A Tradeoff between Compositionality and Complexity in the 
Semantics of Dimensional Adjectives 
Geoffrey Simmons 
Graduiertenkolleg Kognitionswissenschaft 
Universit£t Hamburg 
Bodenstedtstr. 16 
D-W-2000 Hamburg 50 
Germany 
e-maih simmons@bosun2.informatik.uni-hamburg.de 
Abstract 
Linguistic access to uncertain quantita- 
tive knowledge about physical properties 
is provided by dimensional adjectives, 
e.g. long-short in the spatial and tempo- 
ral senses, near-far, fast-slow, etc. Seman- 
tic analyses of the dimensional adjectives 
differ on whether the meaning of the dif- 
ferential comparative (6 cm shorter than) 
and the equative with factor term (three 
times as long as) is a compositional func- 
tion of the meanings the difference and fac- 
tor terms (6 cm and three times) and the 
meanings of the simple comparative and 
equative, respectively. The compositional 
treatment comes at the price of a meaning 
representation that some authors (\[Pinkal, 
1990\], \[Klein, 1991\]) find objectionally un- 
parsimonious. In this paper, I compare 
semantic approaches by investigating the 
complexity of reasoning that they entail; 
specifically, I show the complexity of con- 
straint propagation over real-valued inter- 
vals using the Waltz algorithm in a system 
where the meaning representations of sen- 
tences appear as constraints (cf. \[Davis, 
1987\]). It turns out that the compositional 
account is more complex on this measure. 
However, I argue that we face a tradeoff 
rather than a knock-down argument against 
compositionality, since the increased com- 
plexity of the compositional approach may 
be manageable if certain assumptions about 
the application domain can be made. 
TOPIC AREAS: semantics, AI-methods in com- 
putational linguistics 
1 Introduction 
In the past decade, the field of knowledge represen- 
tation (KR) has seen impressive growth of sophis- 
tication in the representation of uncertain quantita- 
tive knowledge about physical properties in common- 
sense reasoning and qualitative physics. The input 
to most of these systems is entered by hand, but 
some of them, especially those with commonsense 
domains involving spatial and temporal knowledge, 
are amenable to interaction by means of a natural 
language interface. Linguistic access to knowledge 
about properties such as durations, rates of change, 
distances, the sizes of the symmetry axes of objects, 
and so on, is provided by dimensional adjectives 
(e.g. long-short in the spatial and temporal senses, 
fast-slow, near-far, tall-short). In this paper, I will 
investigate two aspects of their semantics that have 
an impact on the quality of a KR system with an NL 
interface. One aspect is the complexity of reason- 
ing entailed by their semantic interpretations. As an 
example, suppose that we have a text about the in- 
stallation of new kitchen appliances that contains the 
following sentences: 
(1) a. The refridgerator is about 60 cm wide. 
b. The cupboard is about as deep as the 
refridgerator is wide. 
c. The kitchen table is about 5 cm longer 
than the cupboard is deep. 
d. The oven is about twice as high as 
the table is long. 
We may view the relations expressed by these sen- 
tences as constraints on the measurements of the ob- 
ject axes (the width of the fridge, the depth of the 
cupboard, and so on), which are represented as pa- 
rameters in a constraint system. Then constraint 
propagation, along with some knowledge about the 
348 
sizes that are typical for object categories, should 
allow us to derive the following sentences (among 
others) from (1): 
(2) a. The cupboard is about 60 cm deep. 
b. The kitchen table is longer than 
the refridgerator is wide. 
c. The kitchen table is short 
(for a kitchen table). 
d. The oven is about 70 cm higher than 
the cupboard is deep. 
e. The oven is high (for an oven). 
The inferences from (1) to (2) are rather simple, 
but reasoning can become very complicated if a large 
number of parameters and constraints must be ac- 
counted for. As we will see below, the computational 
properties of this kind of reasoning are dependent on 
the types of relations that appear in the knowledge 
base. Thus in the present paper, I investigate the 
kinds of relations that appear in formal theories of 
the meanings of the following morphosyntactic con- 
structions of dimensional adjectives: 
(3) a. Positive 
The board is long/short. 
b. Comparative 
The board is (6 cm) longer/shorter than 
the table is wide. 
c. Equative 
The board is (three times) as long as 
the table is wide. 
d. Measurement 
The board is 50 cm long. 
This brings us to the second issue: the compo- 
sitionality of meaning representations proposed for 
the sentences in (3). It is appealing from the view- 
point of theoretical linguistics to regard each of the 
morphosyntactic categories (positive, etc.) as lexical 
items with their own semantics, and to assume that 
the semantics of each sentence in (3) is a composi- 
tional function of the semantics of the morphosyntac- 
tic category and the semantics Of the adjective stem. 
Compositional meaning representations may also be 
computationally more advantageous, since they can 
be computed very efficiently from syntactic represen- 
tations (e.g. in unification-based formalisms). 
Most formal theories of the meanings of adjectives 
attempt to fulfill this criterion of compositionality, 
but as we will see, they differ on a more far-reaching 
criterion: whether the meaning of the differential 
comparative (6 cm shorter than) and the equative 
with factor term (three times as long as) is a compo- 
sitional function of the meanings the difference and 
factor terms (6 cm and three times) and the meanings 
of the simple comparative and equative, respectively. 
Although compositionality is generally regarded as a 
virtue in and of itself, some authors (\[Pinkal, 1990\], 
\[Klein, 1991\]) have objected to compositional treat- 
ments of difference and factor terms on the grounds 
that they introduce an excessive amount of mathe- 
matical structure into our linguistic models. 
In section 3, I will compare semantic representa- 
tions that do and do not foresee a compositional 
treatment of difference and factor terms by analyzing 
the complexity of reasoning that they entail. In par- 
ticular, I will investigate the complexity of constraint 
propagation in a system where the meaning repre- 
sentations appear as constraints. In this paradigm, 
uncertain quantitative knowledge is accounted for 
with real-valued intervals, a popular choice in KR 
systems, and constraint propagation is performed by 
the Waltz algorithm (which gets its name from 
David Waltz \[1975\]). Ernest Davis \[1987\] shows in 
his detailed analysis that the Waltz algorithm is one 
of the best choices for this task, for reasons that I 
will explain in section 3.1 
It turns out that the constraint propagation with 
the Waltz algorithm under the compositional ap- 
proach is more complex; thus, we apparently face 
a tradeoff between compositionality and com- 
plexity. I argue in section 4 that this is indeed 
a tradeoff, since the non-compositional formation of 
meaning representations may be expensive, and the 
increased complexity of the compositional approach 
may be manageable, especially if certain assumptions 
can be made about the domain of physical properties 
being represented. 
2 Compositionality in the Semantics 
of Adjectives 
There is a vast amount of linguistic data on which 
a formal semantics of adjectives can be evaluated, 
such as the interaction of comparative and equative 
complements with scope-bearing operators: quanti- 
tiers, logical connectives, modal operators and neg- 
ative polarity items (e.g. John is taller than I will 
ever be). A good theory must also account for the 
phenomenon of markedness, i.e. the semantic asym- 
metry of the antonyms (see \[Lyons, 1977, Sect. 9.1\]). 
However, I will ignore these issues in order to focus 
on the matter of compositionality. Thus I classify the 
existing theories of adjective meaning very coarsely 
as 'compositional' or 'non-compositional'. Note that 
these labels indicate only whether or not the treat- 
ment of difference and factor terms is compositional 
(in other respects, all of the theories mentioned be- 
low are compositional). 
To begin with, I presuppose a component of di- 
mensional designation that determines which prop- 
erty of an object is described by an adjective, thus 
1I have only recently become acquainted with Eero 
Hyv5nen's "tolerance propagation" (TP) approach to 
constraint propagation over intervals (see \[Hyv6nen, 
1992\]), which in some circumstances can compute solu- 
tions that are superior to those of the Waltz algorithm, 
but at the price of increased complexity. I comment on 
this briefly in section 3.2. 
349 
Semantic Analyses of Dimensional Adjectives 
Formal interpretations of (3) 
a. Positive amount(length(board)){'q / r--}Nc(length(board)) 
b. Comparative amount(length(board)){-q / F)amount(width(table)) 
c. Equative amount(length(board)) ~ amount(width(table)) 
d. Measurement amount(length(board))= (50, cm) 
Table 1: Non-compositional approach 
a. Positive amount(length(board)){'q / r}D + We(length(board)) 
b. Comparative amount(length(board)){~ / f-}D rl: amount(width(table)) 
c. Equative amount(length(board)) ~_ n x amount(width(table)) 
d. Measm-ement amount(length(board)))=(50, em) 
Table 2: Compositional approach 
determining that short conference describes a dura- 
tion but short stick describes the length of the stick's 
elongated axis. Each class of properties (duration, 
length, etc.) is assumed to be associated with a set 
of degrees reflecting their magnitudes. I will sim- 
ply use the function expression amount(p(x)) to de- 
note the degree to which entity x exhibits property 
p. Each set of degrees is assumed to be ordered, 
and I will use the symbols I- and E for the ordering 
relation. Most authors assume measurement theory 
(\[Krantz et al., 1971\]) as the axiomatic basis in the 
formal semantics of linguistic measurement expres- 
sions (cf. \[Klein, 1991\]). For measurement expres- 
sions such as 3 cm, I simply use a tuple (3, cm) de- 
noting a degree. Finally, I follow \[Bierwisch, 1989\] 
in using the symbol We(a) for the 'norm' expected 
for amount a in context C. This reflects the usual 
assumption that the positive expresses a relation to 
a context-dependent standard. In this paper, I will 
restrict my attention to norms that are typical for 
the categories named in the sentence, such as tall for 
an adult Dutchman, slow for a sports car, etc. 2 
The class of theories that I am referring to as 'non- 
compositional' include those of \[Cresswell, 1976\], 
\[Hoeksema, 1983\] and \[Pinkal, 1990\], who propose 
formulas similar to those in Table 1 as interpreta- 
tions of the sentences in (3). The relation used in 
place of the expression {-\] / \[-'} is -1 for the un- 
marked case (e.g. tall) and 1- for the marked case (short) .3 
2Clearly, there are many other kinds of norms. Jan 
is tall may mean tall for his age, taller than I expected, 
etc. \[Sapir, 1944\] is still one of the best surveys of the 
norms employed in natural language, while Bierwisch has 
a more modern analysis. 
3Of course, Tables 1 and 2 are strong simplifica- 
I call this approach non-compositional because in- 
terpretations of the differential comparative (6 cm 
longer than) and of the equative with factor term 
(three times as long as) are not derivable from the 
formulas shown in lines (b) and (c) (the same can be 
said of \[Kamp, 1975\] and \[Klein, 1980\]). 
The compositional approach is taken by \[Hellan, 
1981\], \[von Stechow, 1984\] and \[nierwisch, 1989\], 
whose renderings of (3) are, in simplified form, some- 
thing like those in Table 2. The symbol '+' is + in 
the unmarked case and - in the marked case, and 
'x' stands for scalar multiplication. 4 
In the case of the positive and the ordinary com- 
parative, the difference term D is existentially quan- 
tified, as is the factor term n in the case of the ordi- 
nary equative (with the additional condition that n 
is greater than or equal to one). But if the difference 
or factor term is realized in the sentence surface, then 
its contribution to (b) and (c) in \]?able 2 is embedded 
compositionally. 5 
tions that fail to reflect important differences between 
the authors mentioned that are unrelated to the issue of 
compositionaiity. 
4In measurement theory, the '+' operation is inter- 
preted as concatenation in the empirical domain, and 
scalar multiplication is interpreted as repeated concate- 
nation. Krantz et at. \[1971\] show that under proper ax- 
iomatization, concatenation is homomorphic to addition 
on the reals. 
SBierwisch \[1989\] differs from the other authors ad- 
vocating a compositional approach in that he does not 
assume the interpretation of the equative shown in Ta- 
ble 2. He points out (p. 85) that this analysis does not 
account for the fact that the equative is norm-related in 
the unmarked case: Fritz is as short as Hans presup- 
poses that Fritz and Hans are short. Moreover, it is not 
clear whether this approach can capture the duality of 
350 
For the computational analysis, we will need to 
classify the relations shown in Tables 1 and 2, since 
these relations form the input to a knowledge base. 
But to do so, we must first decide what sorts of en- 
tities the difference and factor terms denote. I as- 
sume that they do not denote constants, since we 
may be just as uncertain of their magnitudes as we 
are of the other magnitudes mentioned in the sen- 
tences. Thus it should be possible to treat each of 
the mini-discourses in (4)-(6) in a similar fashion: 
(4) a. The board is 90 to 100 cm long. 
b. In fact, it is about 95 cm long. 
(5) a. The board is longer than the table is wide. 
b. In fact, it is about 6 cm longer. 
(6) a. The board is five to ten times as long as 
the table is wide. 
b. In fact, it is about seven times as long. 
The information given in (b) in (4)-(6) can be ac- 
counted for by simply modifying the terms intro- 
duced in (a). Hence, the difference and factor terms, 
like the 'amount' terms in Tables 1 and 2, denote 
uncertain quantities whose magnitude may be con- 
strained by sets of sentences. I will refer to these 
terms generally as 'parameters'. 
With this assumption, we can classify the relations 
in Tables 1 and 2 as follows: 
(7) Non-compositional 
a. Ordering relations 
(Positive, Comparative, Equative) 
b. Linear relations 
of the form amount(x) + D ~ amount(y) 
(Differential Comparative) 
c. Product relations 
of the form n x amount(x) ~_ amount(y) 
(Equative with factor term) 
(8) Compositional 
a. Linear relations 
(Positive, Comparative, Differential Comparative) 
b. Product relations 
(Equative with & without factor term) 
In both approaches, measurements simply serve to 
identify the degree to which an object exhibits the 
property in question. 
Under the compositional approach, it is possible to 
assume a single semantic representation in the lex- 
icon for each adjective stem and each morphosyn- 
tactic category such that the formulas in Table 2 
are generated from those lexical entries. Bierwisch 
\[1989\], for example, proposes lexical entries of the 
following form for each dimensional adjective: 
~c~x\[amount(p(x) ) = (v :t: c)\] 
comparatives and equatives: Fritz is taller than Hans 
should be semantically equivalent to Hans is not as tall 
as Fritz. However, Bierwisch does assume a representa- 
tion like this for equatives with realized factor terms. 
where c is a difference value and v is a comparison 
value (see \[nierwisch, 1989\] for details). 
But the elegance of the compositional approach 
comes at the price of lexicM semantic representations 
that include addition and multiplication operators~ 
which is precisely what Pinkal \[1990\] and Klein \[1991\] 
have criticized: they find the assumption of math- 
ematical operations as basic constituents of lexical 
meaning uncomfortably strong. This is one of the 
reasons why Pinkal proposes separate lexical entries 
for each morphosyntactic form of an adjective. 
3 The Complexity of Constraint 
Propagation 
The objection to the complexity of the lexical mean- 
ing representations required for the compositional 
approach appeals to intuitions of parsimony, and is 
in part a matter of philosophical opinion that may 
be difficult to resolve. Perhaps a decision could be 
made on the basis of psycholinguistic experimenta- 
tion, but I will pose a more utilitarian question in 
this section by examining whether the increase in 
representational complexity in the transition from 
Table 1 to Table 2 entails an increase in the com- 
putational complexity of reasoning for a knowledge 
base containing those representations. The reasoning 
paradigm to be investigated is constraint propaga- 
tion (sometimes called constraint satisfaction) over 
real-valued intervals. 
Intervals are intended to account for uncertainty 
in quantitative knowledge. For example, the mea- 
surement of a parameter at 20 units on some scale 
with a possible measurement error of +0.5 units is 
represented as \[19.5, 20.5\], to be interpreted as mean- 
ing that the unknown measurement value in ques- 
tion lies somewhere in the set {x119.5 <_ x <_ 20.5}. 
Additional knowledge about the relations that hold 
between parameters constrains their possible values 
to smaller sets (hence the term 'constraints' for the 
propositions in a knowledge base expressing such re- 
lations). 
Constraint propagation over intervals has been ap- 
plied in spatial reasoning (\[McDermott and Davis, 
1984; Davis, 1986; Brooks, 1981; Simmons, 1992\]), 
temporal reasoning (e.g. \[Dean, 1987; Allen and 
Kautz, 1985\]) and in systems of qualitative physics 
(see \[Weld and deKleer, 1990; Bobrow, 1985\]). In- 
tervals have a very obvious weakness in that the 
highly precise choice of endpoints can rarely be well- 
motivated in natural domains such as these. In par- 
ticular, the reasoner may draw very different infer- 
ences, e.g. about whether two intervals overlap, if 
the endpoint of some interval is changed by what 
seems to be an insignificant amount. Thus, as Me.- 
Dermott and Davis\[1984\] note, such a system must 
not only be able to report whether they overlap, but 
also "how close" they come to overlapping. 
If they do come close ..., then ...\[the 
351 
reasoner\] must decide whether to act on the 
suspect information or work to gather more, 
which is really the only interesting decision 
in a case like this. Eventually, when all 
possible information has been gathered, if 
things are still close to the borderline then a 
decision maker must just use some arbitrary 
criterion to make a decision. We don't see 
how anyone can escape this. \[McDermott 
and Davis, 1984, p. 114\] 
A formalism such as fuzzy logic attempts to al- 
leviate the problem of sharp borderlines by using 
infinitely many intermediate truth values for vague 
predicates. I happen to have reservations about the 
adequacy of fuzzy logic for this task 6, but I have cho- 
sen to study constraint propagation mainly because 
its computational properties are well-researched and 
are attractive for applications in which the potential 
overprecision of endpoints can be tolerated. Thus it 
provides a sound basis for comparing the semantic 
analyses presented in section 2. 
3.1 Syntax and Semantics 
In the following, I briefly review some definitions 
from \[Davis, 1987, Appendix B\] (with slight modi- 
fications) 
Syntax 
Assume a set of symbols X = {XI,..., X v} called 
parameters. A label is written \[z_, x+\] with real 
numbers 0 < z_ <__ z:~; the symbol oo may also be 
used for z_ and z+. A labelling L for X is a 
function from parameters to labels. If L is under- 
stood, we write Xi - \[z_, z+\] for L(Xi) = \[z_, z+\]. 
A constraint is a formula over parameters in X 
in some accepted notation (e.g. X1 x X2 = )(3 or 
p _< -XI + X2 + )(3 <_ q). A constraint system 
C = (X, C, L / consists of a set X of parameters, a 
set C of constraints over X, and a labelling L for X. 
Semantics 
A valuation V for X is a function from the 
parameters to reals. The denotation of a label 
\[z_,z+\] is the set D(\[z_,z+\]) = {z\[z_ < z _< z+} 
if z+ # oo, D(\[z_,co\]) = {z\]z_ _< z} if z_ # oo, 
D(\[oo, oo\]) -- {oo) otherwise. A labelling L is in- 
terpreted as restricting the set of possible valua- 
SThis is not because I object to the notion of truth 
measurement, but rather because I believe that the fuzzy 
logicians' assumption that the connectives of a logic of 
vagueness are truth functional is contradicted by the 
facts of human reasoning about vague concepts (as ar- 
gued by \[Pinkal, to appear\]). In my opinion, a formalism 
for truth measurement would have to be more like prob- 
ability theory. 
TI assume the non-negative reals for simplicity, be- 
cause most of the physical properties mentioned in the 
examples have non-negative measurement scales. Even 
some of the exceptions, such as the common temperature 
scales, ate in fact equivalent to a scale of non-negative 
values. 
tions for X to those V such that for all Xi E X, if 
L(XI) = \[x_,z+\], then V(X~) E D(\[x_,z+\]). Thus 
we may view L as denoting a set of valuations on the 
parameters; we refer to this set as V(L). 
A constraint C i denotes the largest set of valua- 
tions that are consistent with the relation expressed 
by Cj; call this set V(Cj). 
3.2 Constraint Propagation Algorithms 
The task of a constraint propagation algorithm 
(CPA) is to tighten the interval labels in an attempt 
to either (1) find a labelling that is just tight enough 
to be consistent with the constraints and initial la- 
belling, or (2) signal inconsistency. Constraint prop- 
agation separates a stage of assimilation, during 
which intervals are tightened, from querying, dur- 
ing which the tightened values are reported. It is 
also possible to infer previously unknown relations 
between the parameters in the querying stage by in- 
specting the tightened intervals. This method of rea- 
soning may be applied in the linguistic application 
under study, for example to derive the sentences in 
(2) above from (1). 
A CPA is sound if V(Cl)n...VIV(Cn)nV(LI) C_ V(L) 
for every labelling L returned by the algorithm, 
where {el,...,Ca} is the set of constraints in the 
system and L1 is the initial labelling. It is complete 
if V(L) C V(Cl) n ... VI V(Cn) N V(L1) for every 
L that it returns. In other words, the algorithm is 
sound if it does not eliminate any values that are 
consistent with the starting state of the system, and 
complete if it returns only such values. 
As we will see, CPA's for intervals can only be 
complete under very restricted circumstances. Thus 
Davis defines a weaker form of completeness for the 
assimilation process. A CPA is complete for as- 
similation if every labelling L that it returns as- 
\[z_,x+\] such that if Vi(Xi) e signs labels Xi - i i 
D(\[zi.., z~.\]), then l~ • Y(C1) n... N Y(Cn). That 
is, the label assigned to each parameter accurately 
reflects the range of values it may attain given the 
constraints in the system. 
The Waltz algorithm, which is stated below, is su- 
perior to many other CPA's in these respects. It is 
a sound algorithm, unlike the Monte Carlo method 
used by \[Davis, 1986\] and the hill-climber used by 
\[McDermott and Davis, 1984\]. Moreover, for con- 
straint systems containing restricted types of con- 
straints, the Waltz algorithm is complete for assim- 
ilation and terminates very quickly. In contrast, 
Davis reports that the h{ll-climbers used by \[McDer- 
mott and Davis, 1984\] were prohibitively slow and 
unreliable. 
The algorithm is based on an operation called re- 
finement, defined as follows. Given a constraint Cj, 
a parameter Xi appearing in Cj, and labelling L de- 
fine: 
REFINE(Q, Xi, L) = {Y'(Xi)\]Y' • V(L)rW(Cj)} 
352 
Relation 
Order O(pc) 
Unit Linear O(pS)* 
Inequality 
Product O(pS) t 
Time Complexity Completenessll 
Assimilation 
Incomplete 
Incomplete 
Complexity of 
Complete Solutions 
O(p ~) 
As hard as 
linear programming 
NP-hard 
Table 3: Complexity of the Waltz algorithm for various systems of relations 
(from \[Davis, 1987\] and \[Simmons, 1993\]) 
p = number of parameters, c = number of constraints 
S = size of the system (the sum of the lengths of all of the constraints) 
* May not terminate if the system is inconsistent 
tTerminates in arbitrarily long (finite) time if the system is inconsistent 
tMay not terminate if the solution is inadmissible (see text) 
This is the set of values of Xi that consistent with 
both the labelling and the constraint. 
The two refinement operators for a constraint 
Cj and parameter Xi are functions from labellings 
to labellings, written R-(Xi,Cj) and R+(Xi,Cj). 
If L(Xi) = \[x/_,x~\], then R-(Xi,Q)(L)is formed 
by replacing x/__ in L with the lower bound of 
REFINE(Cj, Xi, L), and R+(Xi, Cj)(L) is formed 
by replacing x~ in L With the upper bound of 
REFINE(Cj, Xi, L). We say that these refinements 
are based on Cj. If the upper and lower bounds 
of REFINE are computable, then refinement is by 
definition a sound operation. 
For a constraint system C = (X, {C1,..., Ca}, L), 
L is quiescent for a set of refinement operators R = 
{R1,...,R,} if RI(L) = ...= R,~(L) = L. The 
solution to C (if it exists) is the labelling L' denoting 
the largest set of valuations V(L') C_ V(L)N V(Ct)N 
• .. f'l V(C,~) such that L' is quiescent for any set of 
refinements based on the constraints in the system. 
If no such solution exists, then C is inconsistent. 
The Waltz algorithm repeatedly executes refine- 
ments until the system is quiescent, and returns the 
solution (or signals inconsistency) if it terminates (cf. 
\[Davis, 1987, p. 286\]). 
procedure WALTZ 
L *-- the initial labelling 
Q *-- a queue of all constraints 
while Q ~ @ do 
begin remove constraint C from Q 
for each Xi appearing in C 
if REFINE(X~, C, L) = 
then return INCONSISTENCY 
else L *-- the result of executing 
R-(Xi, C) and n+ ( xi , C) on L 
for each Xi whose label was changed 
for each constraint C' ~ C in which Xi appears 
add C I to Q 
end 
Since refinement is a sound operation, the Waltz 
algorithm is sound. The completeness, termination 
and time complexity of the algorithm depends on 
what kinds of relations appear as constraints in the 
system, and on the order in which constraints are 
taken off the queue. The results for systems consist- 
ing exclusively of one of the three kinds of relations 
mentioned in (7)-(8) in section 2 are given in Table 
3, under the assumption that constraints are selected 
in FIFO order or a fixed sequential order (other or- 
derings lead to worse results). Time complexity is 
measured as the number of iterations through the 
main loop of the algorithm. For comparison, Table 
3 also gives the best known times for complete solu- 
tions to systems of such relations, s 
In the linguistic application proposed here, the 
term S in Table 3 (the sum of the lengths of all of 
the constraints) is proportional to c (the number of 
constraints), since there are no more than three pa- 
rameters in each constraint. Hence, O(pS) is O(pc) 
in this application. 
Note that Table 3 gives results for linear inequali- 
ties with unit coefficients (of the form p < )'~ Xi - 
~j Xj < q, where no coefficients differ from 1 or 
-1). These are the only kind of linear inequalities 
under consideration in the linguistic application. In 
general, the Waltz algorithm breaks down if the sys- 
tem contains more complex relations, such as linear 
inequalities with arbitrary coefficients or product re- 
lations, since it may go into infinite loops even if the 
starting state of the system was consistent. Con- 
sider, for example, the set of constraints {nl x X = 
Y, n2 x X = Y} with the starting labels nl - \[1; 1\], 
n2 --" \[2, 21, X - \[0,100\] and Y - \[0,100\]. The sys- 
tem continually bisects the upper bounds of X and Y 
without ever being able to reach the solution, which 
SHyvSnen's \[HyvSnen, 1992\] tolerance propagation 
(TP) approach is similar to the Waltz algorithm, but 
it uses a queue of solution functions from interval arith- 
metic \[Alefeld and Herzberger, 1983\] rather than refine- 
ment operations. The "global TP" method computes 
complete solutions, but at the price of increased com- 
plexity. In the "local" mode, tolerance propagation is 
very similar to the Waltz algorithm in its computational 
properties. 
353 
is X - \[0, 0\] and Y -" \[0, 0\]. Similarly, if the starting 
labels are X - \[1, ~\] and Y - \[1, c~\], then the the 
lower bounds are continually doubled without reach- 
ing the solution X - leo, ~\] and Y - \[oo, oo\]. 
However, it is shown in \[Simmons, 1993\] that this 
happens only if the solution contains labels of this 
kind. Define a label as admissible if it is not equal 
to \[0, 0\] or \[0% oo\]; otherwise, it is inadmissible. A 
labelling L is admissible if it only assigns admissible 
labels; otherwise, L is inadmissible. Then it can be 
shown that if a system of product constraints is con- 
sistent and its solution is admissible, then the Waltz 
algorithm terminates in O(pS) time. Moreover, if 
the system is inconsistent, the algorithm will find 
the inconsistency in finite but arbitrarily long time. 
Unfortunately, the proof is too long to include in the 
present paper, but a brief outline of the argument is 
given in the Appendix. 
Systems with linear inequalities or product con- 
straints are liable to enter infinite or very long loops 
if the starting state is inconsistent (or if the solution 
is inadmissible in the case of products). Davis \[1987, 
p. 305-306\] suggests a strong heuristic for detecting 
and terminating such long loops: stop if we have been 
through the queue p times (for p parameters). He is 
not clear on what he means by "having been through 
the queue z times", but I interpret him as meaning 
that we should stop if any constraint has been taken 
off the queue more often than p times. The rationale 
is the observation that in practice, most systems that 
do terminate normally seem to do so before this con- 
dition is fulfilled, much sooner than the worst-case 
time predicted by the complexity analysis. The reli- 
ability of such a heuristic is one of the topics of the 
next subsection. 
3.3 Empirical Testing 
The analytic results given in the previous subsection 
have left two important questions open: 
• What is the complexity of constraint propaga- 
tion if the system contains different kinds of con- 
straints? 
• How reliable is Davis' heuristic for terminating 
infinite (or very long) loops? 
The first question lends itself to an analytic an- 
swer, but the results are not known at present. But 
we can seek empirical evidence by running the al- 
gorithm on mixed systems of constraints to see if 
the time to termination is significantlY greater than 
the complexity expected for systems containing just 
the most complex type of relation in the system. If 
this does not happen for a number of representa- 
tive systems, we may conjecture that the combina- 
tion of constraints has not made the problem more 
complex. The second question can only be answered 
empirically, by testing whether the heuristic tends 
to terminate the algorithm too soon (i.e. whether 
it terminates refinement of systems that might have 
terminated normally in a short time). 
Empirical investigations of these questions are re- 
ported in \[Simmons, 1993\], and described briefly 
here. To investigate the first question, the algorithm 
was run on a number of large, consistent constraint 
systems with admissible solutions in which the three 
types of constraints shown in Table 3 appeared in 
approximately equal numbers. On each run, the con- 
straints in the initial queue were permuted randomly 
to suppress the possible effects of ordering. None of 
these runs required more time to termination than 
is predicted by the O(pS) result for systems con- 
taining just unit linear inequalities or just product 
constraints. 
To investigate the second question, I attempted 
to build consistent constraint systems with admissi- 
ble solutions that are terminated by Davis' heuris- 
tic sooner than they would have been normally. It 
turns out that the algorithm runs to completion on 
almost all systems that were tested long before any 
constraint is taken off the queue p times, although 
there are systems for which refinement is terminated 
too soon on this heuristic. If the limit is increased 
by a constant factor, e.g. if assimilation is stopped 
after some constraint is processed 2p times, then the 
risk of early termination is greatly reduced. 
In all, the empirical results on the open questions 
mentioned above have been encouraging. It is an 
admitted weakness of these tests, however, that they 
were performed on systems built by hand, not on 
constraint systems that occur "naturally" as part of 
an NL interface to a KR system. 
4 Conclusions 
The results of the previous section yield Tables 4 
and 5 as the complexity of reasoning with the Waltz 
algorithm under the non-compositional and compo- 
sitional approaches, respectively. These results de- 
pend in part on the fact that there is a maximum 
number of parameters in each constraint in the lin- 
guistic application. Measurements are modelled as 
predicate constraints, i.e. they simply impose inter- 
val bounds on some parameter. Intervals are also 
assumed to model the range of measurement values 
for the physical property that is typical for members 
of a category (e.g. the typical width of refridgera- 
tots), thus accounting for the norm used in the in- 
terpretation of positives. An important property of 
such "norm intervals" is that they may not be re- 
fined, at least not too much. This may be achieved 
by adding constraints imposing absolute upper and 
lower bounds on their ranges (cf. \[Simmons, 1992\]). 
Although the worst-case time complexity in all 
cases turns out to be the same, the compositional 
approach is more complex for two reasons. First, 
the system is prone to enter infinite loops under the 
compositional approach if the starting state is incon- 
sistent, or if the solution is inadmissible. Consistency 
cannot generally be guaranteed in the linguistic ap- 
plication under consideration, since the sentences in 
354 
Non-compositional I Morphosyntactic Relation 
Category II 
Measurements 
Positive 
Comparative 
Equative 
Differential 
comparative 
Equative w/ 
factor term 
Predicate 
Order 
Order 
Order 
Linear 
Inequality 
Product 
Time Complexity 
trivial OIpc} 
pc OIpc O~ 
O(pe), 
o(pc)t 
Completeness 
Complete 
Assimilation 
Assimilation 
Assimilation 
Incomplete 
Incomplete 
Table 4: Complexity of reasoning under the non-compositional approach 
I Morphosyntactic Category II 
Measurements 
Positive 
Comparative 
Equative 
Differential 
comparative 
Equative w/ 
factor term 
Compositional Relation 
Predicate 
Linear Inequality 
Linear Inequality. 
Product 
Linear 
Inequality 
Product 
Time Complexity 
trivial O(pc), 
O(pc), 
O(pc)t O(pc), 
O(pc) t 
Completeness 
Complete 
Incomplete 
Incomplete 
Incomplete 
Incomplete 
Incomplete 
Table 5: Complexity of reasoning under the compositional approach 
p = number of parameters, c = number of constraints 
• May not terminate if the starting state is inconsistent 
tTerminates in arbitrarily long (finite) time if the system is inconsistent 
fMay not terminate if the solution is inadmissible 
a text may contain errors. Second, reasoning under 
the compositional approach is incomplete in all but 
the trivial case of measurements, whereas the non- 
compositional approach guarantees at least assimi- 
lation completeness for a subset of the parameters 
in the system. This means that under the compo- 
sitional approach, the reasoner does not refine some 
intervals as tightly as it could have under the non- 
compositional approach. 
These results may be taken as grounds for reject- 
ing the compositional approach to the semantics of 
dimensional adjectives in the design of an NL in- 
terface to a KR system for quantitative knowledge. 
However, I do not believe that the compositional ap- 
proach is contraindicated for all conceivable systems. 
In addition to the general theoretical appeal of com- 
positional semantics, the compositional formation 
of meaning representations may be computationally 
more attractive in some cases (e.g. in unification- 
based formalisms). Thus if the non-compositional 
formation of semantic representations turns out to 
be too expensive, it may defeat the computational 
advantage gained in the reasoning process. 
This is especially true if the weaknesses of the com- 
positional approach do not turn out to be highly 
relevant in the specific application. For example, if 
the domain of physical properties being represented 
is such that a set of constraints requiring some pa- 
rameter to be set to \[0, 0\] or \[c~, co\] is unlikely to 
be encountered, and hence the solution is likely to 
be admissible, then the risk of infinite loops is re- 
duced. Moreover, if Davis' heuristic for terminating 
infinite loops turns out to be reliable (which might 
be determinable by experimentation within the spe- 
cific application), then inconsistencies need not be 
very damaging. 
The incompleteness of reasoning under the com- 
positional approach is unacceptable for an applica- 
tion if it is crucial that the inferred intervals con- 
tain precisely those values that are warranted by the 
constraints and the initial labelling. If a superset of 
those values can be accepted, however, then the com- 
positional approach can be taken. Both approaches 
suffer a lack of what Davis calls query complete- 
ness: if the value of a term T is to be determined 
during the querying stage (i.e. after assimilation), 
355 
the system may return a superset of the values for T 
that are warranted by the constraints. 
Thus an engineer building an NL interface to a 
system for reasoning about uncertain quantitative 
knowledge of physical properties must make a num- 
ber of design decisions: 
• How important are difference and factor terms 
in the linguistic material to be processed? 
If difference and factor terms are so marginal 
that they may not occur at all, then the non- 
compositional approach is probably the better 
choice, due to its guarantee of termination and as- 
similation completeness. 
• Does the compositional generation of lexical se- 
mantic representations have a significant advan- 
tage (computational or otherwise) over the non- 
compositional approach? 
• Is it possible or likely for the measurement of 
some physical property to be exactly zero? 
While there is probably no natural application in 
which the magnitude of some property can be in- 
finitely large, there are different philosophies about 
the treatment of zero. In a system of temporal rea- 
soning, for example, saying that some event has zero 
duration may be a way of saying that the event does 
not exist. But another policy might be to insist that 
no physical property is represented if it is not exhib- 
ited to a positive degree. If this assumption can be 
made, then the intervals \[0, 0\] and \[c¢, oo\] are truly 
inadmissible, and hence one weakness of the compo- 
sitional approach is diminished. 
• Is it important that the precise range of permis- 
sible measurement values be inferred for each 
parameter, or can a superset of those values be 
useful? 
If a superset of the possible values is acceptable, then 
the compositional approach can be chosen. Other- 
wise, the non-compositional approach must be taken. 
By weighing the various answers to these ques- 
tions, an engineer can stake out a position on the 
tradeoff and design a system with the power and ef- 
ficiency most appropriate to his or her needs. 
Acknowledgements 
Thanks to Carola Eschenbach, Claudia Maienborn, 
Andrea Schopp, Heike Tappe and the referees for 
their comments on earlier versions of this paper. 
Thanks also to Longin Latecki for discussions about 
constraint propagation, and to Christopher Habel for 
encouraging me to pursue this work. 
Appendix 
In the following, the proof of the following theorem 
(from \[Simmons, 1993\]) is briefly outlined: 
Theorem 1 If a system of product constraints is 
consistent and its solution is admissible, then the 
Waltz algorithm brings it to quiesenee in time O(pS). 
Recall that a product constraint is of the form 
~i Xi = Y, and that a labelling is admissible if it 
does not assign \[0, 0\] or \[c~, oo\] to any parameter. 
First we need some terminology defined in \[Davis, 
1987, Appendix B\] (recall the definition of refine- 
ment operators in section 3.2 above). 
For a refinement operator R, let OUT(R) be the 
bound affected by R, and let ARGS(R) be the set 
of bounds other than OUT(R) that enter into the 
computation of OUT(R). Given a labelling L, R is 
active on L if it changes L, i.e. if L ~ R(L). 
A series of refinement operators T~ = (RI,..., Rm) 
is active if each refinement in T~ is active. We say 
that Ri is an immediate predecessor of Rj in 7~ 
if i < j, OUT(Ri) E ARGS(Rj), and for all k such 
that i < k < j, OUT(Rk) # OUT(I~). In other 
words, some argument of P~ has been set most re- 
cently in the series by Rj. We say that Ri depends 
on Rj if either i = j or Ri depends on Rk and Rj 
is an immediate predecessor of Rk. Thus the depen- 
dence relation is the transitive and reflexive closure 
of the immediate precedence relation. We say that 
Ri depends on bound B if for some Rj, Ri depends 
on Rj and B E ARGS(Rj). 
The series of refinements T~ = (R1,..., R~) is self- 
dependent if Rn depends on OUT(Rn), its own out- 
put bound. In other words, a series is self-dependent 
if the last bound affected by the series is also an argu- 
ment to the first refinement in a chain of refinements 
in the precedence relation, as illustrated below. 
(OUT( Rn ~OUT( R, }~-~OUT( R2 } . . . ~ 
Davis shows that such self-dependencies are po- 
tential infinite loops: 
Theorem 2 Any infinite sequence of active refine- 
ments contains an active, self.dependent subsequence 
(\[Davis, 1987, Lemma B.15\]}. 
In \[Simmons, 1993\], it is shown that if any self- 
dependent sequence 7~ is active on the labelling of a 
system of product constraints, then a certain sub- 
sequence T~' of ~ will be active infinitely many 
times. Moreover, on the rn-th execution of each re- 
finement Ri in ~', there is a term 7~n/, where each 
T/m > T/m-1 > 1, such that OUT(Ri) is multiplied 
by: 
(T~) -1, if OUT(e,) is an upper bound 
sty-, if OUT(R~) is a lower bound 
It follows that upper bounds are refined so as to 
become arbitrarily small (asymptotically approach- 
ing zero), and that lower bounds become arbitrarily 
large, up to infinity. 
Thus if there is any constraint Ci in the system 
that imposes a lowest value greater than zero on an 
356 
upper bound that is affected by a refinement oper- 
ator in ~', that bound will be refined often enough 
until it becomes inconsistent with Ci. Similarly, if 
any constraint Cu imposes a largest finite value on 
a lower bound that is affected by a refinement in 
7U, then that bound will be refined until it becomes 
inconsistent with Cu. In both cases, the system is 
inconsistent. 
If there are no such constraints, then it is consis- 
tent for upper bounds affected by T~' to be asymp- 
totically close to zero and for lower bounds affected 
by T~' to be arbitrarily large. This can only be con- 
sistent if, in the case of upper bounds, the solution 
assigns \[0, 0\] to the parameter in question, and in the 
case of lower bounds, the solution assigns \[co, oo\] to 
its parameter. Hence, the solution is inadmissible. 
But according to Davis' result (Theorem 2), in- 
finite loops must contain an active, self-dependent 
subsequence such as 7~. It follows that if a system 
of product constraints is consistent and its solution 
is admissible, then the Waltz algorithm finds its so- 
lution in finite time. The time complexity result is a 
straightforward extension of Davis' analysis of unit 
linear inequalities (see \[Simmons, 1993\]). 
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