M O N O T O N I C
S E M A N T I C I N T E R P R E T A T I O N *
H i y a n A l s h a w i a n d R i c h a r d C r o u c h SRI International
Cambridge Computer Science Research Centre 23 Millers Yard
Cambridge CB2 1RQ, U.K. hiyan~cam, sri. corn rc~cam, sri. c o r n
A B S T R A C T
Aspects of semantic interpretation, such as quan- tifier scoping and reference resolution, are often realised computationally by non-monotonic opera- tions involving loss of information and destructive manipulation of semantic representations. The paper describes how monotonic reference resolu- tion and scoping can be carried out using a re- vised Quasi Logical Form (QLF) representation. Semantics for QLF are presented in which the de- notations of formulas are extended monotonically as QLF expressions are resolved.
1. I N T R O D U C T I O N
The monotonicity property of unification based grammar formalisms is perhaps the most impor- tant factor in their widespread use for grammatical description and parsing. Monotonicity guarantees that the grammatical analysis of a sentence can proceed incrementally by combining information from rules and lexical entries in a nondestructive way. By contrast, aspects of semantic interpreta- tion, such as reference and quantifier scope reso- lution, are often realised by non-monotonic opera- tions involving loss of information and destructive manipulation of semantic representations. A 'two- level' approach to semantic interpretation tends to
result (Bronneberg el al. 1980), where an initial,
underspecified representation is transformed into a separate, specified, representation.
The goal of the work described here is to pro- vide a model for semantic interpretation that is fully monotonic in both linguistic and contextual aspects of interpretation, and which employs just one level of semantic representation - - Quasi Log- ical Form (QLF). Contextual resolution of under-
*This work on the Core Language Engine was carried out u n d e r C L A R E , a collaborative project involving B P Research, British Aerospace, British Telecom, Cambridge University, SRI International a n d the UK Defence Research Agency. T h e project is funded in p a r t by the UK Depart- m e n t of Trade and Industry.
specified QLF expressions involves the instantia- tion of QLF meta-variables. The semantics for the QLF formalism makes the denotation of a QLF formula a partial function to truth-values, with resolution leading to a monotonic extension of the denotation function. We believe that there are several advantages to the approach taken, includ- ing:
• Order independence of resolution operations
• Production of partial interpretations
• Simpler interactions between phenomena
• Reversibility for synthesis/generation
The QLF formalism is a development of Alshawi 1990. As before, underspecified QLFs are pro- duced on the basis of a unification grammar. Pre- viously, QLF resolution was only partially mono- tonic; full monotonicity required changes to the original QLF formalism and the resolution and scoping processes. These changes have been im- plemented in a further development of the Core Language Engine (Alshawi 1992), although we will ignore most implementation issues in the present paper.
The paper is organized as follows. Section 2 provides the syntax of the QLF language and Sec- tion 3 gives some illustrative examples of mono- tonic QLF resolution. Sections 4 and 5 present the semantics of the QLF formalism. Section 6 dis- cusses the relationship between monotonic inter- pretation, Pereira's categorial semantics (Pereira 1990), and context change approaches to seman- tics. Section 7 mentions some benefits of using QLF-like representations in implementing natural language systems.
2. S Y N T A X O F M O N O T O N I C Q L F
We give here a syntactic description of the QLF constructs for terms and formulas 1.
A Q L F t e r m m u s t be one of the following
• a t e r m variable: X, Y, . . .
• a t e r m index: +i, +j, . . .
• a constant term: 7, m a r y l , . . .
• an expressions of the form:
term ( Idx, Cat, Re str, Quant, Reft )
T h e t e r m index, I d x , uniquely identifies the t e r m expression. Cat is a list of feature-value equations, for example < t y p e = p r o , n u m = s i n g , . . . >. R e s t r is a first-order, one-place predicate. For a resolved t e r m , Quant will be a generalized quantifier (a car- dinality predicate holding of two properties) and R e f t , the t e r m ' s 'referent', will be a constant or t e r m index. For an 'unresolved' t e r m , Quant and R e f t m a y be m e t a - v a r i a b l e s ( _ x , _ y , . . . ) . ( Q L F t e r m s m a y also be functional applications, though we will ignore these here).
A Q L F f o r m u l a m u s t be one of the following
• the application of a predicate to arguments: P r e d i c a t e (Argument 1 , . . . , A r ~ m e n t n )
• an expression of the form:
f o r m ( C a t e g o r y , R e s t r i c t i o n , l~es o l u t i o n )
• a f o r m u l a with scoping constraints: Scope : F o r m u l a
P r e d i c a t e is a first or higher-order predicate, in- cluding the usual logical o p e r a t o r s and, n o t , etc. An a r g u m e n t m a y be a term, a f o r m u l a or a l a m b d a abstract. L a m b d a a b s t r a c t s take the form V a r ' B o d y where Body is a f o r m u l a or an a b s t r a c t and Vat is a variable ranging over individuals or relations. R e s t r i c t i o n is a higher-order predi- cate. R e s o l u t i o n is a f o r m u l a (the 'referent' of the f o r m expression), or is a meta-variable if the f o r m expression is unresolved. Scope is either a m e t a - variable when scoping information is underspeci- fled or a (possibly e m p t y ) list of t e r m indices e.g. [ + i , + j ] if t e r m + i outscopes +j. T h e terms iden- tified by the indices m u s t occur within Formula.
T h e degree to which a Q L F is unresolved cor- responds a p p r o x i m a t e l y to the extent to which m e t a - v a r i a b l e s (appearing above as Quant, R e S t ,
Scope, and Resolution) a r e i n s t a n t i a t e d to the a p p r o p r i a t e kind of object level expressions (though see Section 5 for an explicit characteri- zation of unresolved Q L F s and partial interpreta- tions.)
3 . E X A M P L E Q L F R E S O L U T I O N S
Resolution of Q L F s through the instantiation of recta-variables has been applied to a wide range
of p h e n o m e n a . T h e s e include pronouns, definite descriptions, implicit or vague relations, ellipsis and t e m p o r a l relations (see Alshawi 1990 for an account of some kinds of reference resolution in an earlier Q L F formalism). For concreteness, we present a few illustrative e x a m p l e s of monotonic Q L F resolution 2. We do not a t t e m p t to describe the m e c h a n i s m b y which the resolutions are cho- sen.
It will b e c o m e evident t h a t the n o t a t i o n is closer to (the s y n t a c t i c s t r u c t u r e of) n a t u r a l language t h a n is the case for t r a d i t i o n a l logical formalisms. For example, t e r m s usually correspond to noun phrases, with information a b o u t w h e t h e r e.g. they are pronominal, quantified or p r o p e r names in- cluded in the t e r m ' s category. T h i s makes the Q L F r e p r e s e n t a t i o n easier to read t h a n it might seem at first, once its initial unfamiliarity is over- c o m e .
Q u a n t i f i c a t i o n : Every boy met a tall girl illus-
t r a t e s the r e p r e s e n t a t i o n of quantification. T h e basic Q L F analysis m i g h t be (ignoring tense):
_s:meet(term(+b,<type=q,lex=every>,boy,_q,x), term(+g,<type=q,lex=a>,
Y'and(girl(Y),tall(Y)),_r,_y)).
A resolved s t r u c t u r e could be o b t a i n e d by instan- tinting the quantifier m e t a - v a r i a b l e s _q and _r to f o r a l l and e x i s t s 3, and the scoping m e t a - variable s to [ + b , + g ] for the 'Y3' reading:
[+b,+g]:
meet(term(+b,<type=q,lex=every>, boy,forall,+b), term(+g,<type=q,lex=a>,
Y'and(girl(Y),tall(Y)),exists,+g)).
In a r e s t r i c t i o n - b o d y n o t a t i o n for generalized quantifiers, the t r u t h conditional content of this resolved expression corresponds to
forall(B,boy(B),
exists(G,and(girl(G),tall(G)), meet(B,G))).
A n a p h o r a : Every boy claims he met her illus-
t r a t e s the t r e a t m e n t of a n a p h o r a (in a context
2 A l t h o u g h t h e Q L F f r a m e w o r k c a n s u p p o r t a v a r i e t y of a l t e r n a t i v e s e m a n t i c a n a l y s e s for specific p h e n o m e n a , to p r o v i d e c o n c r e t e i l l u s t r a t i o n s o n e or o t h e r a n a l y s i s n e e d s to b e c h o s e n . I n t h e following e x a m p l e s , it s h o u l d b e p o s s i b l e to s e p a r a t e p a r t i c u l a r a n a l y s e s f r o m t h e g e n e r a l p o i n t s we w i s h to m a k e a b o u t m o n o t o n i c i n t e r p r e t a t i o n .
where M a r y is assumed to be salient) 4
Unresolved:
_sl:claim(
term(+b,<type=q,lexfevery>,boy,_ql,_x), _s2:meet(term(+hl,<type=pro,lex=he>,
male,_q2,_y),
term(+h2,<type--pro,lex=her>, female,_q3,_z))).
Resolved: [+b]:claim(
term(+b,<type=q,lex=every>, boy,forall,+b),
[+hl]:meet(term(+hl,<type=pro,lex=he>, male,exists,+b), term(+h2,<type=pro,lex=her>,
female,exists,mary))).
T h e pronominal t e r m for her is resolved so t h a t it existentially quantifies over female objects ident~ cal to mary. T h e ' b o u n d variable' pronoun he has
a referent coindexed with its antecedent, +b. T h e scope of +h2 is leK unspecified, since exactly the same t r u t h conditions arise if it is given wide or
narrow scope with respect to every boy or he.
V a g u e R e l a t i o n s : An unresolved Q L F expres-
sion representing the noun phrase a woman on a
bus might be a t e r m containing a form t h a t arises f r o m the the prepositional phrase modification:
term(+w,<lexsa,..>, X'and(woman(X),
form(<type=prep,lex=on>,
R'R(+w,term(+b,<lex=a,..>,
b u s , _ q 2 , _ b ) ) ,
_f)),
_ q l , _ w ) .
Informally, the f o r m is resolved by applying its re- striction, R ' R ( . . . ) to an appropriate salient pred- icate, and instantiating the form's meta~variable, f , with the result. In this case, the appropriate predicate might be inside, so t h a t _f is instant~ ated to
inside(+w,term(+b,<lex=a,..>,bus,_q2,_b)).
Tense: One way of t r e a t i n g tense is by means of a t e m p o r a l relation f o r m in the restriction of an event t e r m . For John slept we might have:
_ s : s l e e p ( t e r m ( + e , < t y p e = e v e n t > ,
E - f o r m ( < t y p e = t r e l , t e n s e = p a s t > , R ' a n d ( e v e n t ( E ) , R ( E ) ) , _ t ) ,
_ q l , _ e ) ,
term(+j,<type=name>,
J ' n a m e ( J , ' J o h n ' ) , _ q 2 , _ j ) ) .
4 Here we simplify the issues arising out of tile semantics
of intensional, sentential complement verbs like claim.
Since the tense on the temporal relation category is past, the resolution says t h a t the event occurred before a particular speech time, t 7 :
[+el : s l e e p (
t e r m ( + e , < t y p e = e v e n t > ,
E~f orm (<type=trel, t enseffipast >, R'and(event (E) ,R(E) ),
and (event (E), precede (E, t7) ) ), exists ,+e),
t erm(+j, <typefname>,
J'name (J, ' John ' ), exists, j ohnl ) ).
T h e resolution a n d ( e v e n t ( E ) , p r e c e d e ( E , t T ) ) is the result of applying the form's restriction K ' a n d ( e v e n t ( E ) , R(E)) to a contextually derived predicate, in this case E l ' p r e c e d e ( E l , t T ) .
Q L F is not c o m m i t t e d to an event based treat- ment of tense. An alternative t h a t has also been implemented is to t r e a t the verbal predication s l e e p ( . . . ) as a t e m p o r a l f o r m , whose category specifies tense and aspect information.
E l l i p s i s : A more complex example, involving el-
lipsis and quantification, is provided by
Each boy claimed he was clever, and so did John.
A partially resolved Q L F , b u t one in which the ellipsis is still unresolved, might be as follows (ig- noring tense and event variables):
and(
c l a i m ( t e r m (+b, < l e x = e v e r y > , boy , e x i s t s , + b ) , c l e v e r (term (+h, < l e x f h e > ,
male, exists ,+b) ) ), f orm (<type=vpellipsis>,
P ' P (term (+j ,<typefname>, J ' n a m e ( J , ' J o h n ' ) , e x i s t s , j o h n ) ) ,
_ e ) ) .
This is a conjunction of the Q L F for the an-
tecedent clause (Each boy claimed he was clever
under a b o u n d pronoun reading) with a form ex- pression for the verb phrase ellipsis. Solutions for instantiating the meta~variable _e for the ellipsis are the result of applying a p r o p e r t y Pl, derived from the antecedent clause, to the t e r m with in- dex +j. T h e sentence has two readings: a sloppy reading where John claims t h a t he is clever, and a strict one where J o h n claims t h a t each of the boys is clever. T h e choice between a strict or sloppy
reading depends on how the term he is reinter-
preted in the ellipsis resolution. Intuitively, strict identity involves referring to the same object as before, whereas sloppy identity involves referring to a relevantly similar object.
X ' c i a i r . (X, c l e v e r ( + h ) ) .
C o n s t r a i n t s on legitimate scoping (Section 5) force +b and +h to t a k e wide scope over b o t h the an- tecedent a n d ellipsis. T h e sloppy reading results f r o m re-indexing the ellipsis pronoun so t h a t it has the same restriction and category as the original, b u t is resolved to +j and has a new index +hl. This corresponds to taking P1 to be:
X~claim (X, c l e v e r ( t erm ( + h l , <lex=he> male, exists,+j))).
More generally, in Crouch and Alshawi 1992 we explore the claim t h a t solutions to verb phrase el- lipsis have the general form:
P 1 = X l ' . . X i ' S [ X l / s l . . . . X i / s i . . . . t n / s n ] .
T h a t is, P1 is formed out of an antecedent clause Q L F S by a b s t r a c t i n g over the 'parallel e l e m e n t s ' s l . . s i , p e r h a p s with some additional substitu- tions for t e r m s s i + l . . s n in S ( E [ a / b ] is the ex- pression E with a s u b s t i t u t e d for b). This seems to be sufficient to cover the range of examples t r e a t e d by Dalrymple, Shieber and Pereira (1991), but t h a t is a specific linguistic claim a b o u t verb phrase ellipsis in English and not central to the present paper.
4 . S E M A N T I C S F O R Q L F
In this section we outline the semantics of the Q L F language in a way t h a t is as close as possible to classical approaches t h a t provide the semantics in t e r m s of a function f r o m models to t r u t h values. T h e m a i n difference is t h a t denotation functions will be partial functions for some unresolved Q L F formulas, reflecting the intuition t h a t these are ' p a r t i a l i n t e r p r e t a t i o n s ' . T h e denotation of a Q L F expression will be extended monotonically as it is further resolved, a fully resolved f o r m u l a receiving a t o t a l function as its denotation. T h e semantics is not intended to describe the resolution process.
Before giving evaluation rules for the Q L F lan- guage, we first present a simplified version of the semantics for fully i n s t a n t i a t e d Q L F expressions. This is for expository purposes only; the full Q L F semantics does not depend on the simplified ver- sion.
4.1 S I M P L I F I E D S E M A N T I C S
We will use the n o t a t i o n [[~.]]m for the t r u t h value of an expression ~. with respect to a model m (but will leave m implicit), m includes an i n t e r p r e t a t i o n function I for m a p p i n g constants and predicates into domain individuals and relations. Also left implicit is a function assigning values to variables, which is required for the evaluation of l a m b d a ab- s t r a c t s as characteristic functions.
C o n s t r u c t s in the ' s t a n d a r d ' p r e d i c a t e logic sub- set of Q L F receive their semantics with the usual evaluation rules, for example:
• [[P(al . . . an)]] = 1 iff I ( a l ) . . . I(an) are in the relation I(P), and 0 otherwise.
• [[and(F1,F2)]] = 1 iff [[F1]]=I and [[F2]]=l, a n d 0 otherwise.
T h e evaluation rule for a f o r m u l a F with a scop- ing variable i n s t a n t i a t e d to [ I , J . . . . ] and con- taining a t e r m T - - - - t e r m ( I , C , R , Q , A ) is as follows:
• [ [ [ I , J . . . . ] : F ] ] = I iff [[Q(R' , F ' ) ] ] = I , and 0 otherwise, where
R' is X ' ( a n d ( R ( X ) , X = A ) ) [ X / I ] , and F ' is X ' ( [ J . . . . ] :and(F,X=A))[X/T, X/I]
T h i s evaluation rule s t a t e s t h a t a f o r m u l a with a scoping constraint list m a y be e v a l u a t e d by 'dis- charging' the t e r m for the first index on the list with respect to a f o r m u l a with a reduced scop- ing constraint. T h e rule discharges the t e r m by a b s t r a c t i n g over occurrences of the t e r m and its index, and a p p l y i n g the generalized quantifier Q to the t e r m ' s restriction a n d the a b s t r a c t derived f r o m the formula. In Section 5 we will say more a b o u t the ramifications of a d o p t i n g this t y p e of quantifier evaluation rule. Note t h a t this rule is also applicable to resolved t e r m s such as pronouns for which q has been resolved to e x i s t s and T is a constant or a scoped variable.
T h e d e n o t a t i o n assigned to a resolved f o r m u l a f o r m ( C , R, F ' ) in which the resolution variable has been i n s t a n t i a t e d to a f o r m u l a F ' is simply:
• [ [ f o r m ( C , R , F ' ) ] ] = l iff [ [ F ' ] ] = I , and 0 other- wise.
4 . 2 Q L F S E M A N T I C S
As mentioned earlier, t h e d e n o t a t i o n of a f o r m u l a F in the Q L F language will be a possibly par- tial function ([[... ]]) f r o m models to t r u t h values. Again we use the n o t a t i o n [[F]]m for the t r u t h value of a f o r m u l a F with respect to a model m (explicit reference to a variable assignment func- tion is again suppressed). For i n t e r p r e t a t i o n to be monotonic, we w a n t [[G]] to be an extension of [[F]] whenever G is a m o r e resolved version of F, and in particular for [[G]] to be t o t a l if G is fully resolved.
• [[F]]m=l iff W(F,m,1) but not W(F,m,0); • [[F]]m:0 iff W(F,m,0) but not W(F,m,1); • [[F]]m undefined iff W(F,m,1) and W(F,m,0).
Henceforth we will leave the model argument m implicit. T h e evaluation rules for W will generally take the form
W(F,v) if
W(F',v)
where F ' contains one fewer unresolved expression t h a n F (so t h a t it is possible for the process of rule application to t e r m i n a t e ) . T h e use of
if
r a t h e r t h a niffin
these rules means t h a t it is possible for rules producing more t h a n one value v to apply and hence for [IF]] to be partial.T h e model provides an i n t e r p r e t a t i o n function I m a p p i n g constants and predicates to individual and relations. We will also need to assume a rela- tion S(C,H) (for 'salient') between Q L F categories C and Q L F expressions H standing for individuals, quantifiers, or predicates, b u t the precise nature of the salience relation and the way it changes during a discourse are not i m p o r t a n t for the evaluation rules for Q L F given here. T h e intuitive motiva- tion for S is t h a t the category in an unresolved Q L F expression restricts the set of possible refer- ents for t h a t expression. S is discussed further in Section 5. We are now in position to present the evaluation rules, which we n u m b e r Q1, Q2, etc.
For s t a n d a r d connectives we have the obvious evaluation rules, for example,
Q1 W(and(F,G),I) if W(F,1) and W(G,1).
Q2 W(and(F,G),0) if W(F,0) or W(G,0).
Q3 W(not (F) ,l) if W(F,0).
Q4 W(not(F),0) if W(F,1).
T w o rules applicable to a formula F containing a t e r m with u n i n s t a n t i a t e d referent and quantifier meta-variables:
Q5 W ( F , v ) i f
W(F[existsl_q,h/_z],v)
W(RCA) ,1), where:
F is a formula containing the t e r m T = t e r m ( I , C , R , _ q , _ r ) , and h is t e r m such t h a t S(C,A).
and
Q6
W(F,v) if W(F[Q/_q, I / _ r ] , v ) , where:F is a f o r m u l a containing the t e r m
T=term(l,C,R,_q,_r),
andQ is a quantifier such t h a t S(C,Q).
( T h e substitutions for the meta-variables _ r and _q are to be read as p a r t of the evaluation rule.)
A rule applicable to a formula F in which a (pos- sibly unscoped) quantified t e r m occurs:
Q7
W(F,v) if W(Q ( R ' , F ' ) ,v), where:F is a formula containing the t e r m T = t e r m ( I , C , R , Q , A ) ,
R' is X" ( a n d ( R ( X ) , X=A) ) IX/I], and F ' is X'(a_nd(F,X=A))[X/T, X/I].
A rule applicable to a formula with an instantiated seoping constraint
Q8
W ( E I , J . . . . ] :F,v) if W(Q(R' , F ' ) , v ) , where:F is a formula containing the t e r m
T=term(I,C,R,Q,h),
R' is X ' ( a n d ( R ( X ) , X = A ) ) [ X / I ] , and F ' is X ' ( [ J . . . . ] :and(F,X=A))[X/T, X/I].
We also need a trivial rule for a formula with an uninstantiated scoping constraint so that it re- duces to application of other rules:
Q9 W(_s:F,v) if W(F,v).
T w o rules are applicable to f o r m expressions, cor- responding to the cases of an uninstantiated or instantiated resolution meta-variable:
Q10 W(F,v) if W(F[R(P)/_r],v)
where:F is a formula f o r m ( C , R , _ r ) P is a predicate such t h a t S(C,P).
Q l l W ( f o r r a ( C , R , F ' ) , v ) i f W(F',v) where F ' is a Q L F formula.
In a more complete description of the semantics we would also have to state t h a t the evaluation rules provided give the only way of determining membership of the relation W.
5. N O T E S O N T H E S E M A N T I C S
M o n o t o n l c l t y : In this paper we are using
m o n o t o n i c i t y in two senses which (by design) turn out to be consistent. T h e first is a syntactic no- tion for Q L F representations (instantiation rather t h a n destructive manipulation), while the second is semantic:
1.
2.
F1 is a more resolved version of F2 if F1 can be obtained by instantiating zero or more meta- variables in F2.
F1 is a less partial interpretation than F2 if [IF1]] is an extension of [[F2]].
S c o p i n g C o n s t r a i n t s : The quantification rules, (Q7) and (Q8), (i) select a t e r m from a for- mula, (ii) discharge all occurrences of the t e r m and its index in the formula and the term's restriction, replacing t h e m by a variable, and (iii) apply the term's quantifier to the discharged restriction and formula. T h e difference between (QT) and (Q8) is simply t h a t the latter also discharges the head of the scoping list, in this case by removing it r a t h e r than b y replacing it. (Keep in mind t h a t the dis- charge and replacement operations take place at the level of the evaluation rules for QLF; they are not applied to the Q L F expressions representing natural language meanings themselves).
As with Lewin's scoping algorithm, (Lewin 1990), there are no constraints built explicitly into the Q L F semantics on where a quantification rule for a t e r m m a y be applied, or indeed on the num- ber of times it may be applied. However, several
constraints arise out of (a) the absence of any se-
mantic rules for evaluating isolated terms, t e r m indices or scope lists, and (b) the requirement t h a t a t e r m be selected from a formula so t h a t its quan- tifier is known.
T h e emergent conditions on legitimate scoping are
1. No t e r m m a y be quantified-in more t h a n once: T h e first application of the quantifier rule dis- charges the term. Subsequent applications of the rule lower down in the evaluation would fail to select an undischarged term.
2. When a term's index occurs in a scope list, the quantifier rule for the term must be applied at t h a t point: It must be applied to discharge the head of the scope list, and by (1) above cannot additionally be applied anywhere else.
3. All occurrences of a term's index must oc- cur within the scope of the application of the term's quantifier rule: T h e quantification rule will only discharge indices within the formula to which it is applied. Any occurrences of the index outside the formula will be undis- charged, and hence unevaluable.
4. If a term R occurs within the restriction of a t e r m H, and R is to be given wide scope over the restriction, then R must also be given wide scope over H: Otherwise, suppose H is given wide scope over R. T e r m H will first be discharged, replacing the term, and with it its restriction, in the formula to which the rule is applied. T h e n the quantification rule for R needs to be applied to the discharged formula, but the formula will not contain an occurrence of the term R, making the rule inapplicable.
T h e last two constraints have often been at- tributed to restrictions on free variables and vacu-
ous quantification. T h e a t t r i b u t i o n is problematic since open formulas and vacuously quantified for- mulas are b o t h logically well defined, and without suspect appeal to the s y n t a x of the logical formal- ism they cannot be ruled out as linguistically ill- formed. By contrast, Q L F makes these violations semantically unevaluable.
U n s c o p e d T e r m s : When a t e r m ' s index is not mentioned in any scope list, the t e r m may be quantified in at any point within the formula. For anaphoric terms whose referent has been resolved to some individual constant, it does m a t t e r where the quantification rule is applied; since the term existentially quantifies over things identical to a single object, the scope of the quantification is im- material. It is thus convenient to leave anaphoric terms like this unscoped in QLF. Although this makes the Q L F look (syntactically) as though it is not fully resolved, semantically it is. For other un- scoped terms, alternative applications of the quan- tifier rule may well lead to distinct t r u t h condi- tions, and in these cases the Q L F is genuinely un- resolved.
C o n t e x t D e p e n d e n c e : Fully resolved QLFs are context-independent in the same sense that holds for closed formulas in traditional predicate logic (i.e. if the i n t e r p r e t a t i o n of the constant
symbols in the language is fixed). Unresolved
QLFs behave more like open formulas, and there is an analogy between assignments to u n b o u n d vari- ables in predicate logic and possible resolutions of meta-variables a d m i t t e d by the salience relation S. S(C,H) should be t h o u g h t of as providing Q L F expressions whose denotations are possible refer- ents for unresolved expressions with category C. (It would have been possible to define S as a direct relation between categories and referents, but this complicates the s t a t e m e n t of its role in resolution and in the semantic definitions.) We used S above in the definition of Q L F semantics, but it is also central to NL processing: being able to compute S can clearly play an i m p o r t a n t role in the process of reference resolution during NL interpretation and in the process of building descriptions during NL synthesis. (The c o m p u t a t i o n a l analogue of S was implemented as a collection of 'resolution rules' in Alshawi 1990.)
An i m p o r t a n t question is what to allow as possi- ble expressions in the range of S. One observation is t h a t as the range is widened, more NL resolu- tion p h e n o m e n a are covered. A rough s u m m a r y is as follows:
• constants: intersentential pronouns
• quantifiers: vague determiners
• indices: b o u n d variable, intrasentential pro- n o u n s
• predicates built from NP restrictions: one- a n a p h o r a
• predicates built from previous QLFs: inter- sentential ellipsis
• predicates built from current QLF: intrasen- tential ellipsis
6 . R E L A T E D A P P R O A C H E S
Viewed from a slightly different perspective, m o n o t o n i c i n t e r p r e t a t i o n has a n u m b e r of points of c o n t a c t with Pereira's categorial semantics (Pereira 1990). P u t briefly, in categorial seman- tics, semantic evaluation is represented as deduc- tion in a functional calculus t h a t derives the mean- ings of sentences from the meanings of their parts. Considerable emphasis is placed on the n a t u r e of these semantic derivations, as well as on the fi- nal results of the derivations (the 'logical forms' of sentences).
One significant advantage of this approach is t h a t constraints on legitimate scoping emerge nat- urally f r o m a consideration of permissible deriva- tions of sentence meaning, r a t h e r t h a n arising arti- ficially from syntactic constraints imposed on log- ical forms. Derivations involving quantified terms first introduce an assumption t h a t allows one to derive a simple t e r m from a quantified term. This assumption is later discharged b y the application of a quantifier. Conditions on the appropriate in- t r o d u c t i o n and discharge of assumptions in natu- ral deduction systems impose restrictions on the way t h a t quantifiers m a y legitimately be applied. For example, a quantifier assumption m a y not be discharged if it depends on further assumptions t h a t have not themselves been discharged. This prevents the occurrence of free variables in logical form, b u t w i t h o u t appeal to the s y n t a x of logical form.
T h e discharge of terms and t e r m indices when evaluating Q L F closely parallels the discharge of quantifier assumptions in categorial semantics. In- deed, the t e r m s and the indices are precisely the assumptions i n t r o d u c e d by quantified expressions, and which need to be discharged. Furthermore, the different orders in which quantifier assump- tions m a y be discharged in categorial derivation correspond to the choices t h a t the quantifier rules p e r m i t for discharging quantified terms.
W h e r e m o n o t o n i c i n t e r p r e t a t i o n and categorial semantics p a r t c o m p a n y is on the degree of ex- plicitness with which semantic derivations are rep- resented. In categorial semantics, derivation is a
background process t h a t builds up logical forms, but is not explicitly represented in the semantic formalism. By contrast, the a n n o t a t i o n of QLFs with scope lists provides an e x t r a level of informa- tion a b o u t how the derivations proceed. In partic- ular, they indicate which evaluation rules should be applied where.
Q L F thus provides a (usually partial) specifica- tion of a semantic derivation, showing (a) what the initial 'premises' are (roughly, lexical meanings, al- though these too m a y only be partially specified), and (b) the rules by which the 'premises' are com- bined. Q L F resolution a m o u n t s to further instan- tiating this specification. This view of Q L F can be contrasted with Logical Form as it is normally un- derstood, which represents the results of carrying out a semantic derivation.
T h e difference between specifying a derivation and carrying it out is what makes resolution order independent in monotonic interpretation. Making a resolution to Q L F only specifies when and how an expression should be evaluated during seman- tic derivation; it does not carry out t h a t part of the derivation. Where no distinction is drawn be- tween making a resolution and carrying out the corresponding step of the derivation, the order of resolution can be i m p o r t a n t . Thus, for Dalrymple, Shieber and Pereira (1991), where this distinction is not drawn, the precise interleaving of scope and ellipsis resolution determines the interpretation of the sentence. In QLF, resolutions dictate the order in which various steps of the derivation are carried out, b u t the resolution order does not reflect the derivation order.
Distinguishing between specifying and perform- ing a derivation also means t h a t a monotonic t r e a t m e n t of ellipsis resolution does not need to
resort to higher-order unification. Dalrymple,
Shieber and Pereira use higher-order unification to 'unpick' the composition of constituent mean- ings obtained in the semantic derivation from the ellipsis antecedent. Some of these meanings are then put back together to produce a predicate that can be applied to the ellipsis arguments. Since monotonic resolution does not carry out the final composition of meanings, b u t merely sets out con- ditions on how it is to take place, there is no need to unpick composed meanings and put t h e m back together again.
monotonic interpretation thus refers to the way in which alternative derivations of sentence meanings may be chosen, b u t not to the semantic effects of those sentence meanings.
7. I M P L E M E N T A T I O N
B E N E F I T S
A description of the language processing mecha- nisms to which we have applied the monotonic semantics model is beyond the scope of this pa- per. However, we believe t h a t the Q L P represen- tation presented here brings significant advantages to implementing mechanisms for reference resolu- tion, scoping, preference and generation.
R e f e r e n c e a n d S c o p i n g : T h e order indepen- dence of resolution operations allows for a variety of control structures in implementing a resolution mechanism. We find it convenient to make a bot- tom up pass through QLFs making reference res- olutions, followed b y a stage of scoping resolution, and to iterate over this should any of the resolu- tions introduce further unresolved expressions.
T h e salience relation S can be implemented as procedures t h a t search for properties, objects or indices in context. Scoping proceeds simply by the non-deterministic instantiation ofscoping con- straints, subject to the restrictions imposed on evaluable QLFs (Section 5), plus techniques for ignoring logically equivalent scopings, as for ex- ample described by Moran (1988).
P r e f e r e n c e a n d D i s a m b i g u a t i o n : A resolved QLF preserves all the information in the original unresolved QLF, and also records the correspon- dence between resolved and unresolved expres- sions. This makes it possible to define preference metrics that can be used for ranking alternative interpretations independently of the search strate- gies used to derive them. For example, in the case of scoping, these metrics can combine information about how far a quantifier was 'raised' with infor- mation a b o u t the surface form of its determiner. Preference ranking over alternative resolutions fa- cilitates automatic disambiguation of input. Inter- active disambiguation can make use of generation from resolved QLFs for confirmation by a user.
G e n e r a t i o n : T h e r e is a strong connection be- tween monotonicity and reversibility in language processing systems. Monotonicity of unification means that algorithms such as head-driven gener-
ation (Shieber et al 1990) can be applied to gram-
mars developed for analysis. We use a variant of this algorithm for generating from QLFs, and the monotonicity of semantic interpretation means
t h a t the g r a m m a r used for generating from un- resolved QLFs (the normal ' o u t p u t ' of the gram- mar) can also be used for generation from resolved QLFs.
In parallel to the distinction between g r a m m a t - ical analysis (of NL into unresolved QLFs) and interpretation, we make the distinction between
grammatical synthesis (of NL from QLFs) and de-
scription. Description is the process of deriving a Q L F from which synthesis proceeds by taking a fact (e.g. a database assertion) as input. We hope to report on our approach to description else- where. However, one of the principles of QLF- based description is t h a t while interpretation in- stantiates referent fields in underspecified QLFs, description involves instantiating category and re- striction fields for QLFs in which referent fields are already instantiated. T h e preference metrics applied to rank alternative interpretations can be applied equally well to ranking resolved QLFs pro- duced by a nondeterministic description process, so there is a sense in which the preference mecha- nism can also be made reversible.
R E F E R E N C E S
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