On Interpreting F-Structures as UDRSs 
Josef van Genabith 
School of Computer Applications 
Dublin City University 
Dublin 9 
Ireland 
j osef@compapp, dcu. ie 
Richard Crouch 
Department of Computer Science 
University of Nottingham 
University Park 
Nottingham NG7 2RD, UK 
rsc@cs, nott. ac. uk 
Abstract 
We describe a method for interpreting ab- 
stract fiat syntactic representations, LFG f- 
structures, as underspecified semantic rep- 
resentations, here Underspecified Discourse 
Representation Structures (UDRSs). The 
method establishes a one-to-one correspon- 
dence between subsets of the LFG and 
UDRS formalisms. It provides a model 
theoretic interpretation and an inferen- 
tial component which operates directly 
on underspecified representations for f- 
structures through the translation images 
of f-structures as UDRSs. 
1 Introduction 
Lexical Functional Grammar (LFG) f-structures 
(Kaplan and Bresnan, 1982; Dalrymple et al., 1995a) 
are attribute-value matrices representing high level 
syntactic information abstracting away from the par- 
ticulars of surface realization such as word order 
or inflection while capturing underlying generaliza- 
tions. Although f-structures are first and foremost 
syntactic representations they do encode some se- 
mantic information, namely basic predicate argu- 
ment structure in the semantic form value of the 
PRED attribute. Previous approaches to provid- 
ing semantic components for LFGs concentrated on 
providing schemas for relating (or translating) f- 
structures (in)to sets of disambiguated semantic rep- 
resentations which are then interpreted model the- 
oretically (Halvorsen, 1983; Halvorsen and Kaplan, 
1988; Fenstad et al., 1987; Wedekind and Kaplan, 
1993; Dalrymple et al., 1996). More recently, (Gen- 
abith and Crouch, 1996) presented a method for 
providing a direct and underspecified interpretation 
of f-structures by interpreting them as quasi-logical 
forms (QLFs) (Alshawi and Crouch, 1992). The ap- 
proach was prompted by striking structural similar- 
ities between f-structure 
\['PRED ~COACH ~ \] SUBJ NUM 
SG 
/SPEC EVERY 
PRED 'pick (T SUB J, T OBJ)' \[PRED 'PLAYER'\] 
L°B: iN'M s/ J LSPE¢ 
and QLF representations 
?Scope : pick (t erm(+r, <hUm= sg, spec=every>, 
coach, ?Q, ?X), 
term (+g, <num=sg, spec=a>, 
player, ?P, ?R) ) 
both of which are fiat representations which allow 
underspecification of e.g. the scope of quantifica- 
tional NPs. In this companion paper we show that 
f-structures are just as easily interpretable as UDRSs 
(Reyle, 1993; Reyle, 1995): 
coach(x) layer(y) 
I pick(x,y) I 
We do this in terms of a translation function r from 
f-structures to UDRSs. The recursive part of the def- 
inition states that the translation of an f-structure is 
simply the union of the translation of its component 
parts: 
'F1 71 ... 
T( PRED I-\[(~ rl,...,l l~n) ) 
r, ..... T r.)) u u... u 
While there certainly is difference in approach and 
emphasis between f-structures, QLFs and UDRSs 
402 
the motivation foi" flat (underspecified) representa- 
tions in each case is computational. The details of 
the LFG and UDRT formalisms are described at 
length elsewhere: here we briefly present the very 
basics of the UDRS formalism; we define a language 
of wff-s (well-formed f-structures); we define a map- 
ping 7" from f-structures to UDRSs together with a 
reverse mapping r -1 and we show correctness with 
respect to an independent semantics (Dalrymple et 
al., 1996). Finally, unlike QLF the UDRS formal- 
ism comes equipped with an inference mechanism 
which operates directly on the underspecified rep- 
resentations without the need of considering cases. 
We illustrate our approach with a simple example 
involving the UDRS deduction component (see also 
(KSnig and Reyle, 1996) where amongst other things 
the possibility of direct deductions on f-structures is 
discussed). 
2 Underspecified Discourse 
Representation Structures 
In standard DRT (Kamp and Reyle, 1993) scope re- 
lations between quantificational structures and op- 
erators are unambiguously specified in terms of the 
structure and nesting of boxes. UDRT (Reyle, 1993; 
Reyle, 1995) allows partial specifications of scope 
relations. Textual definitions of UDRSs are based 
on a labeling (indexing) of DRS conditions and a 
statement of a partial ordering relation between the 
labels. The language of UDRSs is based on a set 
L of labels, a set Ref of discourse referents and a 
set Rel of relation symbols. It features two types of 
conditions: 1 
1. (a) if/E L and x E Refthen l : x is a condition 
(b) if 1 E L, R E Rel a n-place relation and 
Xl, ..,Xn E Ref then l : P(Xl, ..,Xn) is a 
condition 
(c) if li, lj E L then li : '~lj is a condition 
(d) if li, lj, Ik E L then li : lj ::¢, l~ is a condition 
(e) if l, ll,...,ln E L then l: V(ll,...,ln) is a 
condition 
2. if li, Ij E L then li < lj is a condition where _< is 
a partial ordering defining an upper semi-lattice 
with a top element. 
UDRSs are pairs of a set of type 2 conditions with 
a set of type 1 conditions: 
• A UDRS /C is a pair (L,C) where L = (i,<) 
is an upper semi-lattice of labels and C a set of 
conditions of type 1 above such that if li : ~lj E 
1The definition abstracts away from some of the com- 
plexities in the full definitions of the UDRS language 
(Reyle, 1993). The full language also contains type 1 
conditions of the form 1 : a(ll,...,ln) indicating that 
(/1,..., In) are contributed by a single sentence etc. 
Cthenlj :< li E £ and ifli : lj ~ lk E C then 
lj < li,lk < li E £.2 
The construction of UDRSs, in particular the speci- 
fication of the partial ordering between labeled con- 
ditions in £, is constrained by a set of meta-level 
constraints (principles). They ensure, e.g., that 
verbs are subordinated with respect to their scope 
inducing arguments, that scope sensitive elements 
obey the restrictions postulated by whatever syn- 
tactic theory is adopted, that potential antecedents 
are scoped with respect to their anaphoric potential 
etc. Below we list the basic cases: 
• Clause Boundedness: the scope of genuinely 
quantificational structures is clause bounded. 
If lq and let are the labels associated with the 
quantificational structure and the containing 
clause, respectively, then the constraint lq < let 
enforces clause boundedness. 
• Scope of Indefinites: indefinites labeled li may 
take arbitrarily wide scope in the representa- 
tion. They cannot exceed the top-level DRS IT, 
i.e. li < IT. 
• Proper Names: proper names, 7r, always end 
up in the top-level DRS, IT. This is specified 
lexically by IT : r 
The semantics is defined in terms of disambiguations 
& It takes its cue from the definition of the conse- 
quence relation; in the most recent version (Reyle, 
1995) with correlated disambiguations 8t 
V61( r~, D M') 
resulting in a conjunctive interpretation of a goal 
UDRS. 3 In contrast to other proof systems the 
UDRS proof systems (Reyle, 1993; Reyle, 1995; 
Kbnig and Reyle, 1996) operate directly on under- 
specified representations avoiding (whenever possi- 
ble) the need to consider disambiguated cases. 4 
3 A language of well-formed 
f-structures 
The language of wff-s (well-formed f-structures) is 
defined below. The basic vocabulary consists of five 
disjoint sets: GFs (subcategorizable grammatical 
functions), GF,~ (non-subcategorizable grammatical 
functions), SF (semantic forms), ATR (attributes) 
and ATOM (atomic values): 
2This closes Z: under the subordination relations in- 
duced by complex conditions of the form -~K and Ki =~ Kj. 
38 is an o~eration mapping a into one of its disam- 
biguations c~ . The original semantics in (Reyle, 1993) 
took its cue from V~i3/ij(F 6i ~ v~ 6j) resulting in a dis- 
junctive semantics. 
4 Soundness and completeness results are given for the 
system in (Reyle, 1993). 
403 
• CFs = {SUB J, OBJ, COMP, XCOMP,...} 
• GFn -~ {ADJUNCTS,RELMODS,...} 
• SF = {coach(}, support(* SUB J, 1" OUJ},...} 
• ATR "~ {SPEC,NUM,PER, GEN...} 
• ATOM = {a, some, every, most,..., SG, PL, . . .} 
The formation rules pivot on the semantic form 
PRED values. 
* if\[10 E SF then \[PRED lI 0 \]~ e wff-s 
• if ~o1~,...,~o,,\[\] e wff-s and H{T F1,...,* 
rn} e SF then ~ e wff-s where ~ is of the 
form 
PRgD \[1(* I~1,...,1" FN) ~\] ~ ~ff-8 
r. 
where for any two substructures ¢~\] and ¢r~1 
occurring in ~d~\], 1 :~ m except possibly where ¢-¢.s 
• if a E ATR, v E ATOM, ~o E wff-s where 
~\]isoftheform \[PRED.,. II(...)\]~\]andc~ 
dom(~\]) then 
ED n(...) ~1 e wl/-s 
The side condition in the second clause ensures 
that only identical substructures can have identi- 
cal tags. Tags are used to represent reentrancies 
and will often appear vacuously. The definition cap- 
tures f-structures that are complete, coherent and 
consistent.6 
4 An f-structure - UDRS return trip 
In order to illustrate the basic idea we will first give 
a simplified graphical definition of the translation r 
from f-structures to UDRSs. The full textual defini- 
tions are given in the appendix• The (U)DRT con- 
struction principles distinguish between genuinely 
SWhere - denotes syntactic identity modulo permu- 
tation of attribute-value pairs. 
6Proof: simple induction on the formation rules for 
wff-s using the definitions of completeness, coherence and 
consistency (Kaplan and Bresnan, 1982). Because of lack 
of space here we can not consider non-subcategorizable 
grammatical functions. For a treatment of those in 
a QLF-style interpretation see (Genabith and Crouch, 
1996). The notions of substructure occurring in an .f- 
structure and dom(~o) can easily be spelled out formally. 
The definition given above uses textual representations 
of f-structures. It can easily be recast in terms of hier- 
archical sets, finite functions, directed graphs etc. 
quantificational NPs and indefinite NPs. 7 Accord- 
ingly we have 
F2 ~o2 
,.. • r(lPXED II<Trl,...,TFN) ) 
:= 
/Lr. .' ~ 
T1(~01) T2(~2) ..... Tn(~On) 
II(zl, "2,.-., x~) 
\[sP c \]) • r'(LPRED ~II() 
:=~ 
\[SPEC every \] 
• ri(iVRE D H0 ) := 
The formulation of the reverse translation r- 1 from 
UDRSs back into f-structures depends on a map be- 
tween argument positions in UDRS predicates and 
grammatical functions in LFG semantic forms: 
I1( ~1, ~2, ..., ~, ) 
I I I I n( ,rl, tru, ..., ,r~ } 
This is, of course, the province of lexical mapping 
theories (LMTs). For our present purposes it will be 
sufficient to assume a lexically specified mapping. 
• r-l( re1 g2 To. ):= 
n(zl, x2,..., x~) I 
rl r-1(~1) 
r2 r-1 (7¢2) 
n{r rl,T r2,...,, rN) 
• := LPRE D 110 
• := 
sPzc every \] 
PRED no J 
7Proper names are dealt with in the full definitions 
in the appendix. 
404 
I coach( x\[~\]) ~ yer(y~) 
Figure 1: The UDRS rT-(~l) =/C~ 
If the lexical map between argument positions in 
UDRS predicates and grammatical functions in LFG 
semantic forms is a function it can be shown that for 
all ~ E wff-s: 
~-l(r(~)) = 
Proof is by induction on the complexity of ~. This 
establishes a one-to-one correspondence between 
subsets of the UDRS and LFG formalism. Note that 
7" -1 is a partial function on UDRS representations. 
The reason is that in addition to full underspecifica- 
tion UDRT allows partial underspecification of scope 
for which there is no correlate in the original LFG 
f-structure formalism. 
5 Correctness of the Translation 
A correctness criterion for the translation can be de- 
fined in terms of preservation of truth with respect 
to an independent semantics. Here we show correct- 
ness with respect to the linear logic (a)s based LFG 
semantics of (Dalrymple et al., 1996): 
\[r(~)\] --- \[~(~)\] 
Correctness is with respect to (sets of) disambigua- 
tions and truthfl 
{ulu = 6(r(~))} - {ll~(~ ) ~, l} 
where 6 is the UDRS disambiguation and b'u the lin- 
ear logic consequence relation. Without going into 
details/f works by adding subordination constraints 
turning partial into total orders. In the absence of 
scope constraints l° for a UDRS with n quantifica- 
tional structures Q (that is including indefinites) this 
results in n! scope readings, as required. Linear logic 
deductions F-u produce scopings in terms of the order 
SThe notation a(~a) is in analogy with the LFG a - 
projection and here refers to the set of linear logic mean- 
ing constructors associated with 99. 
9This is because the original semantics in (Dalrymple 
et al., 1996) is neither underspecified nor dynamic. See 
e.g. (Genabith and Crouch, 1997) for a dynamic and 
underspecified version of a linear logic based semantics. 
Z°Here we need to drop the clause boundedness 
constraint. 
in which premises are consumed in a proof. Again, 
in the absence of scope constraints this results in 
n! scopings for n quantifiers Q. Everything else be- 
ing equal, this establishes correctness with respect 
to sets of disambiguations. 
6 A Worked Example 
We illustrate our approach in terms of a simple ex- 
ample inference. The translations below are ob- 
tained with the full definitions in the appendix. 
\[~ Every coach supported a player. 
Smith is a coach. 
Smith supported a player. 
Premise ~ is ambiguous between an wide scope and 
a narrow scope reading of the indefinite NP. From \[-fl 
and \[\] we can conclude Ii\] which is not ambiguous. 
Assume that the following (simplified) f-structures 
!a\[~\], ¢\[\] and ~\[i\] are associated with \[-fl, \[\] and \[if, 
respectively: 
\[ \[PRED tCOACH'\] suBJ 
LsPEc EVERY j\[\] 
'SUPPORT (~" \['f\] J PRED SUBJ,T OBJ)' 
L TM L sPEc \[PRED 'PLAYER' \] A \[~ 
SUBJ \[PRED 'SMITH'\]~\] \] 
PRED 'COACH (~ SUB J)' \] \[\] 
SUBJ 
PRED 
OBJ 
We have that 
'SUPPORT (r SUS.J,I" OS.O' / 
\[PRED 'PLAYER' \] | \[\]'\] \[SPEO A \]\[\] J 
({t~: z®, v~® %~,%: ~\],z~ : ~oa~h(~), 
t~ : ~G\] ' l~ : pt~,~,e,( ~m ), Zmo : s,,pport( ~® , ~)}, 
405 
the graphical representation of which is given in Fig- 
ure 1 (on the previous page). For (N\] we get 
= 
({IT : z~\],lr :smith(z~),l\[-g\]o: coach(xM} , {lNo < Iv}) I 1} 
_~ smith(z~) = IC\[~\] 
$ I co ch( M) l 
In the calculus of (Reyle, 1995) we obtain the UDRS 
K:Ii I associated with the conclusion in terms of an 
application of the rule of detachment (DET): 
l' : support(x~, x~\])}, {l~\]. < IT, l~\] ° < l~\] l~ < IT }) 
smith( x~ ) 
p uer(@ 
$ 
l 
F SUBJ PRED 
7"T( L TM 
\[PRED 'S IT.' \] \] 
'SUPPORT (\[ SUB J,'\[ OBJ)' / 
\[PRED 'PLAYER' "1 | \[SPEC A \]\['ffl J M) 
which turns out to be the translation image under r 
of the f-structure ~\[i\] associated with the conclusion 
~.la Summarizing we have that indeed: 
rr ( lil) 
which given that 7- is correct does come as too much 
of a surprise. The possibility of defining deduction 
rules directly on f-structures is discussed in (KSnig 
and Reyle, 1996). 
l XNote that the conclusion UDRS K;\[I l can be "col- 
lapsed" into the fully specified DRS 
zy smith(z) 
player(y) support(x, y) 
7 Conclusion and Further Work 
In the present paper we have interpreted f-structures 
as UDRSs and illustrated with a simple example how 
the deductive mechanisms of UDRT can be exploited 
in the interpretation. (KSnig and Reyle, 1996) 
amongst other things further explores this issue and 
proposes direct deduction on LFG f-structures. We 
have formulated a reverse translation from UDRSs 
back into f-structures and established a one-to-one 
correspondence between subsets of the LFG and 
UDRT formalisms. As it stands, however, the level 
of f-structure representation does not express the 
full range of subordination constraints available in 
UDRT. In this paper we have covered the most basic 
parts, the easy bits. The method has to be extended 
to a more extensive fragment to prove (or disprove) 
its mettle. The UDRT and QLF (Genabith and 
Crouch, 1996) interpretations of f-structures invite 
comparison of the two semantic formalisms. With- 
out being able to go into any great detail, QLF 
and UDRT both provide underspecified semantics 
for ambiguous representations A in terms of sets 
{col, ..., COn } of fully disambiguated representations 
COi which can be obtained from A. For a simple core 
fragment (disregarding dynamic effects, wrinkles of 
the UDRS and QLF disambiguation operations/)~ 
and 79q etc.) everything else being equal, for a given 
sentence S with associated QLF and UDRS repre- 
sentations Aq and A~, respectively, we have that 
Dq(Aq) = {COl,..., q CO~} and "D~,(Au) = {CO?,..., CO,I} 
and pairwise \[CO/q \] = \[\[CO u\] for 1 < i < n and 
col 6 ~)q(Aq) and COl' e 7)~(A=). That is-the QLF 
and UDRT semantics coincide with respect to truth 
conditions Of representations in corresponding sets 
of disambiguations. This said, however, they differ 
with respect to the semantics assigned to the un- 
derspecified representations Aq and An. \[\[Aq~ is de- 
fined in terms of a supervaluation construction over 
{CO q .... , CO q} (Alshawi and Crouch, 1992) resulting 
in the three-valued: 
\[Aq\] = 1 ifffor all co~ E ~)q(Aq), \[COq\] ~. 1 
\[Aq\]\] 0 ifffor no COl E :Dq(Aq), \[COl\] = 1 
\[Aq\] = undefined otherwise 
The UDRT semantics is defined classically and takes 
its cue from the definition of the semantic conse- 
quence relation for UDRS. In (Reyle, 1995): 
+' A +') 
(where IE e+ =COi E :D,,(\]E)) which implies that a goal 
UDRS is interpreted conjunctively: 
\[A~,~ 95 = 1 ifffor all CO u E 7:)~,(A~,), \[COr~ 9s = 1 
\[Au\]gs = 0 otherwise 
while the definition in (Reyle, 1993): 
+' A 
results in a disjunctive interpretation: 
406 
\[A.\] 93 = 1 ifffor some O}' E V.(A,~), \[0~\]93 = 1 
\[Au\]\]93 = 0 otherwise 
It is easy to see that the UDRS semantics \[o~\] 95 and 
\[\[od\] 93 each cover the two opposite ends of the QLF 
semantics \[\[%\]\]: \[o=\] 95 covers definite truth while 
\[\[Ou\] 93 covers definite falsity. 
On a final note, the remarkable correspondence be- 
tween LFG f-structure and UDRT and QLF repre- 
sentations (the latter two arguably being the ma- 
jor recent underspecified semantic representation 
formalisms) provides further independent motiva- 
tion for a level of representation similar to LFG f- 
structure which antedates its underspecified seman- 
tic cousins by more than a decade. 
8 Appendix 
We now define a translation r from f-structures to 
UDRSs. The (U)DRT construction principles distin- 
guish between genuinely quantificational NPs, indef- 
inite NPs and proper names. Accordingly we have 
• ~(\[pRED n(t rl,...,t r~) \[i\]):= 
/-'" 
kr. ~.\[\] 
uYmo: n(N2,..., %\])} where 
{ x\[~\] iff FiE{SUBJ,OBJ,...} 
7~\] := l\[~\]o iff ri E {COMP, XCOMP} 
* T.\[~(\[SPEC EVERY \] ffRrD nO m) := 
: 'm,Wmtm ,/ml : : -< l\[3\], l~o ~- lm2} 
\[3"\], \[SPEC A \] 
" r=t/PREDL HO J \]\]\]) := 
: tm z z tin) 
. T~\](\[PRED l-I 0 \]~) := 
{tT : xm,tT : n(xm),lmo _< l~} 
The first clause defines the recursive part of the 
translation function and states that the translation 
of an f-structure is simply the union of the trans- 
lations of its component parts. The base cases of 
the definition are provided by the three remaining 
clauses. They correspond directly to the construc- 
tion principles discussed in section 2. The first one 
deals with genuinely quantificational NPs, the sec- 
ond one with indefinites and the third one with 
proper names. Note that the definitions ensure 
clause boundedness of quantificational NPs {l\[/\] < 
l\[\] } , allow indefinites to take arbitrary wide scope 
{1\[\]\] <_ h-} and assign proper names to the top level 
of the resulting UDRS {iv : z~,/v : H(zffj)} as re- 
quired. The indices are our book-keeping devices for 
label and variable management. F-structure reen- 
trancies are handled correctly without further stipu- 
lation. Atomic attribute-value pairs can be included 
as unary definite relations. 
For the reverse mapping assume a consistent UDRS 
labeling (e.g. as provided by the v mapping) and 
a lexically specified mapping between subcategoriz- 
able grammatical functions in LFG semantic form 
and argument positions in the corresponding UDRT 
predicates: 
II( gel, ~g2, .'', Xn ) 
I I I I n( Try, Tr2, ..., tr, ) 
The scaffolding which allows us to ire)construct a 
f-structure from a UDRS is provided by UDRS sub- 
ordination constraints and variables occurring in 
UDRS conditions) 2 The translation recurses on 
the semantic contributions of verbs. To translate 
a UDRS ~ = (£:,C) merge the structural with the 
content constraints into the equivalent ~t = E U C. 
Define a function 0 ("dependents") on referents, la- 
bels and merged UDRSs as in Figure 2. 0 is 
constrained to O(qi, IV.) C \]C. Given a discourse 
referent x and a UDRS, 0 picks out components 
of the UDRS corresponding to proper names, in- 
definite and genuinely quantificational NPs with x 
as implicit argument. Given a label l, 0 picks 
out the transitive closure over sentential comple- 
ments and their dependents. Note that for sim- 
ple, non-recursive UDRSs \]C, 0 defines a partition 
{{/: II(xl,...,xn)},O(xi,~),..., O(~cn,~)} of/(;. 
s ifIg = {/~o : 1-I(~1,... ,~,)}t~7~ then r-l(\]C) := 
PREp n(t F1,...,T FN) IN\] 
SPEC EVERY \] 
PRED II 0 \[\] 
12The definition below ignores subordination con- 
straints. It assumes proper UDRSs, i.e. UDRS where 
all the discourse referents are properly bound. Thus the 
definition implements the "garbage in - garbage out" 
principle. It also assumes that discourse referents in 
"quantifier prefixes" are disjoint. It is straightforward 
to extend the definition to take account of subordina- 
t~ion constraints if that is desired but, as we remarked 
above, the translation image (the resulting f-structures) 
cannot in all cases reflect the constraints. 
407 
{la, : Th,la, : II(rh)} U {.~ < l¢,,l()~ < la,) E E} if T/i e Ref 
O(o~,/~):= {l,~, l,~.Voil~,,~,l,~,, :~?,,1,~. :II(o~},U{A<_I,~,~I(A<I,~,~)E~} if rliE Ref 
{l,, I\]('y~,...,7,~)}OD(7~,K.),...,D(%,If. ) if ~EL 
Figure 2: The "dependents" function 0 (where 0(~i, K:) C_/C). 
. T-a({/. :x,l~ :n(x)}~Sub):= sPEc A \] 
PRED I-i() \[\] 
° T-I({IT : X, IT : II(x)}~S~b):= \[PREp n0 \]\[\] 
Note that r -1 is a partial function from UDRSs to 
f-structures. The reason is that that f-structures do 
not represent partial subordination constraints, in 
other words they are fully underspecified. Finally, 
note that r and r -1 are recursive (they allow for ar- 
bitrary embeddings of e.g. sentential complements). 
This may lead to structures outside the first-order 
UDRT-fragment. As an example the reader may 
want to check the translation in Figure 3 and fur- 
thermore verify that the reverse translation does in- 
deed take us back to the original (modulo renaming 
of variables and labels) UDRS. 
9 Acknowledgements 
Early versions of this have been presented at Fra- 
CaS workshops (Cooper et al., 1996) and at \]MS, 
Stuttgart in 1995 and at the LFG96 in Grenoble. 
We thank our FraCaS colleagues and Anette Frank 
and Mary Dalrymple for discussion and support. 

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