FRANK G. PAGAN 
CONSTRUCTIBLE REPRESENTATIONS 
FOR TWO SEMANTIC RELATIONS 
1. INTRODUCTION 
This paper is concerned with semantic (or lexical) structures of 
the type used in programs which involve natural language processing, 
such as question-answering systems. These structures, which generally 
take the form of graphs representing semantic relations defined on 
word senses, must satisfy some rather self-evident requirements re- 
lating to their linguistic significance and adequacy and to their suita- 
bility for computational implementation. In addition, if the lexical 
subsets in question are to be nontrivial in size, the structures must be 
constructible in some systematic, consistent way, preferably with the 
aid of a computer. The structures which have been used in existing 
experim.ental systems, such as those reported in M. R. QUILLIAN (1969) 
and K. M. SCaWARCZ, J. F. BtraGrR, K. F. SIMMONS (1970), have gene- 
rally been very restricted, and it has been argued (S. Y. S~D~OW, 
1972) that it is their lack of constructibility which has precluded the 
possibility of extending them. 
The aim of this paper is to describe two examples of the use of 
constructibility as a design criterion for semantic structures. The se- 
mantic relations of hyponymy and compatibility are introduced in the 
following section, and suitable representations for them are developed 
in sections 3 and 4, respectively. Throughout the discussion, the con-- 
structibility of a representation is to be interpreted as its amenability 
to the use of a computational discovery algorithm which would build 
the structure and have the properties of semi-automatic operation, 
simple input data, efficiency in the quantity of input, consistency main- 
tenance, and monotonic refinement of the growing structure. The 
meaning of this terminology will be made more clear in the course 
of the discussion. 
30 FRANK G. PAGAN 
It is far beyond the scope of the paper tO consider the development 
and implementation of practical programs for building large semantic 
structures. It should also be emphasized that the aim is not the de- 
velopment of linguistic theories or the presentation of particular lexical 
structures. While the linguistic positions adopted in the following 
section are consistent with much of the existing literature, they are 
primarily intended to provide a sample context in which the role of 
constructibility may be illustrated. 
2. HYPONYMY AND COMPATIBILITY 
The term hyponymy has been used by J. LYoNs (1968) to denote 
the common but somewhat vague notion of "semantic inclusion" 
A hyponymy p of a word sense q is a word sense whose meaning is a 
"subset " of the meaning of q; for example, scarlet is a hyponym of 
red, and tulip is a hyponym of flower. Intuitively, ifp is a hyponym of 
q, then it makes sense to say that p is a kind of q. Synonymy of word 
senses may be defined in terms of hyponymy: p is a synonym of q 
if and only if p is a hyponym of q and q is a hyponym of p. The 
hyponymy relation is evidently reflexive and transitive. 
The importance of hyponymy has been generally recognized by 
workers in computational semantics (see, for example, M. 1~. QuII.- 
I.IAN, 1969, and P,.. M. SCHWARCZ, J. F. BrinGeR, 1~. F. SIMMONS, 1970). 
Quillian's system (M. R. QUILI.IAN, 1969) is a generalization of a scheme 
for representing lexical definitions of word senses in terms of supersets 
and properties, where the superset relation is simply the inverse of hypo- 
nymy. When the superset of some sense is modified by the relevant 
properties, an expression is obtained which is a definition of paraphrase 
of the given sense. As a highly simplified example, suppose that the 
superset of bachelor is man, with the corresponding property unmar- 
ried, and that the superset of youth is also man, with the corresponding 
property young. A simple calculation would then mark the expressions 
zoung bachelor and unmarried youth as paraphrases, since both are equiv- 
alent to the expression young unmarried man. Although I have not 
mentioned many of the details and problems involved (such as the 
possible forms properties may take), it seems clear that a representation 
for hyponymy is necessary for such important semantic processes as 
transformation between equivalent linguistic expressions. 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS 31 
Turning to the compatibility relation, the simplest case involves 
combinations of adjective senses and noun senses to form expressions 
which may be either meaningful (hungry man) or meaningless (hungry 
theory). The two senses may be said to be compatible if their combina- 
tion is meaningful. It turns out that little can be done with regard to 
the representation of this relation unless its definition is extended as 
follows: two adjective senses are compatible if and only if there are 
noun senses compatible with both. It is useful here to think of noun 
senses as "atoms " or "constants " and adjective senses as "predicates " 
which take noun senses as arguments (this is similar to Fillmore's 
approach \[C.J. FII.LMOI~, 1969\]). The second definition of compatibility 
then involves the intersections of the domains of predicates. This re- 
lation, which is reflexive and symmetric, will be the more important 
as far as constructibility is concerned. 
• As for the use of compatibility, on the other hand, the first defini- 
tion is the important one. First, in a linguistic or computational system 
involving the generation or analysis of sentences, it would be useful 
in excluding meaningless constructions of word senses, and it can be 
argued that this capability is a fundamental one for any such system. 
Secondly, the relation provides the ability to disambiguate words in the 
context of their combination with other words. For example, suppose 
that hard-has two senses written as hard (difficult) and hard (not soft), 
and that ball has the two senses ball (round object) and ball (social event). 
Suppose further that hard (di~cult) is compatible with question and 
incompatible with both senses of ball, and that hard (not soft) is incom- 
patible with question and ball (social event) but compatible with ball 
(round object). Then, in the expression• hard question, hard must mean 
hard (diffcult). Similarly, in the expression hard ball, hard must mean 
hard (not soft) and ball must mean ball (round object), since the other 
three combinations are all incompatible. 
3. STRUCTURES. FOR HYPONYMY 
To illustrate some classes of structures for the hyponymy relation, 
a hypothetical set of seven word senses a, b .... , g will be used. Their 
hyponymy relationships are given in Fig. 1 in the form of a matrix: 
a sense x is a hyponym of a sense y if and only if there is a 1 in the po- 
sition whose row corresponds to x and whose column corresponds 
to y. 
32 FRANK G. PAGAN 
a b c d e f g 
1 0 1 1 1 1 1 
0 1 1 1 1 0 0 
0 0 1 1 1 0 0 
0 0 0 1 1 0 0 
0 0 0 0 1 0 0 
0 0 0 1 1 1 1 
0 0 0 1 1 1 1 
Fig. 1. Hyponymy matrix. 
The graph structure which directly corresponds to this matrix is 
given in Fig. 2. 
e 
b 
4 • ct I 
o. 
Fig. 2. F-graph. 
In an F-graph such as this, each node corresponds to a sense, and 
the interpretation of the structure is that x is a hyponym of y if and 
only if there is an arc from x's node to y's node (stricdy speaking, 
since hyponymy is reflexive, there should be a loop at each node, 
but these may be omitted without loss of information). 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS 33 
An obvious simplification is otbained by making use of the transi- 
tivity of hyp0nymy in the interpretation of the structure. In a G-graph 
(Fig. 3), then, the nodes are unchanged from the corresponding F- 
graph, but now x is a hyponym of/, if and only if there is a path from 
x's node to/s node. 
e 
I~ G ~ g 
Fig. 3. G-graph. 
An F-graph is the transitive closure of its corresponding G-graph, 
and the G-graph has no "redundant " arcs; i.e., arcs from x to y and 
from y to z imply that there is no arc from x to z. This, together with 
the fact that synonyms are represented by separate nodes, implies that 
G-graphs are not necessarily unique; for example, the G-graph in Fig. 4 
is equivalent to that in Fig. 3. 
@ 
I - 
- f 
Fig. 4. An equivalent G-graph. 
34 FRANK G. PAGAN 
This non-uniqueness may be avoided by transforming the G-graph 
into an H-graph, in which synonyms are associated with the same node 
but the interpretation is otherwise unchanged. All H-graphs are acydic, 
since all the senses on a cycle would have identical hyponymy relation- 
ships with all other senses, i.e., they would be synonyms. Thus the 
nodes form a partially ordered set, and/-/-graphs may be represented 
by lattice-like Hasse diagrams (D. E. P,.OTr~RrORD, 1965) where each 
edge is assumed to be directed towards the top of the page, although 
the arrows are not explicitly present (Fig. 5). 
e 
b a 
f,g 
Fig. 5. H-graph. 
It may easily be seen that H-graphs are in one-to-one correspond- 
ence with the set of all possible hyponymy matrices (transitive, square, 
Boolean matrices). They are thus a fully general class of structures, and 
the question arises as to whether the set of linguistically significant 
structures is actually more restrictive than this. Judging from the ex- 
sting literature in pure and applied linguistics, there would seem to be 
a consensus that hyponymy graphs should in fact be tree-structured. 
This is apparently the case in Quillian's system (M. P~. QmLLIAN, 
1969), for example, and some experimental support for the psycholog- 
ical reality of this type of structure is given in A. M. CotuNs, M. 
1~. QUttHAN (1969). T. G. B~.W.R and P. S. R.OSSNRAU~ (1970) have 
also argued for the adequacy of tree structures in this context. In 
terms of H-graphs, the necessary restriction is obtained by requiring 
that any word sense may be an immediate hyponym of at most one 
other sense; i.e., it may not be a hyponym of two " disjoint" senses. 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS 35 
The class of structures so obtained may be termed H-forests, since they 
are a proper subset of the/-/-graphs. 
Although it has been generally adopted in existing models, the 
restriction to H-forests is not without its dit~iculties. The H-graph 
in Fig. 6 represents an intuitively correct set of hyponymy relationships 
which is inconsistent with the H-forests. 
child 
VOllllin 
~other faBher 
Fig. 6. Apparent inadequacy of H-forests. 
An interesting but open question is whether such examples can be 
considered to be rare in natural language, occurring only in very special 
cases such as kinship terminology. A related question concerns the 
validity of the assumption that a given sense is necessarily an immediate 
hyponym of at most one other sense, so that hyponymy, defined on 
this basis, must be tree-structured. It is clear that with the present state 
of our knowledge a linguistic case can be made for both H-graphs and 
/-/-forests as the appropriate class of structures for the hyponymy re- 
lation. 
Turning now to the constructibility of the various possible classes 
of hyponymy structures, it is not necessary to consider F-graphs and 
G-graphs, since they m:iy be dismissed in favor of H-graphs purely 
on the grounds of their mathematical properties and economy of rep- 
resentation. Both H-graphs and H-forests are constructible insofar 
as one can devise semi-automatic discovery procedures which will, 
in principle, construct them from basic data obtained from an informant. 
36 FRANK G. PAGAN 
This "basic data " would consist of judgments of the relationships 
between particular pairs of senses: either the first is a hyponym of the 
second, the second is a hyponym of the first, they are synonyms, or 
they are related in none of these ways. 
The H-graph or H-forest would be constructed by a process of 
monotonic refinement; i.e., after the incorporation of each additional 
word sense, the new structure would be a recognizable generalization 
of the previous structure. More specifically, suppose that Hi and Hi+l 
are the current structures after i and i + 1 senses, respectively, x and 
y are any two senses previously processed, and the distance between 
two senses on the same path is the number of arcs separating them. 
Then the following statements would hold: 
a) If x and y are at the same node in H i, then they are at the 
same node in Hi+l; 
b) x and y are not at the same node in H i iff they are not at 
the same node in H~+I; 
c) x is above y in Hi iff x is above y in Hi+l; 
d) if x is a distance d above y in Hi, then x is a distance d or 
d q-1 above y in Hi+l. 
e) if x is a distance d above y in Hi+l, then x is a distance d or 
d-1 above y in Hi; 
f) x and y are not on the same path in Hi iff they are not on 
the same path in Hi+l. 
The remaining requirements for constructibility are those of input 
efficiency and consistency maintenance. An obvious point of reference 
for efficiency is the maximum number of inputs which would be re- 
quired irrespective of the algorithm and the set of word senses used. 
1 N 2, where N is the number of senses, This is approximately given by 
and would be reached in cases where inputs must be obtained for all 
pairs of senses. The transitivity of hyponymy would allow H-graphs 
to be constructed with better efficiency than this; for example, if a 
sense z were found to be a hyponym of y, and y were known to be 
a hyponym of x, then it could be inferred or predicted that z is a hyponym 
of x with no additional input. H-forests could be constructed with 
better efficiency than H-graphs since they are a much more predictive 
class of structures; for example, if z were found to be " unrelated" 
at the root of a tree, then it would be predicted to be "unrelated" 
everywhere in the tree. 
Because the H-graphs are a fully general class of structures, incon- 
sistency of a set of inputs with respect to them could be due only to 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS - 37 
a violation of transitivity, and the procedure could check for such 
occurences automatically. In the case of H-forests, there is more scope 
for inconsistency, since any valid H-graph which is not also an H- 
forest represents a set of information which is inconsistent with H- 
forests. The procedure would have to obtain redundant inputs in order 
to maintain a consistency check in this case. The details of testing for 
and correcting inconsistencies would depend upon the properties of 
the particular discovery procedure used. 
To summarize this section, I have attempted to place in perspective 
the plausible classes of structures for hyponymy. Both H-graphs and 
H-forests satisfy the basic requirement for constructibility. The con- 
struction of the latter, however, would be more efficient, so that it 
would be disadvantageous to use a discovery procedure for H-graphs 
in general if H-forests alone are linguistically adequate. 
4. STRUCTURES FOR COMPATIBILITY 
The variety of structure classes for the compatibility relation turns 
out to be richer than in the case of hyponymy. To illustrate some of 
these representations, suppose that the compatibility relationships among 
the seven-word senses a, b ..... g are as shown in the matrix of Fig. 7. 
a b c d e f g 
1 (symmetric) 
1 1 
1 1 1 
1 0 0 1 
1 0 0 1 1 
1 0 0 1 1 1 
1 0 0 1 0 0 
Fig. 7. Compatibility matrix. 
Since the relation is symmetric, it is quite natural to represent it as 
an undirected graph (an A-graph) where the nodes correspond to senses 
and the edges indicate compatibility. The A-graph corresponding to 
Fig. 7 is given in Fig. 8. 
38 FRANK G. PAGAN 
b 
Fig. 8. A-graph. 
For large sets of senses, A-graphs would clearly be unwieldy and 
uneconomic, and a more compact representation would be highly 
desirable. One particularly interesting transformation of the matrix 
or A-graph is obtained by grouping senses with the same compatibility 
properties with respect to all the other senses and placing them at the 
same node. The interpretation of such a C-graph is that two senses are 
compatible if and only if they are associated with the same or adja- 
cent nodes. The C-graph for the present example is given in Fig. 9. 
It can be easily shown that every C-graph is unique and that this class 
of structures is fully general. 
b,c T 
e,f g 
tl 
Fig. 9. C-graph. 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS 39 
Turning now to more restrictive structures, a more compact rep- 
resentation can sometimes be obtained by transforming the undi- 
rected C-graph into a directed D-graph with the same set of nodes 
but with the interpretation that two senses are compatible if and only 
if either they are at the same node or there exists a path between their 
nodes. In analogy to the situation for/-/=graphs, the nodes of a D- 
graph are partially ordered. Fig. 10 shows a D-graph for the example 
in the form of a Hasse diagram. The D-graphs are not a fully general 
class of structures; i.e., there are C-graphs with no corresponding 
D-graphs. Neither are they necessarily unique: it can be shown that 
some C-graphs have more than one equivalent D-graph. 
• ' ~" d 
b 
e,f g 
Fig. 10. D-graph. 
The linguistic significance of these structures lies in their strong 
similarity to systems of semantic markers or features (corresponding to 
nodes) related by redundancy rules (corresponding to edges) (J. J. KATZ, 
J. A. FODOR, 1963). In this case, a feature is defined as the set of senses 
which, in other formulations, would be said to have the feature. The 
relationships among features, and hence the compatibility properties of 
word senses, have often been represented by structures similar to/9= 
graphs; moreover, a survey of the relevant theoretical and computa- 
tional literature (e,g., tL. C. SCHANK, 1969; F. SOMMF.RS, 1963) would 
reveal that tree-like structures have nearly always been considered to 
be adequate for this purpose. The graph of Fig. 10 is in fact such a 
D-forest. A given set of compatibility relationships can be represented 
by at most one D-forest. 
The constructibility criterion also bears strongly on the relative 
merits of these structure classes. Considering the C-graphs first, it 
40 FRANK G. PAGAN 
may easily be shown that this class satisfies all the requirements of con- 
structibility except the crucial one of effciency. Assuming the availa- 
bility of basic data in the form of yes-or-no judgments of the compati- 
bility of unordered pairs of word senses, a C-graph could be built 
by a monotonic refinement process: at each stage the current sense would 
either be added to an existing node or form a new node, possibly in 
conjunction with the "splitting " of other nodes. Efficiency, however, 
is the worst possible because the class of structures is fully general, and 
the judgements obtained for the compatibility of the current sense with 
some subset of the previous senses have no predictive value for its com- 
patibility with any of the other senses in the graph. 
The lack of effciency in constructing C-graphs can be regarded as 
a corollary of the fact that inconsistencies are impossible2 They are pos- 
sible in the case of D-graphs, however, and thus the assumption that 
they would not arise would make some judgments unnecessary after 
others became known. But D-graphs fail to be constructible because 
of their non-uniqueness. The D-graphs shown in Fig. 11 and Fig. 12 
are equivalent, 
8. 
c g 
f 
e 
Fig. 11. A non-unique D-graph. 
and that shown in Fig. 13 represents the same information except 
that it contains an additional word sense. 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS 41 
& 
e cl 
e 
Fig. 12. An equivalent D.-£raphl 
a 
:r 
e; 
Fig. 13. A rcfineraent of Fig. 1L 
42 FRANK G. PAGAN 
The latter graph is a simple modification of Fig. 11 but not of Fig. 
12, and there is no graph equivalent to Fig. 13 which is easily obtained 
from Fig. 12. Since a given discovery procedure could construct either 
of the first two graphs depending upon the order in which the senses 
were incorporated, the process would in general involve complete 
reconstruction rather than monotonic refinement. 
The D-forests, finally, are constructible in all respects. Monotonic 
refinement is characterized in this case by the following statements, 
where x and y again are any two senses previously processed and D~ 
and D~+ 1 are the current forests after i and i q- 1 senses, respectively: 
a) if x and y are at the same node in D,., then they are at the 
same or adjacent nodes in Di+l; 
b) if x is a distance d above y in Di, then x is a distance d or 
d -q- 1 above y in D~+I; 
c) if x is a distance d above y in Di+l, then x is a distance d or 
d-- 1 above y in D~; 
d) x and y are not on the same path in Di iff they are not on 
the same path in Di+l. 
A D-forest is a highly predictive structure; i.e., a judgment involving 
one of its word senses will in general predict the outcome of many 
other judgments. A detailed estimation of the efficiency of construction 
would involve the order of the senses, the configuration of the final 
structure, and the node-scanning strategy used in the discovery al- 
gorithm. It is sufficient here to recall, however, that nodes correspond 
to linguistic features, and hence one would expect that if the number 
of senses processed were very large, then the number of nodes would 
be much smaller. Thus all the nodes would appear relatively early 
in the construction process, and this leads to the conclusion that the 
total number of judgments would be a nearly linear function of the 
number of senses. In order to detect inconsistencies, however, some 
of this efficiency would have to be sacrificed and redundant judgments 
taken. 
To sum up this section, the series of structure classes considered for 
representing compatibility has led to one class, the D-forests, which is 
constructible. These structures also have several important properties 
in common with other models in theoretical and computational lin- 
guistics. 
CONSTRUCTIBLE REPRESENTATIONS FOR TWO SEMANTIC RELATIONS 43 
5. CONCLUSION 
This paper has presented two examples of the use of constructibility 
as a design criterion for lexical or semantic structures of the type used 
in natural language processing. This criterion is essential if we ever 
hope to implement computational systems with broad linguistic com- 
petence. It is very encouraging that in the two cases considered the 
approach has led to structures basically equivalent to those which have 
actually been used. Although these previous models have generally 
lacked the property of constructibility, then, it is suggested that this 
is a fault of design which can be overcome. 

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