Direct and Underspecified Interpretations of LFG f-structures 
Josef van Genabith* 
Insitute for Natural Language Processing 
University of Stuttgart 
Azenbergstr. 12 
D-70174 Stuttgart, I?'RG 
j osef@ims, uni-stuttgart, de 
Dick Crouch 1 
SRI International 
23 Millers Yard 
Mill Lane 
Cambridge CB2 \]RQ, UK 
re@cam, sri. tom 
Abstract 
We describe an approach to interpreting 
LFG f-structures (Kaplan & Bresnan, 
1982) truth-conditionally as underspeci- 
fled quasi-logical forms. F-structures are 
either interpreted indirectly in terms of 
a homomorphic embedding into Quasi 
Logical Form (QLF) (Alshawi, 1992; 
Alshawi & Crouch, 1992; Cooper et 
al., 1994a) representations or directly 
in terms of adapting QLF interpreta- 
tion clauses to f-structure representa- 
tions. We provide a reverse mapping 
from QLFs to f-structures and establish 
isomorphic subsets of the QLF and LFG 
formalism. A simple maI)ping which 
switches off QLF contextual resolution 
can be shown to be truth preserving with 
respect to an independently given se- 
mantics (Dalrymple et al., 1995). We 
compare our proposal with approaches 
discussed in the literature. 
1 Introduction 
Ditferent languages express grammatical flmctions 
(such as subject or object) in a variety of ways, e.g. 
by position or by inflection. Functional-structures 
(f-structures) (Kaplan & Bresnan, 1982) are 
attribute-value matrices that provide a syntactic 
level of representation that is intended to abstract 
away from such surface variations while capturing 
what are considered underlying generalisations. 
Quasi-Logical Forms (QLFs) (Alshawi & Crouch, 
1992; Cooper et al., 1994a) provide the seman- 
tic level of representation employed in the Core 
Language Engine (CLE) (Alshawi, 1992). The 
two main characteristics of the formalism are un- 
derspeeification and monotonic contextual reso- 
lution. QLFs give (partial) descriptions of in- 
*Present address: Dublin City University, Dublin 
9, Ireland; josefOcompapp, dcu. ie 
l Present address: Speech Research Unit, DRA 
Malvern, St Andrews Road, Great Malvern, Worcs 
WR14 3PS, UK; croueh0signal, dra. hmg. gb 
tended semantic compositions. Contextual reso- 
lution monotonically adds to this description, e.g. 
by placing fln'ther constraints on the meanings 
of certain expressions like pronouns, or quanti- 
tier scope. QLFs at; all stages of resolution are 
interpreted by a truth-conditional semantics via 
a supervaluation construction over the composi- 
tions meeting the description. F-structures are 
a mixture of mostly syntactic information (gram- 
matical flmctions) with some semantic, predicate- 
argument information encoded via the values of 
PRED features: 
"PRED ~CANDIDATI,~ ~ 
NUM SG L OBJ 
LsPEc. A 
Unresolved QLF gives the basic predicate- 
argument structure of a sentence, mixed with 
some syntactic information encoded in the cate- 
gories of QLF terms and forms: 1 
?Scope : support (term (+r, <nmn=sg, spec=every>, 
representative, ?Q, ?X) , 
term (+g, <num=sg, spec=a>, 
candidate, ?P, ?R) ) 
While there is difference in approach and emphasis 
unresolved QLFs and f-structures bear a striking 
similarity and it ix easy to see how to get from one 
to the other: 
pn~:, n($ r~,.,? r,~> ~ .*Scope : n(~,.,~,~) 
£ ")% 
The core of a mapping taking us from fstructures 
to QLFs places the values of subcategorizable 
grammmatieal fnnctions into their argument posi- 
tions in the governing semantic form and recurses 
on those arguments. I,\]'om this rather general per- 
spective the difference between f-structures mid 
l'l'he motivation for including tiffs syntactic in- 
formation in QLFs is that resolution of such things 
as anaphora, ellipsis or quantifier scope may be con- 
strained by syntactic factors (Alshawi, 1990). 
262 
QLF is one of information packaging rather than 
mGthing else. We tbrmalise this intuition in terms 
of translation functions r. The precise fln'm of 
these mappings depends on whether the Q1,Fs and 
f-structures to be, mapI)ed contain comparable lev- 
els of syntactic information, and in the case, of 
QLF how this inforination is distributed between 
term and form categories and the recursive struc- 
ture of the QLF. The QLF formalisln delitmratcly 
leaves entirely open the amounl; of syntactic in- 
formation that should be encoded within a QLF 
the decision rests on how much syntactic intbr- 
mation is required for successful contextual res- 
olution. The architecture of the LFG and QLF 
formalism are described at length elsewhere (Ks- 
plan & Bresnan, 1982; Alshawi & Crouch, 1992; 
Cooper et al., 1994a). l/eh)w we detine, a lan- 
guage of wJ\]:s (well-formed f-struct'tm:s), a (family 
of) translation function(s) r fi:om {-stru(:tures to 
(unresolved) QLFs and an inverse flmction r ~ 
Dom uin'esolved QLFs hack to f-structures, r and 
r -~ determine isolnorphic subsets of the QLF and 
LFG formalism. We eliminate r and give a di- 
rect and underspecified interpretation in terms of 
adapting QLF interpretation rules to fstrueture 
representations. While the initial definition of'r is 
designed to maxilnally exploit 1;he contextual res- 
olution of QLF, later ve, rskms nfininfise resolution 
efl'ecl;s. A simph; version of ~ where the QLF COil- 
l;extual resolution component is "swil;ched off" is 
truth preserving with respect 1;o an independelfl, ly 
given semantics (DMrymple el; al., 1995). 
2 Well-formed f-structures 
We define a language of wff-s (we, ll-fornmd f- 
structures). The basic vocabulary consists of five 
disjoint set;s: GFs = {SUIU, OBJ, OlU2, ore,0,...} 
(subcategorizable grmnnml;ical flmc- 
tions), Gl"~, := {AmS, I, MODS, AMOI)S,...} (noIl- 
subcategorizahle gralnmatical ftlnctioIlS), SI,': 
{candidate0, marY0, support(j" suns, j" oB,,},...} 
(semantic forms), A'/<: {SI'I4C, NUM, 1H,;II,...} (a/,;- 
tributes) and AV= {~,;vl,mY, MOST, el,, FEM,...} 
(atomic values). The tirst two forlnation clauses 
pivot on the semantic form PILED values. The two 
tinM clmlses cow;r non-subcategorizMfle granmml;- 
ical fulmtions aim what we call alomic attrilmte- 
wdue pairs. Tags i\[i\] are used to repre, sent reen- 
trancies and often appear vacuously, q'he side 
condition in the second and third clause ensure, s 
that only identical substructures can have identi- 
cal tags: 
. if 1\[{> 6 5'F then \[P,u,;,) 11{> \] ~ c= wJJ-s 
• if ~o~E\],..., ~,~\[~ < ,,¢.~:, a,,d n(¢ l',,..., 1- r,,) c- 
S1" then ~\[i\] (: wJf-s where 9'2\[i\] :~ 
11'~ ~\] P~u,:,, l~(tp,,...,t ",,,) BI 
and tbr any ¢~ and qS\[i~ occurring in ~\[~, 1 ~- m 
except possibly where 'gJ =- @. 
• if ~o~,...,~o,~\[~,*/;~ ~ wff-s, where ~/2~ \[;,\]~,.:,n<...>\]~\],rce;,<a,,lrCd,,,,4,,ffiA)... 
\[" {~'~\], ' ,~°'~} l ;,L,:~ u<...> \[~ 
and for any ¢\[1\] and XI~ occurring in (\[~, 1 -¢ m 
except possibly where 4) ~ X. 
• if (~ C AT, v 6 AV, ~J G wjf-s where 79\[i\] -: 
I;('~":') H('")\] ~ and('¢d°m(~\[ij) lJmn.... I: j 
,'l,,,:,) n<...) El e ~,JJ-.~ 
Proposition: tim detinition specilies f-structures 
that are (',omph%e, coherent and consistent. 2 
3 How to QLF an f-structure: 
3.1 A Basic Mapping 
Non-recursive f-structures are mapped to QLF 
terms and recursive f-structm'es to QI,F forms 
by metals of a two place flmction r detined below: 
"(tl 1)1 
e, T(F, I'RE1, l\[ 0 \[I\]) : 
L(-tn 'on 
term(I,<gf=F,c~l ::- 'ol,... ~(tn = v,>,I1, 
?Q_I,ZR_I) 
l ib ~ol\[N (tt ~)1 
" r(F' /:'\]u'"\])i; \]\]<J" \]'1) ' ' ' '* l'a> (D/~ K~ \] \[~) :- " 
L(bm 't)n~ 
?Scope:form(I,<gf=l),pred=II(F1, ... , Fn) 
(~l ZVl ~ . . . ~ G~mZ?)m>, P~ PO-(v~, ~,\[~),..., ~(1:,~, ~,~)), 
?F_I) 
where ?Scope is a new QLF mete-variable,, P ~ 
new w~riable and ~i 6_ AT 
~Prool': induction on the formation rules for wff- 
s using the definitions of completeness, coherence atttl 
consistency (KalJan & lbesmm, 1982), The not;ions of 
a'u, bst'r'u, ct'wre occwrrin.q in an f-structure al|d dom,,in of 
an f-struct'urc can easily be spelleA out fol'ntally. ~ is 
syntactic identity modulo permm;ation. The dciinition 
of w\]..f~s uses graphical rel)resen{;ations of t'-struct;ure.s. 
It can e.asily be recast in l;erlns of hierarchical sets, 
finite functions, directed graphs etc. 
263 
To translate an f-structure, we call on r with the 
first argument set to a dummy grammatical flmc- 
tion, SIGMA. The reader may check that given 
SUBJ 
PREI) TENSE 
OBJ 
we obtain 
PRED ~REPRESENTATIVE ~ NUM PI, 
SPEC MOST 
'SUl)port <j" SUB,l,\]" OBJ>' PAST 
IPREI) ICANDIDATE'I NUM PL 
\[\] SPEC TWO 
tile target QLF: 
\[\] 
m 
?Scope:form(+f,<gf=sigma,tense=past, 
pred=support(subj,obj)>, 
P^P(term(+g,<gf=subj,num=sg, 
spec=most>, 
representative,?O_g,?K g), 
term(+h,<gf=obj,num=sg, 
spec=two>, 
candidate,?Q h,?R_h)), 
?F_f). 
The truth conditions of the resulting underspec- 
ified QLF formula are those defined by the QLF 
evaluation rules (Cooper et al., 1994a). The orig- 
inal f-structure and its component parts inherit 
the QLF semantics via r. r defines a simple ho- 
momorphic embedding of f-structures into QLFs. 
It comes as no surprise that we can eliminate r 
and provide a direct underspecified interpretation 
for f-structures. 
Note that r as defined above maximises tile 
use of tile QLF contextual resolution component: 
quantifier meta-variables allow for resolution to 
logical quantifiers diflbxent fl'om surface form (e.g. 
to cover generic readings of indefinites), as do 
predicate variables (in e.g. support verb construc- 
tions) etc. A definition of r along these lines is 
useful in a reusability scenario where an existing 
LFG grammar is augmented with the QLF contex- 
tual resolution component. Alternative definitions 
of r "resolve" to surface form, i.e. minimise QLF 
contextual resolution. Such definitions are useflfl 
in showing basic results such as preservation of 
truth. Below we outline how r can be extended 
in order to capture more then just the basic LFG 
constructs and to allow for different styles of QLF 
construction. 
3.2 F-structure reentrancies 
r respects f-structure reentrancies (indicated in 
terms of identical tag annotations ~\]) without fllr- 
ther stipulation. Consider e.g. the f-st;ructure qo 
associated with the the control construction Most 
representatives persuaded a manager to support 
every subsidiary: 
\[ I'II.ED 'REPItF, SENTATIVE ~ \] NUM PL 
~UBJ |PEa 3 \[\] 
LSPEC MOST 
PRm) 'persuade (} SUBJ,$ on J,i" xcow')' 
0 B J 
XCOMP 
\]I)REI) ~MANAGER~I NUM SG 
SPEC A 
-SUBJ \[I'R'EI) 'MANAGEI\[' 1k SPEc/PElt/NUM 3SOa J/| 
IPREI) 'sell@'sultJ,$ OB.t}' 
PRED 'subsidiary' OBJ NUM SG 
Le~:a 3 
D\] 
\[\] 
where the object \[~ of the matix clause is token 
identical with tile controlled subject \[~ of the em- 
bedded clause. ~o translates into 
?Sl:form(+f,<gf=sigma, 
pred=persuade(subj,obj,xcomp)>, 
P^P(term(+g,<gf=subj,num=pl, pers=3,spec=most>, 
representative,?Q_g,?R_g), 
term(+i,<gf=obj,num=sg, 
pers=3,spec=a>, 
manager,?O_i,?~_i), 
?S2:form(+h,<gf=xcomp, 
pred=support(subj,obj) 
O^Q(term(+i,<gf=subj, 
num=sg,pers=3, 
spec=a>, 
manager,?~_i,ZR_i), 
term(+j,<gf=obj,num=sg, 
pers=3,spec=every>, 
subsidiary,?Q_j,?g_j)), 
?F_h)), 
?F~) 
where the f-structure reentrancy surfaces in terms 
of identical QLF term indices +± and meta- 
w~riables ?0_±,?R i as required. 
3.3 Non-Subcategorizable Grammatical 
l%mctions 
The treatment of modification in both f-structure 
and QLF is open to some flexibility and variation. 
Here we can only discuss some exemplary cases 
such as LFG analyses of N and NP pre- and post- 
modification. We assume an analysis involving the 
restriction operator in the LFG description lan- 
guage (Wedekind & Kaplan, 1993) and selnantic 
form indexing (II<...> @) e.g. by string position) as 
introduced by (Kaplan & Bresnan, 1982). The f- 
structure associated with The company which sold 
APCOM started a new subsidiary is a 
aHere attd in tile following we will sometimes omit 
tags in the f-structure representations. 
264 
-PI{EI) CCOMPANY'(2) 
SPEC TIll'; 
NUM S(I 
GENI) NEUT 
\[Pll'ED '(\]OMI'ANY'(2) \] \] 
SPEC TIlE 
SUBJ SUBJ |NUM S(l 
L(mND NEUT \] 
RM' PR.EI) 'SEI,L (j'S1JB.I,?OB.I>'(4) ' 
TENSE PAST 
OBJ NUM SG 
, k LGEND NEUT 
FRED 'start<? SUUJ,T Ol~a}'<6> 
FPltED ~SUBSIDIAItY'(8 ) 
\] SPEC A 
0BJ / NUM SG 
GENI) NF, UT 
LAM {\[P,mD 'N|:W'<r> \]} 
We extend r as follows: 
l oll Vl I'ItE1) llO(i) • ~(r, N) := \] (~n Ca 
It.M 7~. 
LAM A 
term(l,<gf=F,(h : Vl,.. c~n : ~)~ >,Restr, 
?Q I,?R_I) 
"1~ ~ {~01,... , ~'.,,,},A ~ {#1,... ,#o}, ~91,... , ~.'o, p,l,..., It,~ ~- wJf-s, 
#i ~ \[I'ltED 't/i\] and: 
Restr = ?l' (''" (?}o (,~a;.and(H(a:), T(i),,~ ('P~))))) 
T(i>'a:('\]~) = Z T(II'M'o'i\[\[ I>'1~1';1) lI0<i> <- X\]) 
ojc'~ " " • 
The f-strneture associated with our example sen- 
tence translate.s into 
?SO: 
form(+f, <gf=sigma, pred=start (subj, obj ) >, 
P ~P (term (+g, <gf =subj, num=sg, gend=neut, 
spec=the>, 
x ~ and (company (x) , 
?SI : f orm(+i, <gf=rm, 
pred=sell (subj, obj ) >, 
WQ(x, 
term(+k,<gf=obj, 
num=sg, gend=neut>, 
apcom, ?Q_k, ?R_k) ), 
?F_i) ), 
?Q_g, ?R_g), 
term(+h, <gf=obj, num=sg, 
gend=neut, spec=a>, 
new (subsidiary) , ?Q_h, ?R_h), 
?F_~). 
as required. Note, however, that the translation 
inay overspecify the range. In the f-structure do- 
main modifiers are collected in an unordered set 
while in the range we impose some arbitrary or- 
dering. For intensional adjectives (compare a for- 
mer famous president with a famous former pres- 
ident), this ordering may well be incorrect. Hence 
ordering information should be codable in (or re- 
coverable from) the representations. In LFC this 
is available in terms of f-precedence. A more 
satisfactory translation into QLF complicates the 
treatment of (nominal) Inodification as abstracted 
QLF forms. Modifiers are represente(1 as extra ar- 
guments in the body of the form and take the form 
index of dm restriction as one of their argmnents: 4 
x- ?Scp : form (+r, <gf=np-re str, pred=subs idiar y>, 
P^P(x, 
form(+a, <gf=am, pred=new>, 
Q^Q (+r) ,?h)) ,?R) 
Modifier ordering can then be transferred to reso- 
lution, or encoded in the categories of the rest, r|(> 
Lion and modifiers to filrther constrain the order 
of application selected by resolution. 
4 Direct interpretation 
The core of tile direct interpretation clauses for 
wff-s involve~s a simple variation of the quantifier 
rule and the t)redieation rule of the QLF sentan- 
lies (Cooper et el., 1994a). Consider tile flag- 
meat without N and NP modification. As before, 
t;he semant;ics is detined in terms of a supervalu- 
at|on construction on sets of disambiguated rep- 
resentations. Models, variable assignment flmc- 
lions, generalized quantifier interpretations and 
the QLF definitions for the connectives, abstrac- 
tion and application etc. (see Appendix) carry 
over unchanged. The Ile.W quantification rule D14 
non-det;erlninistically retrieves non-recursive Sll|)~ 
categorizable grammatical fiulctions and entploys 
the vahle of a SI'EC feature in a generalized quaIb 
tiller interpretation: 
D14: if %~#(~) C ',,,if-s, ,sub(%~/,(~)) 
• if" ~/; ~ "£\],:D l\] 0 then \]2o(qo, v ) if 
V q(Q(Ax.I\[(x), X:,:.qo\[¢(~) +- x\]), v), x new 
• if#;~_ \[;",£.1,:1)... II0\] (i.e. SPF, c~dom(',/;)) 
dmn V,(% v) if V,,(~o\[~/,(~\]) < Ill), v) 
The new predication rule 1)10 is defined in terms 
of a notion of nuclear scope f-structure: '~' 
4See (Cooper et al., 1994b) for examples of this 
style of treating VP modification. 
r)A nuclear scope f-structure ~ C nf-s is is an 
f-structure resulting from exhaustive at)plicatiou of 
D14. It can be defined inducdwdy as follows: 
• if 3`i a variable or a constant symbol then 
I F1 3'1 ~ I'RI';I) II(? Pl,..., ? Pn) tK @s 
265 
I F1 Vl DIO: if~o~_ PREDII(J'F1,.. ,\]'P,~} and~Enf-s kK 
then 12~(~, v) if Vg(II(v~ , .,3,~), v) 
To give an example, under the direct interpreta- 
tion the f-structure associated with most repre- 
sentatives supported two candidates is interpreted 
as an underspecified semantic representation in 
terms of the supervaluation over the two gener- 
alized quantifier representations 
most (repr, Ax. two ( cand, .~y. support (x, y))) 
two ( cand, Ay. most( repr , Ax. support ( x, y) ) ) 
as required. The direct underspecified interpre- 
tation schema extends to the modification cases 
discussed above in the obvious fashion. 
5 How to f-structure a QLF 
The reverse mapping from QLFs to LFG f- 
structures ignores information conveyed by re- 
solved recta-variables in QLF (e.g. quantifier 
scope, pronouns antecedents), just as the map- 
ping froIn f-structure to QLF did not attempt to 
fill in values for these recta-variables. For QLF 
terms with simple restrictions (i.e. no modifiers), 
7 --1 is defined as follows: 
• 7--l(term(I,<gf=F,(~l = vi,..., (~n : Vn >,lI, 
_,_)) := \[ 
O~1 Vl 
r PaED II \[\] 
LO~n Vn 
• T -l(_:form(I,<gf=F,pred=l\[(Fi,.. . ,Fm),c~l = 
vi,...,(~j =Vj>,P^P(Loi,... tim),_)) := \[ 
'~1 Vl 
(~ vj 
P /PriED II(tF1,...,tFM) \[\] 
r_~(ol ) 
L T- 1 
As an example the reader may verify that r-~ 
retranslates the QLF associated with Most rep- 
resentatives persuaded a candidate to support ev- 
ery subsidiary back into the f-structure associated 
with the sentence as required. Again, 7 --1 can be 
extended to the non-subcategorizable grammati- 
cal functions discussed above. The extension is 
straightforward but messy to state in full general- 
ity and for reasons of space not given here. 
• if ffi E nf-s, a val-iable or a constant symbol then I 
Fi Vi 
PRED II(t Pl,..-,"1" Pn) E @S 
6 Going back and forth 
Proposition: for an f-structure ~ E wff-s 6 
T--I(T(~p)) = ~t) 
The result establishes isomorphic subsets of the 
QLF and LFG formalisms. For an arbitrary QLF 
¢, however, the reverse does not hold 
T(T--I(~))) ¢ ~/) 
F-structures do not express scope constraints etc. 
7 Preservation of truth 
w assigns a meaning to an f-structure that de- 
pends on the f-structure and QLF contextual reso- 
lution. We define a restricted version T' of ~- which 
"switches off" the QLF contextual resolution com- 
ponent, w' maps logical quantifiers to their surface 
form and semantic forms to QLF formulas (or re- 
solved QLF forms): 
-ill)d,\] D Vl 
SPEC Q 
• ~'(r, " n(> \[i\]):= 
term(I,<gf=F,C~l = Vl,... ,c~n -= v,>,EI,Q,I) 
~tl Vl 
• ~'(r, ~,~, n(tr,,...,tr,d ~):= 
Lf~m Vrn 
?Scope : form(I, <gf=F ,pred=H(F1, . . . , Fn), 
O~ 1 ZVl, . • . ~ ~m zVm > , 
i~(T(r,, ~,~\]),..., T(rn, ~nli~)),m 
Proposition: T' is truth preserving with respect to 
an independent semantics, e.g. the glue language 
semantics of (Dalrymple et al., 1995)• Preserva- 
tion of truth, hence correctness of the translation, 
is with respect to sets of disambiguations. The 
proof is by induction on the complexity of ~7 
The correctness result carries over to the direct 
interpretation since what is eliminated is T'. s 
6Proof: induction on the complexity of y;. 
7Proof sketch: refer to the set of disambiguated 
QLFs resulting from w'(~o) through application of the 
QLF interpretation clauses as \])(T'(~)) and to the 
set of conclusions obtained trough linear logic deduc- 
tion from the premisses of the (r projections of ~p as 
(a(~o))F. Consider the fragment without modification. 
Base case: for So with nonrecursive values of gram- 
matical functions show Y(T'(~)) = (a(W))e. Induc- 
tion: for ~ with possibly reeursive values ~i of gram- 
matical functions on the assumption that for each i: 
V(~'(~i)) = (~(~i)),- (IH) show V(w'(~)) = (a(~))~. 
sIf the results of linear logic deductions are inter- 
preted in terms of the supervaluation construction we 
have preservation of truth directly with respect to un- 
derspecified representations, QLFs and sets of linear 
logic premisses. 
266 
8 Conclusion and Comparison 
We have provided direct and indirect undersl)(!(:- 
ified model theoretic intert)retations for LFG f- 
structures. The interpretal;ions are truth t)rese.rv- 
ing, hence correc% with respect to an indei)en- 
dent semantics. We have established isomorphic 
subsets of the QLF and LFG formalism. Our ap- 
t)roach is in the spirit of but (:ontrasts with at)- 
proaches by (Halvorsen, 1983; ttalvorsen & Ks- 
plan, 1988; Fenstad et al., 1987; Wedekind & Ks- 
plan, 1993; Dalrymple et al., 1995) which are ne.i- 
ther unde.rspecifie.d nor direct. Like (Halvorsen, 
1983; Wedekind & Kaplan, 1993) our approach 
fails int;o the description by analysis 1)aradigln. 
Its limits are determined by what is analysed: 
f-sU'uctm'es. Work is under way to interpret f- 
structures as UDRSs in order to exploit t11(; UDRS 
inference component (Reyh;, 1993). Furl;her work 
recons|;ructs QLF interl)retal;ion in terms of lin- 
ear logic deductioi~s (1)alrymple et al., 1995) and 
provides a scope constraint mechanisln tot such 
deductions. 
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Deduction, In Journal of Semantics, pages 123- 179 
3.Wcdekind and R.Kaplan 1993. Type-Driven Se- 
mantic Interpretatiml of f-Struefiures, In Proceed- 
ings 6th Conference of the European Chapter of the 
Association for Computational Linguistics, pages 
4114 411 
Appendix: Quasi Logical Forms 
iIere we can only outline ttw. parts of the syntax and 
semanti(:s of QLF (for a full account see (Alshawi & 
Crouch, 1.992; Cooper et al., 1994a)) most relevant ior 
(mr present purposes. A QLF term must be an indi- 
vi(tual variable, an individuM constant or a complex 
term expression. A QLF formula must I)e an applica- 
tion of a predicate to arguments possibly with scoping 
constraints or a form expression: 
term ::= x \[ c \[ term(Id,Cat,Restr,Qnt,Ref) 
formula ::= 3cope:Pred(hrg_l,...,Arg_n) 
\[ Scope:form(Id,Cat,Restr,Kes) 
The QLt,' semantics is detined in terms of a su- 
pervaluation construction with standard lfigher order 
models in t;erms of a valuation relation V(¢, v) which 
disambiguates a QI,F (p with respect to a context in 
terms of a salience, relation $(C,P) between syntactic 
category descriptions C and QLF context tel)resents- 
dons P: 
• \[qS~ lv*''j -- 1 iif V(qS, 1) but not 12(¢, 0) 
• ~4)\] M','j = 0 iff V(4), 0) but not V(¢, 1) 
• H M,,, ,.~d,;Ii.,ed itf V(4,, 1) and V(¢, O) 
QI: V,a(and(¢, ~/)), 1) if V.,~(¢, 1) and V,a(~/J, 1) 
Q2: l;,(and(¢,'~b),0)if V,a(4),0 ) or V,~0/),0 ) 
QlO: V,a(p(a'rg~, . . . , at.q,,), P( Arq, ..... Arq,,) ) 
if V~(p,P) and V,a(argt,Argt) and ...and 
l;o (,*rg,~, Arg,~) 
Q12: if 4) is a fl)r- 
mula containing a term term(I ,C,K,?\[~,?R) and 
Q is a quantifier such that 8(C, Q) then V,a(¢, v) 
if v,~(¢\[O/?% ~ /rR\], ~) 
Q14: if ¢ is a formula contain- 
ing a term T = term(I,C,R,Q,h) t.hen l)~a(gb,,v ) 
if F,a(Q (R' ,F'), v) where 
R' is X ^(and(R(x) ,X=A)) \[X/l\], and 
F' is X ^ (and(¢,X=h)) \[X/T,X/I\] 
Q1.5: if \[I,J .... \] :q5 is a formula containing a term 
T = term(I,C,tt,q,h) then 12.,~(\[I,a .... \] :¢,v) 
if 12:/({;1 (it' ,F'), v) wtmre 
R' is X^ (and(R(X) ,X=A)) IX/I\], and 
F' is X'(fa .... \] :and(¢,X=h)) \[X/T,X/I\] 
Q16: G(form(I,C,R,?R),v) if G((R(P),v) where a(c,p) 
Qlr: V,(form(I,cm,¢), v) if k,,((a(~P) ,~) where 4' 
is a QLF expression but not, a reels-variable 
267 
