A General Computational Treatment of Comparatives 
for Natural Language Question Answering 
Bruce W. Ballard 
AT&T Bell Laborotories 
600 Mountain Avenue 
Murray Hill, N.J. 07974 
Abstract 
We discuss the techniques we have developed and 
implemented for the cross-categorial treatment of 
comparatives in TELl, a natural language question- 
answering system that's transportable among both 
application domains and types of backend retrieval 
systems. For purposes of illustration, we shall 
consider the example sentences "List the cars at least 
20 inches more than twice as long as the Century is 
wide" and "Have any US companies made at least 3 
more large cars than Buick?" Issues to be considered 
include comparative inflections, left recursion and 
other forms of nesting, extraposition of comparative 
complements, ellipsis, the wh element "how', and the 
translation of normalized parse trees into logical 
form. 
1. Introduction 
We shall describe a general treatment of 
comparatives that has been implemented in the 
context of TELI, a question-answering system which 
is transportable among both domains of discourse and 
different types of backend retrieval systems.n 
Comparatives are important because of the dramatic 
increase in expressive power they allow; they are 
interesting at least because of the variety of issues 
(from morphology on up) one must deal with in 
order to provide for them. 
1. The examples in this paper illustrate TEL1 us a front-end to 
the Kandor knowledge representation system (Patel-Schneider, 
1984); we will give examples in terms of a knowledge base of 
information about 1987 cars. TELI has produced queries for 
at least four different "backend" systems and has been adapted 
for over a dozen domains of data. 
41 
1.1 Goals 
In seeking to provide TEL1 with general capabilities 
for comparatives, our primary goals have been 
to formulate cross-categorial techniques that treat 
the comparativizations of different syntactic elements 
(e.g. adjectives, quantifiers, and measure nouns) with 
the same mechanisms; 
to allow comparatives to be composed with 
themselves (e.g. "at least 3 more than 3 times as 
many') and with other syntactic features (e.g. wh 
elements); 
to be faithful to what is known from work in 
theoretical linguistics; we draw from Bresnan (1973), 
Cushing (1982), Dik (1980), Jackendoff (1977), 
Sells (1985), and Winograd (1983); 
to account for as many of the specific cases of 
comparatives found in the literatureof implemented 
NL processors as possible. 
1.2 Achievements 
Letting <X> denote a grammatical category to be 
comparativized, we begin by providing for 
comparativized structures C{<X>} of the form 
C{<X>} -.* (<Qmd>) CC{<X>) <Comp> 
<Qua> -'* *tmostlatleutlaolexsctlylg~'dmyljastlealy 
CC{<X>} -=*" (CC{<X>}) (<Measure>) <el> (<X>) <c2> 
<Measure> --* <Number> (<Ordinai>lperc~tltinNs) I 
<onus> --* h~lt~ltUrdsl-- 
<Comp> --0 <NP> <Etcx> 
<el>/<c2> .-4, -er/flum\[less/thu\[ss/us 
where (...) denotes optionality; "/" indicates 
"agreement" between comparative particles; and 
<Etcx> accounts for items parallel to those in the 
matrix clause in which the comparative occurs (e.g. 
"cars that are longer than the Regal (is (wide))'). In 
addition, a variety of extrapositions (i.e. rightward 
and occasional leftward movement) from C{<X>} 
may (and sometimes must) occur. For example, both 
"cars larger than the Century" and "larger cars than 
the Century" are allowed. 
Since we wish to allow C{<X>} structures 
to occur wherever <X> could occur, arbitrarily 
complex interactions with quantifiers (within the 
complement), ordinals, superlatives, raisings, wh 
elements, and other constructs must be provided for. 
In addition to the structures indicated by the BNF 
above, we allow for some simpler expressions not 
conventionally classified as comparatives. Some 
examples are "6 ears" (cf. "as many as 6 cars') and 
"3 inches long" (cf. "as long as 3 inches'). We also 
provide for structures involving the nominal 
counterpart of an adjective, as in "more than 185 
inches in length'. 
To date, we have fully implemented a wide 
variety of comparatives related to adjectives, 
quantifiers, and measure nouns (e.g. "cars that cost 
at least $100 more than the Park Avenue'). Due to 
the commonality among the comparativized syntactic 
structures, our grammar for these three types of 
comparatives is produced by meta-rules suggested by 
the BNF rules shown above. Although the feature 
agreement provided by our parser is used to eliminate 
spurious structures such as "cars more than 3 
(inches/*dollars) long', we avoid conflicts between 
pure numbers and measure phrases that involve a 
unit (e.g. "companies that make more than 3 
(*dollars) cars') by having two (very nearly 
identical) Quantity routines in the grammar. 
1.3 Lhnitatioas" 
In addition to some specific limitations to be stated in 
the remainder of the paper, there are some general 
limitations of our work to date, many of which are 
being rectified by the work mentioned in Section 8.3. 
(1) By analogy with conjunctions, with which 
comparatives share a number of properties (cf. Sager 
1981, pp. 196ff), our comparative particle pairs (- 
er/than etc.) provide for co-ordinate comparatives, in 
contrast to pairs such as so/that, as in "Buick makes 
so many cars that it's the largest company." (2) 
Comparative complements are expected in a limited 
number of places. For example, "Audi makes more 
large cars than Pontiac in France" is recognized but 
"Audi makes more large cars in France than Pontiac" 
is not. This is because we currently propagate the 
evidence of having found a comparative panicle 
("more") to the noun phrase headed by "cars', hence 
the complement ('than ...') can attach there, but not 
to the higher level verb phrase headed by "makes'. 
This limitation also prevents our processing "What 
companies make a larger car than Buick', whose 
exact meaning(s) the reader is invited to ponder. (3) 
Since comparative complements are based on noun 
phrases, neither "Audi makes more large cars in 
France than in Germany" nor "Audi makes large 
ears more in France than in Germany" is recognized. 
(4) We attempt no pragmatic disambiguation of 
semantically ambiguous comparatives. Thus, when 
confronted with "more than 3 inches shorter" or 
"more than 3 fewer cars', we provide the 
compositional interpretation associated with our left 
recursive syntax. Even expressions such as "as many" 
and "as large" are ambiguous between at least and 
exactly. (5) We attempt no anaphora processing, 
and so comparatives without a complement, as in 
"Which cars are larger?', are not processed. (6) We 
provide general conversion of units of measure (e.g. 
"2 feet longer" is the same as "24 inches longer') but 
they are not fully incorporated into the system. 
2. Aa Initial Exmnple 
The mechanisms we shall describe apply a 
conventional series of transformations to sentences 
containing one or more comparatives, ultimately 
resulting in an executable expression. As an example 
of this process, 2 we'll consider the input 
"List the cars at lee.st 20 inches more tlum twice 
as long as the Century is wide" 
which contains a highly comparativized adjective. 
First, this input is scanned and parsed, yielding the 
parse tree shown in Figure 1. Note that each 
COMPAR node has a QUANTITY node and a 
MODE 3 of its own. Also, the MODE of the top 
COMPAR (whose value is "equal') is co-indexed 
(indicated by the subsrcipt i) with the MODE 
feature associate with the panicle ('as') that 
intervenes between the ADJ and its COMPAR- 
ARG; this assures that -er/than, less/than, and as/as 
pairs collocate correctly. Next, we build a 
"normalized" parse tree by reconstructing elements 
that were discontinuous in the surface structure and 
2. A formal account the associated formalisms, including a BNF 
syntax and a denotational semantics for our "normalized parse 
trees" and "algebraic-logical form" language, is given in Ballard 
and Stumberger (1987). 
3. Dashed lines indicate features, as distinct from lcxical items, 
and empty nodes, which result from Whiz-deletion, are denoted 
by'?'. 
42 
by performing other simplifications. This yields the 
following structure, whose 2-place predicate, with P 
(parameter) and A (argument) as variables, 
corresponds to "at least 20 inches more than twice as 
• .. as'. 
Normalized Purse Tree: 
(CAR (NOUN CAR) 
(COMPAR (ADJ LONG) 
(A (P A) (~ P (÷ 20 (. 2 A)))) 
(CAR { = CENTURY) ) 
(ADJ WIDE))) 
Next, user-defined meanings of words and phrases 
are looked up 4 and the comparati~zafion operations 
described in Section 6 are performed, yielding 
Algebraic-Logical Fon~ 
(SET (CAR Pl) 
( ~ (Length-of-Car PI ) 
(+ 20 (~ 2 (Width-of-Car CENTURY\] 
Finally, this representation is converted into the 
executable expression indicated by 
lrmal Executable Exprossiee: 
(SUBSET (X (Pl) 
(~ (KSV PI eS{LENGTH}) 
(÷ 20 
(- 2 (KSV @I(CENTURY} 
BS{WIDTH} ) ) ) ) 
(KI @F{CAR} ) ) ) 
where KSV and KI are primitive retrieval functions 
of the Kandor back-end; @I{...}, @F{...} and @S{...} 
are Lisp objects respectively denoting instances, 
frames, and slots in Kandor's taxonomic knowledge 
base; and >I>/ is a coercion routine supplied by 
TELI to accommodate backend retrieval system that 
produce numbers in disguise (e.g. a Lisp object or a 
singleton set) on which the standard Lisp functions 
would choke. 5 However, since compositionally created 
structures such as the preceding one are often 
intolerably inefficient, optimiz~tions are carried out 
while the executable expression is being formed. In 
the case at hand, the second argument of >I >~ is 
constant, so it is evaluated, producing 
Optimized Executable Exlmressiee: 
(SUBSET (A (Pl) 
(~>/ (KSV P1 @S{LENGTH}) 158)) 
(KI BF{CAR} ) ) 
A second example, which illustrates a comparative 
4. In TELI, meanings may be arbitrary expressions in the 
extended tint-order language discussed in Ballard and 
Stumberger (1987). 
5. Similar functions are also supplied for arithmetic operators. 
quantifier, is given in an appendix where, as a result 
of optimizations analogous to those which produced 
the constant 158 above, the comparative "at least 3 
more large cars than Buick" is eventually processed 
exactly as though it had been "at least 6 cars" (since 
Buick made 3 large cars). 
3. Lexical Provisions for Comparatives 
Our current repertoire of domain-independent lexical 
items associated with comparatives includes "many', 
"few', and "much'; "more', with 3 readings (er, 
er+many, er+much), following Bresnan (1972) and 
similar to Robinson (1982, p. 28); "fewer (er+few); 
"less', with 3 readings (less, er+few 6, less+much); 
several formatives and adverbials ('at', "least', 
"most', "exactlY', "precisely', "only', "just', "half', 
"again', "times', "percent'); and a handful of spelled- 
out ordinals ('thirds" etc.). Though not stored in the 
lexicon, both integers and floating-point numbers (of. 
"3.45 inches') are also involved in comparativization. 
The domain-dependent portion of the lexicon 
includes members of the open categories of 
adjectives, measure nouns, and comparative 
inflections of adjectives. The scanner output for the 
comparative of the adjective A is er +A (e.g. "larger" 
becomes er+large). 
4. Syntax for Comparatives 
The basic syntax for comparatives adheres to the 
meta-rules given in Section 1.2. As indicated in the 
parse tree of Figure 1, COMPAR is never a primary 
tree node but is instead a daughter of the node being 
comparativized. Furthermore, since our grammar 
has recently taken on somewhat of an X-bar flavor 
(cf. Jackendoff, 1977), the complement for a 
comparativized item is found as either its sister or its 
parent's sister. Complex comparatives derive from 
left-recursive structures. 7 Our present grammar for 
comparatives is set up partly by meta-rules 8 and 
partly by hand-coded rules relating to such 
idiosyncracies as "more than 3 inches in length" 
(however, of. "more than 6 in number*). 
6. To the possible horror of the prescriptive grammarian, this 
accounts for such attrecities as "less books'. 
7. Though our parser operates top-down, we've incorporated a 
general mechanism for left recursinn that's also utilized by 
possessives (e.g. "the newest car's company's largest 
compatitor's smallest car'). 
8. Meta-rules are also used to produce the grammar for relative 
clauses, yes-no questions, and a host of other structures (e.g. 
various slash categories) from a hand-coded grammar for basic 
declarative sentences. 
43 
S. Parse Tree Normalization ' 
Letting Node{<X>} denote a node of the 
normalized parse tree associated with an element of 
type <X>, comparatives involve the replacement 
denoted by 
NodelCt<X>}} 
--.* (COMPAR Node{<X>} <Rel> <At\]g> <Etcx>) 
where <Arg> corresponds to an optional noun 
phrase, <Etcx> captures non-elided material 
associated with the matrix clause, and the 2-place- 
relation denoted by <Rel> is the most interesting 
(and by far the most complex) element produced. 
The algorithm that produces it converts "more', 
"less", and "times" respectively into +, -, and *. This 
process is left recursive; the relational operator is 
determined from the highest MODE, and by default 
it is assigned to be _.9 As indicated below, these 
algebraic and arithmetic symbols will be preserved in 
the executable expression unless the word being 
comparativized indicates a downward direction on the 
scale applicable to it (e.g. "fewer', "shorter'), in 
which case they will be reversed (e.g. >i becomes 
and -~ becomes -). Each 2-place-relation is the body 
of a 2-place lambda whose variables, P and A, are 
associated with values obtained from a parameter 
and an argument against which a comparison is 
being made. Some example 2-place-predicates are 
mere than 166 h~les leag 
more than IS feet ling 
at meat 180 inchu king 
~em 
at least u leq as 
1 h~.h ~ger tt~ 
exactly twice as Iomlg as 
3 times as long as 
half agala   leq as 
forty percem kqer t~m 
less thu erie third u leq as 
at least 3 inches mere alma 
twice u leeg u 
(> P 166) 
(> P 180) 
(~ P 18o) 
(> PA) 
(~ PA) 
(- P (. 2~U) 
(;~ P (. 3 A)) 
(~ P (* 1.5 A)) 
(~ P (. (+ (/40 I00) I) A)) 
(< P (. (I 1 3) A)) 
() P (+ 3 (- 2 A))) 
When the measure noun appearing in an English 
input differs from that by which the objects being 
tested are measured, as indicated by the second 
example above, a scalar conversion is required. 
6. Semantics for Comparatives 
The semantics of comparativization involves 
converting a one-place predicate into another one- 
place predicate by performing arbitrarily complex 
operations on it. For example, if "large car" has been 
defined as a car whose length exceeds 190 inches, 
thetl, letting "A" denote a noun phrase complement, 
some examples are 
t0q 
kMq~r tim 180 hm:l~ 
leqcr tlam A 
no lealger than A 
twice as leog as A t- wide 
3 laches mora thaa 
twi~ as long as A 
Lesgth(x) ;~ 190 
Lcegth(x) > 18o 
Leq~(x) > Leq~(A) 
Le,t.m(x) ~ Le~mCA) 
Leqpm(x) ~ 2 • Wldth(A) 
Length(x) > 3 + 2, Length(A) 
where each of these right-hand-sides is the body of a 
one-place predicate whose single variable is x. 
As a second example, comparative quantifiers 
such as "more than 6" are handled by an identical 
process l°, as indicated by Ii 
x has --any y,. Size {y I Jhs(x,y)} ;~ 
x has more tham 6 y's Size {y \[ Has(x,y)\] > 6 
x Im mere y'. em A Size {y I nt, s(x,y)} > Size blt~(A,y)} 
x Im at lem 2 me~ Size {y \[ Hgix,y)} 
y's tim A ~ 2 + Size \[y \] l-I~(A.y)} 
where the initial Constant denotes some arbitrary 
constant. 
In general, comparativizing a one-place 
predicate takes place as follows. 
1. Find (a) an appropriate one-place function and 
(b) an associated relational operator that tells 
which direction on a linear scale indicates 
having "more" of the property. 
2. Apply the relational operator located above to 
the modality of the comparison to determine 
the relational operator that will appear in the 
IR+. If the relational operator of the definition 
being comparativized is either > or >i, use the 
mode occurring in the IR; otherwise, "reverse" 
the mode by doing what would be a negation 
but leaving untouched the - portion of the 
operator. Thus, the reversal of < is >, the 
9. This addresses the inherent ambiguity of as/as structures 
without an adverbial element, such as "exactly" or "at least'. 
Thus, "people with 3 children" is interpreted as people with exactly 
3 children. 
10. That is, we have no special purpose processing for "more than', 
"how many" etc. 
11. We use "has" in these examples for clarity; naturally, the scope 
of a comparative quantifier may contain an arbitrarily complex 
predicate. 
44 
reversal of ~< is />, and so forth. Similarly, +, 
and - are switched. 
3. Determine the argument being compared 
against (possibly a constant). 
4. Link these pieces together. If the argument 
was not constant (e.g. "... longer than at least 3 
foreign cars'), wrap its scope around the 
resulting expression. 
For example, if "short car" has been defined as 
"x is short': Length(x) < 160 
then the 1-place function and relational operator are 
determined in step 1 to be Length and <~, and thus 
we have 
"shorter than A" -"* Leagth(x) < IAalgtk(A) 
"exactly 3 inches shorter than A" 
--* LentO(x) - Izs~(A) - 3 
7. Comparatives Containing a Wh Element 
In addition to recognizing wh elements associated 
with a relative or interrogative clause, 12 TELI 
recognizes the word how when it appears in place of 
a quantity, e.g. "how long" (cf. "6 inches long') and 
"how many more" (of. "6 more't3). Wherever wh 
appears, however, we treat its semantics as roughly 
"solve for wh such that'. In the case of interrogative 
pronouns (e.g. "what'), this leads rather obviously to 
an internal representation asking for a SET. In the 
case of "how', this treatment is also in order since it 
represents a (quantity) NP. For simplicity, we 
produce an expression containing an unbound wh and 
later give it wide scope. 14 In particular, subsequent 
processing involves moving the wh element upward in 
the logical form tree 18 by performing appropriate 
transformations. 
12. To see that wh is less than a "word', consider pairs such as 
what~that, where~there and when~then. The advantage of 
recognizing sub-word units us the primitives on which syntax 
and/or semantic analysis is based should come as no surprise to 
anyone acquainted with the structure of languages other than 
English, which is unusual in coming so close to being treatable 
solely at the word level. 
13. As stated earlier, we have adopted derivations suggested by 
Bresnan (1973) such as -er+many---qnore. In the case at 
hand, we must assume something like Q+many--*Q, where Q denotes a quantity. 
14. The scope is wide but not global because of inputs such as 
"How many cars does each US company make?" 
15. Of course, our algebraic-logical forms, based on operators and 
their associated arguments, amount to being trees. 
For illustration, consider the absurdly 
complicated example 
"Buick makes 3 more than how many percent 
more cars than Audi?" 
the comparative portion of whose internal 
representation t6 is 
(X (P A) (- P (+ (* A (+ 1 (/ WN 100))) 3\] 
At this point, we proceed with semantic processing, 
ignoring for the moment the presence of the unbound 
WH element. In the case at hand, this leads to 
(= (COUNT (SET (CAR Pl) (Make BUICK Pl) ) ) 
(÷ (, (COUNT (SET (CAR Pl) 
(Make AUDI PI) )) 
(+ I 
(/ wH 100))) 
3)) 
after which we "solve for" WH to yield 
(. (- (/ (- (COUNT (SET (CAR PI) 
(Make BUICK PI))) 
3) 
(COUNT (SET (CAR PI) 
(Make AUDI PI)) )) 
I) 
100) 
This process is not dependent on the position in 
which the wh occurred, and thus takes the place of 
sl~:ial-pu~ interpretation routines for "how 
many,, "How <Adjective>', and so forth. 17 
8. Discussien 
Thus far, we have presented an overview of our 
treatment of comparatives, with as much detail as 
we're able to supply in a conference-length paper. 
Although we can offer no substantive empirical 
evidence with TELI (e.g. results of use by non- 
authors), we believe some of the techniques we've 
presented can be put to use by the reader. Further 
information, especially with regard to the interaction 
of comparatives with a variety of other types of 
constructs, can be found in Bailard and Stumberger 
(1987). 
16. The sentence is ambiguous, with readings indicated by "3 more 
than \[how many percent\]" and "\[3 more than how manyl 
percent'. As indicated earlier, we presently take the reading 
that favors the use of left reenrsion. 
17. Problematic situations can arise in which simple algebraic 
operations aren't sufl~cienct. For example, in examples such as 
"Cars were sold to people with how many children?', we must 
move wh past a logical quantifier, rather than the arithmetic 
operators as shown above. 
45 
8.1 Related Work 
Although the literature describing implemented NL 
processors contains many examples of comparative 
constructions (cf. Kirsch (1964) for a wealth of early 
examples), at least two qualifications may be given 
concerning the current "state of the art" of 
comparative treatment. First, the majority of the 
examples appearing in the literature are quite 
simple 18 (e.g. "more than $250") and can be prepared 
for by specifying a 2-place predicate in advance 
that's effectively equivalent to the 2-place predicate 
we construct from an underlying 1-place predicate by 
way of coercion into a 1-place function. This allows 
one to avoid some slippery problems of movement 
(which we have adressed but have certainly not 
disposed of), to ignore morphological subtleties (e.g. 
recognizing the "er" of "larger" or "more" as -er, a 
"word" to be input to the parser), and to take other 
shortcuts. 19 Second, although examples of various 
types of comparatives are not hard to come by, 
accounts of the actual mechatdsms that treat 
comparatives are harder to find, as are specific 
statements of the generality which authors believe 
themselves to have provided for. 
8.2 Levels of Representation 
The architecture of TELI resembles that of similarly 
motivated question answering systems (cf. Grosz et 
al, 1987; Hafncr and Godden, 1985; Bates and 
Bobrow, 1983 and Bates et al 1985) by comprising a 
linear sequence of processing stages which produce 
successively -lower" level representations of the 
input. 2° Although our parse tree format is rather 
conventional, 21 what we have called "normalized 
18. Evidence of the gap between what's been studied and what 
may actually be important is expressed, in the context of 
pronoun resolution, in Hobbs (1978, p. 343) as follows: "There 
are classes of examples from the literature which are not ... 
handled by the algorithm, but they occur rarely in actual texts, 
and in view of the fact that the algorithm fails on much more natural and common examples, there seems to be little point in 
greatly complicating the algorithm to handle them." 
19. The extent to which "shortcuts" are justified, from either a 
psychological or system designer's standpoint, is not clear. As a possibly bizarre example, consider the word "after', which 
could be treated as "-er .aft than', where .aft is the Anglo- 
Saxon root (extant only on I:card ship) from which current 
English word derives. A perhaps even more bizarre 
opportunity may exist for treating "rather" as "-er .rathe', 
where ".rathe" is a Middle English adverb meaning "quickly'. 
20. We're using "low" to refer to level of abstraction. Perhaps 
ironically, successively higher levels of cognitive information 
are involved in producing these "lower" level representation. 
21. The methods whereby TELI produces parse trees are less 
conventional than the trees it produces, due to our provision for 
having the parser enforce agreements automatically while it is 
running, rather than doing subsequent filtering. 
parse tree" and "algebraic-logical form" correspond 
rather loosely to what in the literature are often 
called "logical form" and "meaning representation', 
respectively. Furthermore, in the most recent work 
with TELI, meaningful distinctions between modules 
have become blurred, although the relative order in 
which operations are carried out is largely the same. 
In seeking to compare our formalisms and 
processing strategies with others that have been 
proposed, we have found terms such as "logical form" 
being used in the literature in quite vague and often 
incompatible ways. Furthermore, we know of no 
compelling arguments that suggest that a 
psychologically plausible model of human 
information processing will require intermediate 
levels such as parse trees, logical forms, and the like. 
Is it even clear that there ought be be a finite 
number of successive "levels", whatever they might 
be? We are increasingly doubtful that the trappings 
spawned by linguists and philosophers can be put in a 
bag, sprinkled with Common Lisp, shaken, and 
expected to yield robust natural language processors. 
More of an interdisciplinary effort may be required 
than has yet been seen. 
8.3 Curreat Work 
The representation given in Section 5 fundamentally 
restricts us from handling comparatives whose 
complement is more than one level above the word 
being comparativized (e.g. "John persuaded his 
students to contribute to more museums than Bill 
did'). Our current work involves producing 
normalized parse tree structures of roughly the form 
(COMPAR.2 Ci <Co..p> 
('COMP~-I Ct-) -.) 
where the COMPAR-1 and <Comp> structures 
correspond to the COMPAR structure given in 
Section 5; Ct provides for co-indexing when multiple 
comparativizations are present; and the first "..." 
allows for arbitrarily many levels. This calls upon us 
to modify the semantic processing presented in 
Section 6, making it resemble the treatment given to 
wh elements as described in Section 7. 
46 
9. Conclusions 
We have presented algorithms aimed at the 
morphological, syntactic, and semantic problems 
associated with a large variety of comparative 
structures that arise in the context of question 
answering. We believe the extent of our coverage 
equals in several ways and exceeds in some ways the 
capabilities known to us via the literature. However, 
comparatives operate as a "meta" phenomenon and 
thus cut across many issues; we have ignored certain 
problems and knowingly treated others inadequately. 
Further work is certainly required, and we hope to 
have presented a framework in which (I) some 
interesting and important capabilities can be provided 
for now and (2) further computational studies can be 
carried out. 
10. Acknowledgements 
The author wishes to acknowledge the many insights 
displayed by Mark Jones and Guy Story during a 
number of intense discussions concerning the issues 
discussed in this paper. 
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47 
HEAD 
J 
NOUNNP-TRACE NP/NPVERB/AUX 
I t J J 
CAR TRACE ? AUXIAUX QUALII*Hf, E 
I I I : ? QUAI~'L/~ 
I LEAST 
NIP J 
NP2 
\ 
REL 
AIXI 
COMPAR AI~ 
t 
COMPAIt QUANTITY CMODE AIXI 
QUANH-t~ CMODE NUM TIMES ~ I :. 
NUM MEASURE mere 2 
I I 20 INCH 
COMP~Ait-ARG (:MODE 
NP2 PREDICATE im~ll. I I A 
q--I( LONG NOUNVAL AIXI 
I I CENTURY WIDE 
Figure 1: Parse Tree for The Example of Section 2 .. 
Appendix: Processing a Comparative Quantifier 
gugUsh ~pm: 
"Have any US companies made at least 3 more large cars than Buick?" 
Nonmdized Parse Tree: 
+vP (co.p,~r .~s cAN sxL axL .xL) 
(suaJ (eou,m (a.-"~ ARY) 
(CONPANY (AJDJ US) 
(aoml coNpaJrt)))) 
(OlJ (CAN (CON,AN \[GUANT NAn') () Q (~ CO 3)) 
(COlPaJn' (- B~ZC¢))) 
(CAR (~ L&ItGE) 
(~OUN CAN))))) 
Algebraic.Logical Fore: 
(ooAN~ (co..,.n .1) c> Q 1) 
(O8-Company Pl) 
(~ (eOUIlT (SET (CAN P2) 
(AND (> (Length-of-Car ,2) 190) 
(m (Coml~aY-of-Ca¢ ,1) ,2)))) 
(+ 3 (COUliT (8IT (CAN ,3) 
(&lid (> (~ength-o£-Csr P2) 190) 
(- (COal~ny-of-Ca¢ P2) IUZCE))))))) 
Final Executabb Expression: 
(oPc-soxs "(1 co) 
(X (P1) 
(ANO (KZ? ,1 e,(os-coNp~n')) 
()) (GPC-COOIT (8UBSBT (~ (,2) 
(AND (>> (ESV P2 g8(LSMGTH}) 190) 
(-= (ESV ,2 IS(CONPAIIT)) ,1))) 
(¢x B,(CAN)))) 
(GPC-+ 3 
(EZ OF(CONPMIT)))) 
(GPC-COUNT (SUD8BT (X (P2) 
(AND (>) (ESV P2 OS{LENGTH}) 190) 
(-- (Esv P2 os{conPA~r)) oz(auzc¢)))) 
(¢Z BP(CAN))))))) 
Optimized Executable Expmsion: 
(GPC-SONZ "(1CQ) 
(~ (P1) 
(GPC-a0NZ "(6 CQ) 
(~ (P2) 
(AHD (>) (ESV P2 eS(LBNGTH)) 190) 
(mm (ESV P2 DS{CONPAHY}) Pl)) 
'(eZ(ZWTRGKA) OZ(NOVA} ...)))) 
(El eF{US-CONPAMY))) 
48 
