(B) ~, defined and ;%(B) = 7.(~4). 
Proof: 
The proof is by induction on the structure of the formula. 
The details may be found in Dawar \[Daw88\]. 
2.3 Examples 
We now take a look at the 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 : "~(person : third A number : singular) (1) 
This formula is satisfied by any structure that has an agree- 
ment 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 following 
two structures satisfy the given formula: 
agreement: \[person: second\] 
SEquality here is strong equality (i.e. if .g,x(A\]l) is undefined 
then so is .~',(.4).) 
9Two automata are not unifiable if and only if they do not have a least upper bound 
\[ \[p r,on \] \] agreement : number : plural 
On the other hand, for a structure to contradict formula(1) 
it must have an agreement feature defined for both person 
and number with values third and singular respectively. All 
other automata would have an undefined truth value for 
formula(1). 
Turning to the other example mentioned earlier, the 
formula: 
obj : type : reflexive x/"~(subj : ref ~ obj : re f) (2) 
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 or path-equivalence). 
\[obj. \[type-reflexive \]\] 
\[ obj: \[ ref:\[\] \] \] 
subj : \[ ref :\[\] \] j\] 
type : reflezive 
subj: \[ re1: \[\] \] 
3 Comparison with Other Interpreta- 
tions of Negation 
As we have stated before, the semantics for negation de- 
scribed in the previous section is motivated by the dis- 
cussion of negation in Karttunen \[Kar84\], and that it is 
closely related to the interpretation of Kssper \[Kas88\]. In 
this section, we take a look at the interpretations of nega- 
tion that have been suggested and how they may be related 
to interpretations in a three-valued framework. 
3.1 Classical Negation 
By classical negation, we mean an interpretation in which 
an automaton .4 satisfies a formula -~b if and only if it does 
not satisfy ~b. This is, of course, a two-valued logic. Such 
an interpretation is used by Johnson in his Attribute-Value 
Language \[Joh87\]. We can express it in our framework by 
making ~'~ a total function such that wherever 9re(A) was 
undefined, it is now defined to be False. 
Returning to our earlier example, we can observe that 
for formula(1) the structure 
\[ agreement: \[ person: third\] \] 
has a truth value of .false in the classical semantics but 
has an undefined truth value in the semantics we define. 
This illustrates the problem of non-monotonicity in the 
classical semantics since this structure does subsume one 
that satisfies formula (1). 
21 
3.2 Intultionistic Logic 
In \[MR87\], Moshier and Rounds describe an extension of 
the Rounds-Kasper logic, including an implication opera- 
tor and hence, by extension, negation. The semantics is 
based on intnitionistic techniques. The notion of satisfying 
is replaced by one of forcing. Given a set of automata/C, 
a formula ~b, and .A such that .4 ~ /C, .A forces in IC "~b 
(,4 hn -~b) if and only if for all B ~/C such that A ~ B, B 
does not force ~b in/~. Thus, in order to find if a formula, 
~b, is satisfiable, we have to find a set \]C and an automaton 
~4 such that forces in IC ~. 
Moshier and Rounds consider a version in which forcing 
is always done with respect to the set of all automata, 
i.e. IC*. This means that the set of feature structures 
that satisfy --~b is the largest upward-closed set of feature 
structures that do not satisfy @ (i.e. the set of feature 
structures incompatible with ~b). We can capture this in 
the three-valued framework described above by modifying 
the definition of ~r¢ to make it False for all automata that 
are incompatible (trivially or essentially) with ~b (we call 
this new function ~r~). The definition of ~'~ differs from 
that of ~r+ in the following cases: 
• ~b=a 
~r¢(A) = True 
if A is atomic and A(q0) = a 
~r~(A) = False otherwise 
~'~(~t) = True 
if ~'~(.A) ---- True 
:~(A) = False 
if All is defined and 
vs(wl/! ~_ B =~ ~,,(B) = False) 
~r~(.A) is undefined otherwise. 
~'~(Ft) = True 
if ~+,(.,4) = True and .~+~(.A) = True 
:r~,(A) = False 
if VB(A E_ S =~ 
~r~t(B) # True or Y;2(B) # True) 
~(A) is undefined otherwise ; 
• ~=~v~2 
7;(,4) = Tr.e 
if ~,(.A) = True or ~r~a(A ) = True 
~(A) = False 
if ¥B(.A C B 
~';,(B) # True and Jr;.(B) # True) 
~r~(.4) is undefined otherwise ; 
• ~ = Pl ~ P2 
7;(,4) = True 
if 8(qo, p,) and ~(qo, p2) are defined 
and ~(qo,pl) = 6(qo,p2) 
F~(A) = False 
if A/p1 and .A/p2 are both defined 
and are not unifiable or if .4 is atomic 
~'~(.4) is undefined otherwise . 
In the other cases, the definition of ~'~ parallels that 
of 7+. 
To illustrate the difference between ~'~ and 3r~, we 
define the following (somewhat contrived) formula: 
cb = (11 :avl2 : a) AI2 : b 
We also define the automaton 
,4 = \[11 : b\] 
We can now observe that F~(A) is undefined but 3r~(A) = 
False. To see how this arises, note that in either system, 
the truth value of ,4 is undefined with respect to each of 
the conjuncts of ¢i. This is so because ,4 can certainly be 
extended to satisfy either one of the conjuncts, just as it 
can be extended to contradict either one of them. But, for 
~c'~#(.A) to be False, .4 must have a truth value of False 
for one of the conjuncts and therefore .~'¢(.4) is undefined. 
On the other hand, since .4 can never be extended to sat- 
isfy both conjuncts of ~ simultaneously, it can never be 
extended to satisfy ~b. Hence .4 is certainly incompatible 
with ~, but because this incompatibility is a result of the 
excess of information in the formula itself, we say that it 
is only trivially incompatible with ~. 
To see more clearly what is going on in the above ex- 
ample, consider the formula -~b and apply distributivity 
and DeMorgan's law (which is a valid equivalence in the 
logic described in the previous section, but not in the in- 
tuitionistic logic of this section) which gives us: 
-,~b = (-'la : a A "./2 : a) V -~12 : b 
We can now see why we do not wish .4 to satisfy -~b, which 
would be the case if .~'~#(~4) were False. 
One justification given for the use of forcing sets other 
than /C* is the interpretation of formulae such as -~h : 
NIL. It is argued that since h : NIL denotes all feature 
structures that have a feature labeled h, -,h : NIL should 
denote those structures that do not have such a feature. 
However, the formula -~h : NIL is unsatisfiable both in 
the interpretation given in the last section as well as in the 
/C* version of intuitionistic logic. It is our opinion that the 
use of negation to assert the non-existence of features is 
an operation distinct from the use of negation to describe 
values mad should be described by a distinct operator. The 
present work attempts to deal only with the latter notion of 
negation. The authors expect to present in a forthcomi~ag 
paper a simple extension to the current semantics that will 
deal with issues of existence of features. 
22 
3.3 Karttunen's Implementation of Negation 
As mentioned earlier, our approach was motivated by 
Karttunen's implementation as described in \[Kax84\]. In 
the unification algorithm given, negative constraints are 
attached to feature structures or automata (which them- 
selves do not have any negative values). When the feature 
structure is extended to have enough information to deter- 
mine whether it satisfies or falsifies I° the formula then the 
constraints may be dropped. We feel that our definition 
of the Uset elegantly captures the notion of associating 
constraints with automata that do not have sufficient in- 
formation to determine whether they satisfy or contradict 
a given formula. 
3.4 Kasper's Interpretation of Negation and 
Conditionals 
As mentioned earlier, Kasper ~Kas88\] used the operations 
of negation mad implication in extending Functional Unifi- 
cation Grammar. Though the semantics defined for these 
operators is a classical one, for the purposes of the algo. 
rithm Kasper identified three chases of automata associ- 
ated 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 Tact, Fset and User respectively. For instance, Kasper 
states that an automaton .A satisfies a formula f : v if it 
is defined for f with value v; it is incompatible with f : v 
if it is defined for f with value z (z ~ v) and it is merely 
compatible with f : v if it is not defined for f. 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 Computational Issues 
In this section, we will discuss some computational as- 
pects related to determining whether a formula is satisfi- 
able or not. We will Show that the satisfiability problem is 
NP-complete, which is not surprising considering that the 
problem is NP-complete for the logic not involving nega- 
tion (Rounds-Kasper logic). 
The NP-hardness of this problem is trivially shown 
if we observe that for any formula, ~b, without negation, 
Tset(¢) is exactly the set of automata that satisfy ~ ac- 
cording to the definition of satisfaction given by Rounds 
l°It is not clear whether falsification is equivalent to incomp~- ibility or only essential incompatibility, but from the examples in- 
volvin~ ease and agreement, we believe that only emJential incom- patibihty is intended. 
and Kasper \[KR86, RK86\] in their original logic. Since 
the satisfiabllity problem in that logic is NP-complete, the 
given problem is NP-haxd. 
In order to see that the given problem is in NP, we 
observe that a simple nondeterministic algorithm 11 can be 
given that is linear in the length of the input formula ~b 
and that returns a minimal automaton which satisfies ~b, 
provided it is satisfiable. To see this, note that the size 
(in terms of the number of states) of a minimal automa- 
ton satisfying ~b is linear in the length of ¢ and verifying 
whether a given automaton satisfies ~b is a problem linear 
in the length of ~b and the size of the automaton. The 
details of the algorithm can be found in Dawar \[DawS8\]. 
5 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 vari- 
ety of approaches in their interpretation. In the present 
work, we have presented an interpretation that combines 
the following advantages: it is formally simple as well as 
uniform (it places no special restriction on the negation 
operator); it is motivated by the linguistic applications of 
feature structures; it takes into account the partial na- 
ture of feature structures by preserving the property of 
monotonicity under unification and it is computationally 
no harder than the Rounds-Kasper logic. More signifi- 
cantly, perhaps, we have shown that most existing inter- 
pretations of negation can also be expressed within three- 
valued logic. This framework therefore provides a means 
for comparing and evaluating various interpretations. 
References 
\[Daw88\] 
\[Fit69\] 
Anuj Dawar. The Semantics of Negation in Fea- 
ture Structure Descriptions. Master's thesis, Uni- 
versity of Delaware, 1988. 
Melvin Fitting. Intuitionistic Logic and Model 
Theoretic Forcing. North-Holland, Amsterdam, 
1969. 
\[Joh87\] Mark Johnson. Attribute Value Logic and the 
Theory of Grammar. PhD thesis, Stanford Uni- 
versity, August 1987. 
\[K~84\] Lauri Karttunen. Features and values. In Pro. 
ceedings of the Tenth International Conference on 
Computational Linguistics, July 1984. 
llthis algorithm assumes that the se¢ of atoms is finite. 
23 
\[Kas87\] Robert T. Kasper. Feature Structures: A Logical 
Theory with Application to Language Analysis. 
PhD thesis, University of Michigan, 1987. 
\[Kas88\] Robert T. Kasper. Conditional descriptions in 
Functional Unification Grammar. In Proceedings 
of the ~6th Annual Meeting o\] the Association/or 
Computational Linguistics, pages 233-240, June 
1988. 
\[Kay79\] M. Kay. Functional grammax. In Proceedings of 
the Fifth Annual Meeting of the Berkeley Linguis- 
tics Society, 1979. 
\[Kle52\] S.C. Kleene. Introduction to Metamathematics. 
Van Nostrand, New York, 1952. 
\[KR86\] Robert T. Ka~per and William C. Rounds. A 
logical semantics for feature structures. In Pro- 
ceedings o/the ~th Annual Meeting o.( the Asso- 
ciation for Computational Linguistics, 1986. 
\[MR87\] M. Drew Moshier and William C. Rounds. A 
logic for partially specified data structures. In 
ACM Symposium on the Principles o~ Program. 
ruing Languages, pages 156-167, ACM, 1987. 
\[Per87\] Fernando C. N. Pereira. Grammars and logics 
of partial information. In Jean-Louis Lassez, ed- 
itor, Proceedings o\] the 4th International Con- 
ference on Logic Programming, pages 989-1013, 
May 1987. 
IRK86\] William C. Rounds and Robert T. Kasper. 
A complete logical calculus for record struc- 
tures representing linguistic information. In 
IEEE Symposium on Logic in Computer Science, 
pages 34-43, IEEE Computer Society, June 1986. 
24 
