LOGICAL FORMS IN THE 
CORE LANGUAGE ENGINE 
Hiyan Alshawi & Jan van Eijck 
SRI International Cambridge Research Centre 
23 Millers Yard, Mill Lane, Cambridge CB2 11ZQ, U.K. 
Keywords: logical form, natural language, semantics 
ABSTRACT 
This paper describes a 'Logical Form' target 
language for representing the literal mean- 
ing of English sentences, and an interme- 
diate level of representation ('Quasi Logical 
Form') which engenders a natural separation 
between the compositional semantics and the 
processes of scoping and reference resolution. 
The approach has been implemented in the 
SRI Core Language Engine which handles the 
English constructions discussed in the paper. 
INTRODUCTION 
The SRI Core Language Engine (CLE) is 
a domain independent system for translat- 
ing English sentences into formal represen- 
tations of their literal meanings which are 
capable of supporting reasoning (Alshawi et 
al. 1988). The CLE has two main lev- 
els of semantic representation: quasi logical 
forms (QLFs), which may in turn be scoped 
or unscoped, and fully resolved logical forms 
(LFs). The level of quasi logical form is the 
target language of the syntax-driven seman- 
tic interpretation rules. Transforming QLF 
expressions into LF expressions requires (i) 
fixing the scopes of all scope-bearing opera- 
tors (quantifiers, tense operators, logical op- 
erators) and distinguishing distributive read- 
ings of noun phrases from collective ones, and 
(ii) resolving referential expressions such as 
definite descriptions, pronouns, indexical ex- 
pressions, and underspecified relations. 
The QLF level can be regarded as the nat- 
ural level of sentence representation resulting 
25 
from linguistic analysis that applies composi- 
tional semantic interpretation rules indepen- 
dently of the influence of context. 
Sentence 
~, syntax rules 
Parse trees 
semantic rules 
QLF ezpressions 
~, context 
LF expressions 
The QLF expressions are derived on the ba- 
sis of syntactic structure, by means of se- 
mantic rules that correspond to the syntax 
rules that were used for analysing the sen- 
tence. Having QLFs as a well-defined level of 
representation allows the problems of com- 
positional semantics to be tackled separately 
from the problems of scoping and reference 
resolution. Our experience so far with the 
CLE has shown that this separation can ef- 
fectively reduce the complexity of the system 
as a whole. Also, the distinction enables us to 
avoid multiplying out interpretation possibil- 
ities at an early stage. The representation 
languages we propose are powerful enough 
to give weU-motiwted translations of a wide 
range of English sentences. In the current 
version of the CLE this is used to provide a 
systematic and coherent coverage of all the 
major phrase types of English. To demon- 
strate that the semantic representations are 
also simple enough for practical natural lan- 
guage processing applications, the CLE has 
been used as an interface to a purchase order 
processing simulator and a database query 
system, to be described elsewhere. 
In summary, the main contributions of the 
work reported in this paper are (i) the intro- 
duction of the QLF level to achieve a natural 
separation between compositional semantics 
and the processes of scoping and reference 
resolution, and (ii) the integration of a range 
of well-motivated semantic analyses for spe- 
cific constructions in a single coherent frame- 
work. 
We will first motivate our extensions to 
first order logic and our distinction between 
LF and QLF, then describe the LF language, 
illustrating the logical form translations pro- 
duced by the CLE for a number of English 
constructions, and finally present the addi- 
tional constructs of the QLF language and 
illustrate their use. 
EXTENDING 
FIRST ORDER LOGIC 
As the pioneer work by Montague (1973) sug- 
gests, first order logic is not the most nat- 
ural representation for the meanings of En- 
glish sentences. The development of Mon- 
tague grammar indicates, however, that there 
is quite a bit of latitude as to the scope of the 
extensions that are needed. In developing 
the LF language for the CLE we have tried to 
be conservative in our choice of extensions to 
first order logic. Earlier proposals with simi- 
lar motivation are presented by Moore (1981) 
and Schubert & Pelletier (1982). 
The ways in which first order logic-- 
predicate logic in which the quantifiers 3 and 
V range over the domain of individuals--is ex- 
tended in our treatment can be grouped and 
motivated as follows: 
• Extensions motivated by lack of ex- 
pressive power of ordinary first order 
logic: for a general treatment of noun 
phrase constructions in English general- 
ized quantifiers are needed ('Most A are 
B' is not expressible in a first order lan- 
guage with just the two one-place pred- 
icates A and B). 
• Extensions motivated by the desire 
26 
for an elegant compositional semantic 
framework: 
use of lambda abstraction for the 
translation of graded predicates in 
our treatment of comparatives and 
superlatives; 
use of tense operators and inten- 
sional operators for dealing with 
the English tense and au~liary sys- 
tem in a compositional way. 
• Extensions motivated by the desire to 
separate out the problems of scoping 
from those of semantic representation. 
• Extensions motivated by the need to 
deal with context dependent construc- 
tions, such as anaphora, and the implicit 
relations involved in the interpretation of 
possessives and compound nominals. 
The first two extensions in the list are part 
of the LF language, to be described next, the 
other two have to do with QLF constructs. 
These QLF constructs are removed by the 
processes of quantifier scoping and reference 
resolution (see below). 
The treatment of tense by means of tempo- 
ral operators that is adopted in the CLE will 
not be discussed in this paper. Some advan- 
tages of an operator treatment of the English 
tense system are discussed in (Moore, 1981). 
We are aware of the fact that some as- 
pects of our LF representation give what are 
arguably overly neutral analyses of English 
constructions. For example, our uses of event 
variables and of sentential tense operators say 
little about the internal structure of events or 
about an underlying temporal logic. Never- 
theless, our hope is that the proposed LF rep- 
resentations form a sound basis for the subse- 
quent process of deriving the fuller meaning 
representations. 
RESOLVED 
LOGICAL FORMS 
NOTATIONAL CONVENTIONS 
Our notation is a straightforward extension 
of the standard notation for first order logic. 
The following logical form expression involv- 
ing restricted quantification states that every 
dog is nice: 
quant(forall, x, Dog(x), Nice(x)). 
To get a straightforward treatment of the 
collective/distributive distinction (see below) 
we assume that variables always range over 
sets, with 'normal' individuals corresponding 
to singletons. Properties like being a dog can 
be true of singletons, e.g. the referent of Fido, 
as well as larger sets, e.g. the referent of the 
three dogs we saw yesterday. 
The LF language allows formation of com- 
plex predicates by means of lambda abstrac- 
tion: ,~x,\d.Heavy.degree( z, d) is the predi- 
cate that expresses degree of heaviness. 
EVENT AND STATE VARIABLES 
Rather than treating modification of verb 
phrases by means of higher order predicate 
modifiers, as in (Montague, 1973), we follow 
Davidson's (1967) quantification over events 
to keep closer to first order logic. The event 
corresponding to a verb phrase is introduced 
as an additional argument to the verb pred- 
icate. The full logical form for Every repre- 
sentative voted is as follows: 
quant(forall, x, Repr(x), 
past(quant(exists, e, Ev(e), 
Vote(e,x)))). 
Informally, this says that for every represen- 
tative, at some past time, there existed an 
event of that representative voting. 
The presence of an event variable allows 
us to treat optional verb phrase modifiers as 
predications of events, as in the translation 
of John left suddenly: 
past(quant(exists, e, Ev(e), 
27 
Leave(e, john) ^ Sudden(e))). 
The use of event variables in turn permits 
us to give a uniform interpretation of prepo- 
sitional phrases, whether they modify verb 
phrases or nouns. For example, John de- 
signed a house in Cambridge has two read- 
ings, one in which in Cambridge is taken to 
modify the noun phrase a house, and one 
where the prepositional phrase modifies the 
verb phrase, with the following translations 
respectively: 
quant(exlsts, h, 
House(h) A In_location(h, Cambridge), 
past(quant (exists, e, Ev(e), 
Design( e, john, h ) ) ) ). 
quant(exlsts, h, House(h) A 
past(quant(exists, e, Ev(e), 
Design(e, john, h) ^ 
In_location(e, Cambridge)))). 
In both cases the prepositional phrase is 
translated as a two-place relation stating that 
something is located in some place. Where 
the noun phrase is modified, the relation is 
between an ordinary object and a place; in 
the case where the prepositional phrase mod- 
ifies the verb phrase the relation is between 
an event and a place. Adjectives in pred- 
icative position give rise to state variables in 
their translations. For example, in the trans- 
lation of John was happy in Paris, the prepo- 
sitional phrase modifies the state. States are 
like events, but unlike events they cannot be 
instantaneous. 
GENERALIZED QUANTIFIERS 
A generalized quantifier is a relation Q be- 
tween two sets A and B, where Q is insensi- 
tive to anything but the cardinalities of the 
'restriction set' A and the 'intersection set' 
A N B (Barwise & Cooper, 1981). A gen- 
eralized quantifier with restriction set A and 
intersection set ANB is fully characterized by 
a function AmAn.Q(m, n) of m and n, where 
m = IAI and n = IANB I. In theLFlan- 
guage of the CLE, these quantifier relations 
are expressed by means of predicates on two 
numbers, where the first variable abstracted 
over denotes the cardinality of the restriction 
set and the second one the cardinality of the 
intersection set. This allows us to build up 
quantifiers for complex specifier phrases like 
at least three but less than five. In simple 
cases, the quantifier predicates are abbrevi- 
ated by means of mnemonic names, such as 
exists, notexists, forall or most. Here are 
some quantifier translations: 
• most ",.* Xm,Xn.(m < 2n) \[abbreviation: 
most\]. 
• at least three but less than seven ,,~ 
)tm~n.(n > 3 ^ n < 7). 
• not every .,.* )~m)~n.(m ~ n). 
A logical form for Not every representative 
voted is: 
quant()~mAn.(m # n), x, Rep(z), 
past(quant (exists, e, Ev(e), 
Vote(e,x)))). 
Note that in one of the quantifier examples 
above the abstraction over the restriction set 
is vacuous. The quantifiers that do depend 
only on the cardinality of their intersection 
set turn out to be in a linguistically well- 
defined class: they are the quantifiers that 
can occur in the NP position in "There are 
NP'. This quantifier class can also be char- 
acterized logically, as the class of symmet- 
r/c quantifiers: "At least three but less than 
seven men were running" is true just in case 
"At least three but less than seven runners 
were men" is true; see (Barwise & Cooper, 
1981) and (Van Eijck, 1988) for further dis- 
cussion. Below the logical forms for symmet- 
ric quantifiers will be simplified by omitting 
the vacuous lambda binder for the restric- 
tion set. The quantifiers for collective and 
measure terms, described in the next section, 
seem to be symmetric, although linguistic in- 
tuitions vary on this. 
COLLECTIVES AND 
TERMS 
MEASURE 
Collective readings are expressed by an ex- 
tension of the quantifier notation using set. 
28 
The reading of Two companies ordered five 
computers where the first noun phrase is in- 
terpreted collectively and the second one dis- 
tributively is expressed by the following log- 
ical form: 
quant(set(~n.(n = 2)), x, Company(x), 
quant(~n.(n = 5), y, 
Computer(y), 
past(quant (exists, e, Ev(e), 
Order(e, x, y))))). 
The first quantification expresses that there 
is a collection of two companies satisfying 
the body of the quantification, so this read- 
ing involves five computers and five buy- 
ing events. The operator set is introduced 
during scoping since collective/distributive 
distinctionsmlike scoping ambiguities--are 
not present in the initial QLF. 
We have extended the generalized quanti- 
fier notation to cover phrases with measure 
determiners, such as seven yards of fabric or 
a pound of flesh. Where ordinary generalized 
quantifiers involve counting, amount gener- 
alized quantifiers involve measuring (accord- 
ing to some measure along some appropriate 
dimension). Our approach, which is related 
to proposals that can be found in (Pelletier, 
ed.,1979) leads to the following translation 
for John bought at least five pounds of ap- 
ples: 
quant(amount($n.(n >_ 5), pounds), 
z, Apple(z), 
past(quant(exists, e, Ev(e), 
Buy( e, john , x))))). 
Measure expressions and numerical quanti- 
tiers also play a part in the semantics of com- 
paratives and superlatives respectively (see 
below). 
NATURAL KINDS 
Terms in logical forms may either refer to in- 
dividual entities or to natural kinds (Carlson, 
1977). Kinds are individuals of a specific na- 
ture; the term kind(x, P(x)) can loosely be 
interpreted as the typical individual satisfy- 
ing P. All properties, including composite 
ones, have a corresponding natural kind in 
our formalism. Natural kinds are used in the 
translations of examples like John invented 
paperclips: 
past(quant(exists, e, Ev(e), 
Invent(e, john, kind(p, Paperclip(p) ) ) ). 
In reasoning about kinds, the simplest ap- 
proach possible would be to have a rule of 
inference stating that if a "kind individual" 
has a certain property, then all "real world" 
individuals of that kind have that property as 
well: if the "typical bear" is an animal, then 
all real world bears are animals. Of course, 
the converse rule does not hold: the "typical 
bear" cannot have all the properties that any 
real bear has, because then it would have to 
be both white all over and brown all over, 
and so on. 
COMPARATIVES AND SUPERLA- 
TIVES 
In the present version of the CLE, compara- 
tives and superlatives are formed on the basis 
of degree predicates. Intuitively, the mean- 
ing of the comparative in Mary is nicer than 
John is that one of the two items being com- 
pared possesses a property to a higher degree 
than the other one, and the meaning of a su- 
perlative is that art item possesses a property 
to the highest degree among all the items in 
a certain set. This intuition is formalised in 
(Cresswell, 1976), to which our treatment is 
related. 
The comparison in Mary is two inches 
taller than John is translated as follows: 
quant(amount(An.(n = 2), inches), 
h, Degree(h), 
more()~x Ad. tall_degree(z, d), 
mary, john, h ). 
The operator more has a graded predicate 
as its first argument and three terms as its 
second, third and fourth arguments. The op- 
erator yields true if the degree to which the 
first term satisfies the graded predicate ex- 
ceeds the degree to which the second term 
satisfies the predicate by the amount speci- 
fied in the final term. In this example h is a 
29 
degree of height which is measured, in inches, 
by the amount quantification. Examples like 
Mary is 3 inches less tall than John get sim- 
ilar translations. In Mary is taller than John 
the quantifier for the degree to which Mary 
is taller is simply an existential. 
Superlatives are reduced to comparatives 
by paraphrasing them in terms of the num- 
ber of individuals that have a property to at 
least as high a degree as some specific individ- 
ual. This technique of comparing pairs allows 
us to treat combinations of ordinals and su- 
perlatives, as in the third tallest man smiled: 
quant(ref(the,...), a, 
Man(a) A quant(An.(n = 3), b, 
Man(b)), 
quant(amount(,kn.(n _> 0), units), h, 
more( Az ~d.tall_degree( x, d), b, a, h ), 
past(quant(exists, e, Ev(e), 
Smile(e, a)))))). 
The logical form expresses that there are ex- 
actly three men whose difference in height 
from a (the referent of the definite noun 
phrase, see below) is greater than or equal 
to 0 in some arbitrary units of measurement. 
QUASI LOGICAL FORMS 
The QLF language is a superset of the LF 
language; it contains additional constructs 
for unscoped quantifiers, unresolved refer- 
ences, and underspecified relations. The 
'meaning' of a QLF expression can be 
thought of as being given in terms of the 
meanings of the set of LF expressions it is 
mapped to. Ultimately the meaning of the 
QLF expressions can be seen to depend on 
the contextual information that is employed 
in the processes of scoping and reference res- 
olution. 
UNSCOPED QUANTIPIERS 
In the QLF language, unscoped quantifiers 
are translated as terms with the format 
qterm((quantifier),(number), 
( variable),( restriction) ). 
Coordinated NPs, like a man or a woman, 
are translated as terms with the format 
term..coord( ( operator),( variable), 
(ten)). 
The unscoped QLF generated by the seman- 
tic interpretation rules for Most doctors and 
some engineers read every article involves 
both qterms and a term_coord (quantifier 
scoping generates a number of scoped LFs 
from this): 
quant(exists, e, Ev(e), 
Read(e, 
term_coord(A, x, 
qterm(most, plur, y, Doctor(y)), 
qterm(some, plur, z, Engineer(z))), 
qterm(every, sing, v, Art(v)))). 
Quantifier scoping determines the scopes of 
quantifiers and operators, generating scoped 
logical forms in a preference order. The or- 
dering is determined by a set of declarative 
rules expressing linguistic preferences such 
as the preference of particular quantifiers to 
outscope others. The details of two versions 
of the CLE quantifier scoping mechanism are 
discussed by Moran (1988) and Pereira (A1- 
shawl et al. 1988). 
UNRESOLVED REFERENCES 
Unresolved references arising from pronoun 
anaphora and definite descriptions are rep- 
resented in the QLF as 'quasi terms' which 
contain internal structure relevant to refer- 
ence resolution. These terms are eventually 
replaced by ordinary LF terms (constants or 
variables) in the final resolved form. A dis- 
cussion of the CLE reference resolution pro- 
cess and treatment of constraints on pronoun 
reference will be given in (Alshawi, in prep.). 
Pronouns. The QLF representation of a 
pronoun is an anaphoric term (or a_term). 
For example, the translations of him and 
himself in Mary expected him to introduce 
himself are as follows: 
30 
a_term(ref(pro, him, sing, \[mary\]), 
x, Male(x)) 
a_term(ref(refl, him, sing, \[z, mary\]), 
y, Male(y)). 
The first argument of an a_term is akin 
to a category containing the values of syn- 
tactic and semantic features relevant to ref- 
erence resolution, such as those for the 
reflexive/non-reflexive and singular/plural 
distinctions, and a list of the possible intra- 
sentential antecedents, including quantified 
antecedents. 
Definite Descriptions. Definite descrip- 
tions are represented in the QLF as unscoped 
quantified terms. The qterm is turned into 
a quant by the scoper, and, in the simplest 
case, definite descriptions are resolved by in- 
stantiating the quant variable in the body 
of the quantification. Since it is not possible 
to do this for descriptions containing bound 
variable anaphora, such descriptions remain 
as quantifiers. For example, the QLF gener- 
ated for the definite description in Every dog 
buried the bone that it found is: 
qterm(ref(def, the, sing, Ix\]), sing, y, 
Bone(y) A past(quant(exlsts, e, Ev(e), 
Find(e, a_term(ref(pro, it, sing, \[y,z\]), 
w, Zmv rsonal(w)), y)))). 
After scoping and reference resolution, the 
LF translation of the example is as follows: 
quant(forall, x, Dog(x), 
q uant(exists_one, y, 
Bone(y) A past(quant(exists, e, Ev(e), 
Find(e, x, y))), 
quant(exists, e', Ev( e'), Bury( e', x, y)))). 
Unbound Anaphoric Terms. When an 
argument position in a QLF predication must 
co-refer with an anaphoric term, this is indi- 
cated as a_index(x), where x is the variable 
for the antecedent. For example, because 
want is a subject control verb, we have the 
following QLF for he wanted to swim: 
past(quant(exists, e, Ev(e), 
Want(e, a_term(ref(pro, he, sing, \[ \]), z, 
Male(z)), 
quant(exists, e I, Ev(el), 
Swim( e', a_index(z))))). 
If the a_index variable is subsequently re- 
solved to a quantified variable or a constant, 
then the a_index operator becomes redun- 
dant and is deleted from the resulting LF. In 
special cases such as the so-called 'donkey- 
sentences', however, an anaphoric term may 
be resolved to a quantified variable v outside 
the scope of the quantifier that binds v. The 
LF for Every farmer who owns a dog loves it 
provides an example: 
quant(forall, x, 
Farmer( x )A 
quant(exists, y, Dog(y), 
quant(exists, e, Zv( e ), Own(e, x, y) ) ), 
quant(exists, e ~, Ev(e'), 
Love( e ~, x, a..index(y)))). 
The 'unbound dependency' is indicated by an 
a_index operator. Dynamic interpretation 
of this LF, in the manner proposed in (Groe- 
nendijk & Stokhof, 1987), allows us to arrive 
at the correct interpretation. 
UNRESOLVED PREDICATIONS 
The use of unresolved terms in QLFs is not 
sufficient for covering natural language con- 
structs involving implicit relations. We have 
therefore included a QLF construct (a_form 
for 'anaphoric formula') containing a formula 
with an unresolved predicate. This is eventu- 
ally replaced by a fully resolved LF formula, 
but again the process of resolution is beyond 
the scope of this paper. 
Implicit Relations. Constructions like 
possessives, genitives and compound nouns 
are translated into QLF expressions contain- 
ing uninstantiated relations introduced by 
the a_form relation binder. This binder is 
used in the translation of John's house which 
says that a relation, of type poss, holds be- 
tween John and the house: 
31 
qterm(exists, sing, x, 
a_form(poss, R, House(x) A R(john, x ) ) ). 
The implicit relation, R, can then be deter- 
mined by the reference resolver and instanti- 
ated, to Owns or Lives_in say, in the resolved 
LF. 
The translation of indefinite compound 
nominals, such as a telephone socket, involves 
an a_form, of type cn (for an unrestricted 
compound nominal relation), with a 'kind' 
term: 
qterm(a, sing, s, 
a_form(cn, R, Socket(s) ^ 
R( s, kind(t, Telephone(t)))). 
The 'kind' term in the translation reflects the 
fact that no individual telephone needs to be 
involved. 
One-Anaphora. The a_form construct is 
also used for the QLF representation of 
'one-anaphora'. The variable bound by the 
a_form has the type of a one place predi- 
cate rather than a relation. Resolving these 
anaphora involves identifying relevant (parts 
of) preceding noun phrase restrictions (Web- 
ber, 1979). For example the scoped QLF for 
Mary sold him an expensive one is: 
quant(exists, x, 
a_form(one, P, P( x ) A Expensive(x)), 
past(quant(exists, e, Ev(e), 
Sell(e, mary, z, a_term(...)))). 
After resolution (if the sentence were pre- 
ceded, say, by John wanted to buy a futon) 
the resolved LF would be: 
q uant (exists, z, 
Futon( x ) ^ Expensive(z), 
past(quant(exists, e, Ev(e), 
Sell(e, mary, x, john ) ) ). 
CONCLUSION 
We have attempted to evolve the QLF and 
LF languages gradually by a process of 
adding minimal extensions to first order 
logic, in order to facilitate future work on 
natural language systems with reasoning ca- 
pabilities. The separation of the two seman- 
tic representation levels has been an impor- 
tant guiding principle in the implementation 
of a system covering a substantial fragment 
of English semantics in a well-motivated way. 
Further work is in progress on the treatment 
of collective readings and of tense and aspect. 
ACKNOWLEDGEMENTS 
The research reported in this paper is part 
of a group effort to which the following peo- 
ple have also contributed: David Carter, Bob 
Moore, Doug Moran, Barney Pell, Fernando 
Pereira, Steve Pulman and Arnold Smith. 
Development of the CLE has been carried out 
as part of a research programme in natural- 
language processing supported by an Alvey 
grant and by members of the NATTIE con- 
sortium (British Aerospace, British Telecom, 
Hewlett Packard, ICL, Olivetti, Philips, Shell 
Research, and SRI). We would like to thank 
the Alvey Directorate and the consortium 
members for this funding. The paper has 
benefitted from comments by Steve Pulman 
and three anonymous ACL referees. 
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