Efficient Linear Logic Meaning Assembly 
Vineet Gupta 
Caelum Research Corporation 
NASA Ames Research Center 
Moffett Field CA 94035 
vgupt a@pt olemy, arc. nasa. gov 
John Lamping 
Xerox PARC 
3333 Coyote Hill Road 
Palo Alto CA 94304 USA 
i amping©parc, xerox, tom 
1 Introduction 
The "glue" approach to semantic composition 
in Lexical-Functional Grammar uses linear logic 
to assemble meanings from syntactic analyses 
(Dalrymple et al., 1993). It has been compu- 
rationally feasible in practice (Dalrymple et al., 
1997b). Yet deduction in linear logic is known 
to be intractable. Even the propositional ten- 
sor fragment is NP complete(Kanovich, 1992). 
In this paper, we investigate what has made 
the glue approach computationally feasible and 
show how to exploit that to efficiently deduce 
underspecified representations. 
In the next section, we identify a restricted 
pattern of use of linear logic in the glue analyses 
we are aware of, including those in (Crouch and 
Genabith, 1997; Dalrymple et al., 1996; Dal- 
rymple et al., 1995). And we show why that 
fragment is computationally feasible. In other 
words, while the glue approach could be used 
to express computationally intractable analyses, 
actual analyses have adhered to a pattern of use 
of linear logic that is tractable. 
The rest of the paper shows how this pat- 
tern of use can be exploited to efficiently cap- 
ture all possible deductions. We present a con- 
servative extension of linear logic that allows a 
reformulation of the semantic contributions to 
better exploit this pattern, almost turning them 
into Horn clauses. We present a deduction algo- 
rithm for this formulation that yields a compact 
description of the possible deductions. And fi- 
nally, we show how that description of deduc- 
tions can be turned into a compact underspeci- 
fled description of the possible meanings. 
Throughout the paper we will use the illus- 
trative sentence "every gray cat left". It has 
flHlctional structure 
(1) \[PRED 'LEAVE' 1 PRED 'CAT' 
f: SUBJ g: \[SPEC 'EVERY' 
\[MODS {\[ PRED 'GRAY'\]} 
and semantic contributions 
leave :Vx. ga',-*x --.o fo',-*leave(x) cat :w. (ga VAR)--** ~ (ga RESTR)-,~ Cat(*) 
gray NP. \[Vx. (ga VAtt)---* x --o (g~ RESTR)-,-* P(x)\] -o \[w. (g~ VAR)~, 
every :VH, R, S. \[w. (g~ VAR)--~, --o (g~ RESTR)--~R(z)\] 
®\[Vx. g,,'...*x ~ g-,-*S(x)\] 
--~ H',-* every(R, S) 
For our purposes, it is more convenient to fol- 
low (Dalrymple et al., 1997a) and separate the 
two parts of the semantic contributions: use a 
lambda term to capture the meaning formulas, 
and a type to capture the connections to the 
f-structure. In this form, the contributions are 
leave : 
cat : 
gray : 
every : 
Ax.leave(x) : g,, --o fa 
.,~x.cat(x) : (ga VAR) --o (ga RESTR) 
AP.Ax.gray(P)(x) : ((g~ vAR) --o (ga RESTR)) 
--o (g~ VAn) ~ (ga RESTR) 
AR.AS.every(R, S) : 
VH. (((g~ 'CAR) --o (ga RESTR)) ®(g. ~ H)) 
--oH 
With this separation, the possible derivations 
are determined solely by the "types", the con- 
nections to the f-structure. The meaning is as- 
sembled by applying the lambda terms in ac- 
cordance with a proof of a type for the sen- 
tence. We give the formal system behind this 
approach, C in Figure 1 -- this is a different 
presentation of the system given in (Dalrymple 
464 
et al., 1997a), adding the two standard rules 
for tensor, using pairing for meanings. For the 
types, the system merely consists of the Inear 
logic rules for the glue fragment. 
We give the proof for our example in Figure 2, 
where we have written the types only, and have 
omitted the trivial proofs at the top of the tree. 
The meaning every(gray(cat),left) may be as- 
sembled by putting the meanings back in ac- 
cording to the rules of C and r/-reduction. 
M : A ~-c M / : A 
where M --a,n M ~ 
F,P,Q,A~-c R 
F,Q,P, AF-c R 
r, : A\[B/X\] -o R 
F,M :VX.AF-c R r M: A\[Y/X\] (r new) F t-c M : VX.A 
F ~-c N : A A,M\[N/x\] : B F-c R 
F,A, Ax.M : A --o B ~-c R 
r,y : A be M\[y/x\] : B r F-c Ax.M : A .-.o B (y new) 
F,M :A,N : B I-- R 
r, (M, N) :A® B ~- R 
FF-M:A A~-N:B 
F,A~-(M,N):A®B 
Figure 1: The system C. M,N are meanings, 
and x, y are meaning variables. A, B are types, 
and X, Y are type variables. P, Q, R are formu- 
las of the kind M : A. F,A are multisets of 
formulas. 
2 Skeleton references and modifier ref- 
erences 
The terms that describe atomic types, terms Ike 
ga and (g~ vA1Q, are semantic structure refer- 
ences, the type atoms that connect the semantic 
assembly to the syntax. There is a pattern to 
how they occur in glue analyses, which reflects 
their function in the semantics. 
Consider a particular type atom in the ex- 
ample, such as g~. It occurs once positively in 
the contribution of "every" and once negatively 
in the contribution of "leave". A sightly more 
compIcated example, the type (ga l~nSTR) oc- 
curs once positively in the contribution of "cat", 
once negatively in the contribution of "every", 
and once each positively and negatively in the 
contribution of "gray". 
The pattern is that every type atom occurs 
once positively in one contribution, once nega- 
tively in one contribution, and once each posi- 
tively and negatively in zero or more other con- 
tributions. (To make this generaIzation hold, 
we add a negative occurrence or "consumer" of 
fa, the final meaning of the sentence.) This pat- 
tern holds in all the glue analyses we know of, 
with one exception that we will treat shortly. 
We call the independent occurrences the skele- 
ton occurrences, and the occurrences that occur 
paired in a contribution modifier occurrences. 
The pattern reflects the functions of the lex- 
ical entries in LFG. For the type that corre- 
sponds to a particular f-structure, the idea is 
that, the entry corresponding to the head makes 
a positive skeleton contribution, the entry that 
subcategorizes for the f-structure makes a neg- 
ative skeleton contribution, and modifiers on 
the f-structure make both positive and negative 
modifier contributions. 
Here are the contributions for the example 
sentence again, with the occurrences classified. 
Each occurrence is marked positive or negative, 
and the skeleton occurrences are underlined. 
leave : g_Ka- --o fa+ 
cat : (ga VAtt)- --o (ga ttESWtt) + 
gray : ((ga VAn) + --o (ga aESTR)-) 
---o (ga VAn)- --o (ga RESTR) + 
every : VH. (((ga VAR) + --o (ga RESTR.)-) 
®(g_z.~ ~ --~ g-)) 
---o H + 
This pattern explains the empirical tractabil- 
ity of glue inference. In the general case of 
multiplicative Inear logic, there can be complex 
combinatorics in matching up positive and neg- 
ative occurrences of literals, which leads to NP- 
completeness (Kanovich, 1992). But in the glue 
fragment, on the other hand, the only combina- 
torial question is the relative ordering of modi- 
tiers. In the common case, each of those order- 
ings is legal and gives rise to a different mean- 
ing. So the combinatorics of inference tends to 
be proportional to the degree of semantic am- 
biguity. The complexity per possible reading is 
thus roughly tnear in the size of the utterance. 
But, this simple combinatoric structure sug- 
gests a better way to exploit the pattern. 
Rather than have inference explore all the com- 
binatorics of different modifier orders, we can 
get a single underspecitied representation that 
captures all possible orders, without having to 
465 
cat F- (ga VAR) --o (go RESTR) (go VAR) --O (go RESTR) ~ (go" VAR) ---O (ga RESTR) 
cat, ((ga VAR) ---o (ga RESTR)) --o (ga VAR) ---o (ga RESTR) ~ (ga VAR) ---o (ga RESTR) 
gray, cat ~- (ga VAR) --~ (ga RESTR) leave F- ga --o fo 
gray, cat, leave F ((go VAR) --~ (go RESTR)) ~(ga .--o fa) fo ~- fa 
gray, cat,leave, (((ga VAR) --o (ga RESTR)) ® (go. ---o fo)) --o fo ~- fo 
every, gray, cat, leave F- fa 
Figure 2: Proof of "Every gray cat left", omitting the lambda terms 
explore them. 
The idea is to do a preliminary deduction in- 
volving just the skeleton, ignoring the modifier 
occurrences. This will be completely determin- 
istic and linear in the total length of the for- 
mulas. Once we have this skeletal deduction, 
we know that the sentence is well-formed and 
has a meaning, since modifier occurrences es- 
sentially occur as instances of the identity ax- 
iom and do not contribute to the type of the 
sentence. Then the system can determine the 
meaning terms, and describe how the modifiers 
can be attached to get the final meaning term. 
That is the goal of the rest of the paper. 
3 Conversion toward horn clauses 
The first hurdle is that the distinction between 
skeleton and modifier applies to atomic types, 
not to entire contributions. The contribution of 
"every", for example, has skeleton contributions 
for go, (go VAR), and (ga RESTR), but modifier 
contributions for H. Furthermore, the nested 
implication structure allows no nice way to dis- 
entangle the two kinds of occurrences. When a 
deduction interacts with the skeletal go in the 
hypothetical it also brings in the modifier H. 
If the problematic hypothetical could be con- 
verted to Horn clauses, then we could get a bet- 
ter separation of the two types of occurrences. 
We can approximate this by going to an in- 
dexed linear logic, a conservative extension of 
the system of Figure 1, similar to Hepple's sys- 
tem(Hepple, 1996). 
To handle nested implications, we introduce 
the type constructor A{B}, which indicates an 
A whose derivation made use of B. This is sim- 
ilar to Hepple's use of indices, except that we 
indicate dependence on types, rather than on in- 
dices. This is sufficient in our application, since 
each such type has a unique positive skeletal 
occurrence. 
We can eliminate problematic nested impli- 
cations by translating them into this construct, 
in accordance with the following rule: 
For a nested hypothetical at top level that has 
a mix of skeleton and modifier types: 
M : ( A -o B ) -o C 
replace it with 
x:A, M:(B{A}---oC) 
where x is a new variable, and reduce complex 
dependency formulas as follows: 
1. Replace A{B ---o C} with A{C{B}}. 
2. Replace (A --o B){C} with A --o B{C}. 
The semantics of the new type constructors 
is captured by the additional proof rule: 
F,x:AF-M:B 
F,x : A ~- Ax.M : B{A} 
The translation is sound with respect to this 
rule: 
Theorem 1 If F is a set of sentences in the 
unextended system of Figure 1, A is a sentence 
in that system, and F ~ results from F by applying 
the above conversion rules, then F F- A in the 
system of Figure 1 iff F' F- A in the extended 
system. 
The analysis of pronouns present a different 
problem, which we discuss in section 5. For all 
other glue analyses we know of, these conver- 
sions are sufficient to separate items that mix 
interaction and modification into statements of 
466 
the form S, Jr4, or S -o .h4, where S is pure 
skeleton and M is pure modifier. Furthermore, 
.h4 will be of the form A -o A, where A may be 
a formula, not just an atom. In other words, the 
type of the modifier will be an identity axiom. 
The modifier will consume some meaning and 
produce a modified meaning of the same type. 
In our example, the contribution of "every", 
can be transformed by two applications of the 
nested hypothetical rule to 
every :AR.AS.every(R, S) : 
VH. (ga RESTR){(ga VAR)} 
--o H{gq} -o H 
x :(go VAR) 
Y :ga 
Here, the last two sentences are pure skele- 
ton, producing (g~ VAR) and ga, respectively. 
The first is of the form S -o M, consuming 
(ga RESTR), to produce a pure modifier. 
While the rule for nested hypotheticals could 
be generalized to eliminate all nested implica- 
tions, as Hepple does, that is not our goal, be- 
cause that does remove the combinatorial com- 
bination of different modifier orders. We use the 
rule only to segregate skeleton atoms from mod- 
ifier atoms. Since we want modifiers to end up 
looking like the identity axiom, we leave them 
in the A -o A form, even if A contains further 
implications. For example, we would not apply 
the nested hypothetical rule to simplify the en- 
try for gray any further, since it is already in 
the form A ---o A. 
Handling intensional verbs requires a more 
precise definition of skeleton and modifier. The 
type part of an intensional verb contribution 
looks like (VF.(ha -o F) --o F) -o ga -o fa 
(Dalrymple et al., 1996). 
First, we have to deal with the small 
technical problem that the VF gets in the 
way of the nested hypothetical translation 
rule. This is easily resolved by introducing 
a skolem constant, 5', turning the type into 
((h~ -o 5') --o 5') --o g~ --o f~. Now, the 
nested hypothetical rule can be applied to yield 
(ho -o S) and S{5"{h~}} ---o ga --o fa. 
But now we have the interesting question of 
whether the occurrences of the skolem constant, 
S, are skeleton or modifier. If we observe how 5' 
resources get produced and consumed in a de- 
duction involving the intensional verb, we find 
that (ha --o 5') produces an 5', which may be 
modified by quantifiers, and then gets consumed 
by S { S { ha } } ---o ga -o f~. So unlike a modifier, 
which takes an existing resource from the envi- 
ronment and puts it back, the intentional verb 
places the initial resource into the environment, 
allows modifiers to act on it, and then takes it 
out. In other words, the intensional verb is act- 
ing like a combination of a skeleton producer 
and a skeleton consumer. 
So just because an atom occurs twice in a 
contribution doesn't make the contribution a 
modifier. It is a modifier if its atoms must in- 
teract with the outside, rather than with each 
other. Roughly, paired modifier atoms function 
as f -o f, rather than as f ® f±, as do the S 
atoms of intensional verbs. 
Stated precisely: 
Definition 2 Assume two occurrences of the 
same type atom occur in a single contribution. 
Convert the formula to a normal form consist- 
ing of just ®, ~ , and J_ on atoms by converting 
subformulas A -o B to the equivalent A ± :~ B, 
and then using DeMorgan's laws to push all J_ 's 
down to atoms. Now, if the occurrences of the 
same type atom occur with opposite polarity and 
the connective between the two subexpressions in 
which they occur is ~ , then the occurrences are 
modifiers. All other occurrences are skeleton. 
For the glue analyses we are aware of, this def- 
inition identifies exactly one positive and one 
negative skeleton occurrence of each type among 
all the contributions for a sentence. 
4 Efficient deduction of underspecified 
representation 
In the converted form, the skeleton deductions 
can be done independently of the modifier de- 
ductions. Furthermore, the skeleton deductions 
are completely trivial, they require just a lin- 
ear time algorithm: since each type occurs once 
positively and once negatively, the algorithm 
just resolves the matching positive and nega- 
tive skeleton occurrences. The result is several 
deductions starting from the contributions, that 
collectively use all of the contributions. One of 
the deductions produces a meaning for fa, for 
the whole f-structure. The others produce pure 
modifiers -- these are of the form A --o A. For 
467 
Lexical contributions in indexed logic: 
leave : 
cat : 
gray : 
everyx : 
every2 : 
everya : 
Ax.leave(x) : ga --o fc, ax.eat(x): 
VAR) R .STR) 
: VAR) --o R STR)) VAR) --o RESTR) 
AR.AS.every(R, S) : vg. (g~ RnSTR){(g~ 'CAR)} --o g{ga} ---o H 
z VAR) 
Y :g~ 
The following can now be proved using the extended system: 
gray ~- AP.Ax.gray(P)(x) : ((ga VAR) --o (g~ RESTR)) ----O (g~ VAR) --o (ga RESTR) 
every2, cat, every1 ~- AS.every(Ax.eat(x), S): VH. H{ga} --o H 
everya, leave F- leave(y) : fa 
Figure 3: Skeleton deductions for "Every gray cat left". 
the example sentence, the results are shown in 
Figure 3. 
These skeleton deductions provide a compact 
representation of all possible complete proofs. 
Complete proofs can be read off from the skele- 
ton proofs by interpolating the deduced modi- 
tiers into the skeleton deduction. One way to 
think about interpolating the modifiers is in 
terms of proof nets. A modifier is interpolated 
by disconnecting the arcs of the proof net that 
connect the type or types it modifies, and recon- 
necting them through the modifier. Quantifiers, 
which turn into modifiers of type VF.F ---o F, 
can choose which type they modify. 
Not all interpolations of modifiers are le- 
gal. however. For example, a quantifier must 
outscope its noun phrase. The indices of the 
modifier record these limitations. In the case 
of the modifier resulting from "every cat", 
VH.H{ga} ---o H, it records that it must 
outscope "every cat" in the {ga}. The in- 
dices determine a partial order of what modi- 
fiers must outscope other modifiers or skeleton 
terms. 
In this particular example, there is no choice 
about where modifiers will act or what their rel- 
ative order is. In general, however, there will be 
choices, as in the sentence "someone likes every 
cat", analyzed in Figure 4. 
To summarize so far, the skeleton proofs pro- 
vide a compact representation of all possible de- 
ductions. Particular deductions are read off by 
interpolating modifiers into the proofs, subject 
to the constraints. But we are usually more in- 
terested in all possible meanings than in all pos- 
sible deductions. Fortunately, we can extract a 
compact representation of all possible meanings 
from the skeleton proofs. 
We do this by treating the meanings of the 
skeleton deductions as trees, with their arcs an- 
notated with the types that correspond to the 
types of values that flow along the arcs. Just as 
modifiers were interpolated into the proof net 
links, now modifiers are interpolated into the 
links of the meaning trees. Constraints on what 
modifiers must outscope become constraints on 
what tree nodes a modifier must dominate. 
Returning to our original example, the skele- 
ton deductions yield the following three trees: 
!g 
RESTR) / ~/-/Iga~ tga VAR) ---o 
• ,~Z. \] ga RESTR) leave (go 
RESTR)I gray 
cat lga VAR) --o I g~ (go VAR) ~ I tgo' RESTR) 
y 
leave(y) aS.every(;~x.cat(x),S) aP.ax.gray(P)(x) 
Notice that higher order arguments 
are reflected as structured types, like 
(g~ VAR) ----o (g~ RESTR). These trees are 
a compact description of the possible meanings, 
in this case the one possible meaning. We 
believe it will be possible to translate this rep- 
resentation into a UDRS representation(Reyle, 
1993), or other similar representations for 
ambiguous sentences. 
We can also use the trees directly as an un- 
derspecified representation. To read out a par- 
ticular meaning, we just interpolate modifiers 
into the arcs they modify. Dependencies on a 
468 
The functional structure of "Someone likes every cat". 
PRED 
SUBJ /: 
OBJ 
The lexical entries after 
'LIKE' 
h:\[ pRro 'soMroNE'\] 
PRED 'eAT' \] 
g: SPEC ~EVERY' 
conversion to indexed form: 
like : 
cat : 
someonel : 
someone2 : 
everyl : 
every2 : 
everya : 
Ax.Ay.tike(x, y): (ho ® go) -o/o 
Ax.cat(x): (go VAR) -o (ga RESTR) 
z:hv 
AS.some(person, S) : VH. H{ho) --o H 
AR.AS.every(R, S) : vg. (go RESTR){(go VA1Q) --o H{go) --o H 
x : (go VAR) 
Y:go 
From these we can prove: 
someone1, everya, like ~- like(z, y) : fo 
someone2 F- AS.some(person, S) : VH. H{ho} --o H 
every2, cat, every1 b AS.every(cat, S) : VH. H{go} -o H 
Figure 4: Skeleton deductions for "Someone likes every cat" 
modifier's type indicate that a lambda abstrac- 
tion is also needed. So, when "every cat" mod- 
ifies the sentence meaning, its antecedent, in- 
stantiated to fo{go) indicates that it lambda 
abstracts over the variable annotated with go 
and replaces the term annotated fo. So the re- 
sult is: 
Ifo , every 
RESTR.) A Ax. Y. 
(go RESTR)\] \]fo cat leave 
(go VAR)!: /o 
Similarly "gray" can modify this by splicing 
it into the line labeled (go VAR) --o (go RESTR) 
to yield (after y-reduction, and removing labels 
on the arcs). 
Ifo /ver  
gray leave I 
cat 
This gets us the expected meaning 
every(gray(cat), leave). 
In some cases, the link called for by a higher 
order modifier is not directly present in the tree, 
and we need to do A-abstraction to support 
it. Consider the sentence "John read Hamlet 
quickly". We get the following two trees from 
the skeleton deductions: 
re!fd g/ \ho 
John Hamlet 
read(John, Hamlet) 
I go --o fo 
quickly 
Igo-o fo 
AP.Ax.quickly( P )( x ) 
There is no link labeled ga --o fa to be modi- 
fied. The left tree however may be converted by 
A-abstraction to the following tree, which has a 
required link. The @ symbol represents A ap- 
plication of the right subtree to the left. 
I/o 
Ax. John 
I/o 
read gj \ho 
x Hamlet 
Now quickly can be interpolated into the 
link labeled go --o fo to get the desired 
meaning quickly(read(Hamlet), John), after r/- 
reduction. The cases where A-abstraction is re- 
quired can be detected by scanning the modi- 
fiers and noting whether the links to be mod- 
ified are present in the skeleton trees. If not, 
A-abstraction can introduce them into the un- 
469 
derspecified representation. Furthermore, the 
introduction is unavoidable, as the link will be 
present in any final meaning. 
5 Anaphora 
As mentioned earlier, anaphoric pronouns 
present a different challenge to separating skele- 
ton and modifier. Their analysis yields types 
like f~ --o (f~ ® g~) where g~ is skeleton and f~ 
is modifier. We sketch how to separate them. 
We introduce another type constructor (B)A, 
informally indicating that A has not been fully 
used, but is also used to get B. 
This lets us break apart an implication whose 
right hand side is a product in accordance with 
the following rule: 
For an implication that occurs at top level, 
and has a product on the right hand side that 
mixes skeleton and modifier types: 
Ax.(M, N) : A ---o (B ® C) 
replace it with 
Ax.M : (C)A -o B, N : C 
The semantics of this constructor is captured 
by the two rules: 
M1 : AI~...,M,~ : An ~- M : A 
M1 : (B)A1,...,Mn: (B)A,~ t- M: (B)A 
F, M1 : (B)A, M2 :B~-N :C 
F t, M~:A, M~:B~-N':C 
where the primed terms are obtained by 
replacing free x's with what was applied to 
the Ax. in the deduction of (B)A 
With these rules, we get the analogue of The- 
orem 1 for the conversion rule. In doing the 
skeleton deduction we don't worry about the 
(B)A constructor, but we introduce constraints 
on modifier positioning that require that a hy- 
pothetical dependency can't be satisfied by a 
deduction that uses only part of the resource it 
requires. 
6 Acknowledgements 
We would like to thank Mary Dalrymple, John 
Fry, Stephan Kauffmann, and Hadar Shemtov 
for discussions of these ideas and for comments 
on this paper. 

References 
Richard Crouch and Josef van Genabith. 1997. 
How to glue a donkey to an f-structure, or 
porting a dynamic meaning representation 
into LFG's linear logic based glue-language 
semantics. Paper to be presented at the Sec- 
ond International Workshop on Computa- 
tional Semantics, Tilburg, The Netherlands, 
January 1997. 
Mary Dalrymple, John Lamping, and Vijay 
Saraswat. 1993. LFG semantics via con- 
straints. In Proceedings of the Sixth Meeting 
of the European ACL, pages 97-105, Univer- 
sity of Utrecht. European Chapter of the As- 
sociation for Computational Linguistics. 
Mary Dalrymple, John Lamping, Fernando 
C. N. Pereira, and Vijay Saraswat. 1995. Lin- 
ear logic for meaning assembly. In Proceed- 
ings of CLNLP, Edinburgh. 
Mary Dalrymple, John Lamping, Fernando 
C. N. Pereira, and Vijay Saraswat. 1996. In- 
tensional verbs without type-raising or lexical 
ambiguity. In Jerry Seligman and Dag West- 
erst£hl, editors, Logic, Language and Com- 
putation, pages 167-182. CSLI Publications, 
Stanford University. 
Mary Dalrymple, Vineet Gupta, John Lamp- 
ing, and Vijay Saraswat. 1997a. Relating 
resource-based semantics to categorial seman- 
tics. In Proceedings of the Fifth Meeting on 
Mathematics of Language (MOL5), Schloss 
Dagstuhl, Saarbriicken, Germany. 
Mary Dalrymple, John Lamping, Fernando 
C. N. Pereira, and Vijay Saraswat. 1997b. 
Quantifiers, anaphora, and intensionality. 
Journal of Logic, Language, and Information, 
6(3):219-273. 
Mark Hepple. 1996. A compilation-chart 
method for linear categorical deduction. In 
Proceedings of COLING-96, Copenhagen. 
Max I. Kanovich. 1992. Horn programming in 
linear logic is NP-complete. In Seventh An- 
nual IEEE Symposium on Logic in Computer 
Science, pages 200-210, Los Alamitos, Cali- 
fornia. IEEE Computer Society Press. 
Uwe Reyle. 1993. Dealing with ambiguities by 
underspecification: Construction, representa- 
tion, and deduction. Journal of Semantics, 
10:123-179. 
