Connectivity in Bag Generation 
Arturo Trujillo and Simon Berry* 
School of Computer and Mathem~tical Sciences 
The Robert Gordon University, St Andrew Street 
Aberdeen Al/l 1fig 
Scotland 
{iat,cs5s by } (c~_ sct us. vgu.ac, ul< 
Abstract 
'Fhis l)aper presents a pruning tech- 
nique which can bc used to reduce the 
number of paths searched in rule-based 
b~g generators of the type proposed by 
(Poznafiski el; al., 1!)95) and (l'opowMl, 
1995). Pruning the search space in these 
generators is important given the. com- 
putational cost of bag generation. 'rhe 
technique relies on a connectivity con- 
straint between the semantic indices as- 
sociated with each lexical sign in a ba R. 
Testing the algorithm on a range of sen- 
tences shows reductions in the~ genera- 
tion time and the nmnber of edges con- 
stru cl.cd. 
1 Introduction 
Bag generation is a form of natural language gel> 
er;ttion in which the input is ;~ bag (Mso known as 
a inultiset: a set in which rcpe~ted elements are 
significant) of lexicM elements and the output is a 
grammatical sentence or a statistically most prob- 
able permutation with respect to some. bmguage 
model. 
Bag generation has been considered within the 
st~tistieal and rule-based paradigms of computa- 
tional linguistics, and catch has handled this prob- 
lem differently (Chen and Lee, 1994; Whitelock, 
1994; Popowich, 1995; Tn0illo , 1995). This pa- 
per only considers ruh' based approaches to this 
problem. 
Bag generation has received particulm: atten- 
tion in lexicalist approaches to MT, as exempli- 
tied by Shake-and-Bake generation (Beaven, 1992; 
Whitelock, 1994). One can also envisage applica- 
tions of bag generation to generation fi'om mini- 
*Now at S\[1ARP L~tboral, ories of I"mrope, Ox- 
h)rd Science \[)ark, Oxford OX4 4CA. E-ma~il: 
simon~sh~Lrp.co, nk 
tmdly recursiw', semantic ropresentactions (Cope.s- 
take ct al., 1995) and other semantic fi'ameworks 
which separate scoping fi'om content information 
(l{eyle, 1995). ht these frameworks, the unordered 
natllFe ()f predicate or relation sets makes the aI> 
plict~tion o\[' bag generation techniques attra.ctiw:. 
A notational convention used in the I)al)er is 
that items such as 'dogt' stand for simplitied lex- 
ical signs of the. form (Shieber, 198(0: 
SI'M -- \[ ttl/I,N = dog \] .... 
\[ attc:~l = 1 J 
In such signs, the semantic argument will be re- 
ferred to as an qndex' and will be shown as n 
subscril)t to a lexeme; in the above exmnple, the 
index has been giwm the unique type 1. 
The term index is borrowed rl'Olll IIPSG (Pol- 
lard and Sag, 1994) where indices ~u'e used as ar- 
guments to relations; however these indices tnay 
also be equated with dis(-onrse referents in l)lt:I' 
(Kamp and I{eyle, 1993). As with most lexical- 
ist generators, semantic variables ttttlSl; \[)c distin- 
guished in order to disallow tr;mslationally incor- 
rect permutations of the target bag. We distin- 
guish variables by uniquely typing them. 
Two assumptions are made regarding \[cxieal- 
semantic indexing. 
Assmnption 1 All lea'teal signs must be indexed, 
including fltnetional and nonprcdicative elements 
(Calder cl al., 1989). 
Assumption 2 All le.~ical signs must be con- 
necled to each other. 7'wo lea:ical signs arc con- 
nected if they are directly connected; furthermore, 
the connectivity rclation is h'an.silivc. 
Definition 1 7'wo signs, A, 11, are directly con- 
nccled if there cxisl at least two paths, PathA, 
Palht3, such that A:PathA is token identical with 
B:PathB. 
The indices involved in determining connec- 
tivity arc; specified as pa.rameters for a pro._ 
ticul;tr formalism, l'k)r exanq)le, in tlPSG, 
!01 
play a major role in preventing the generation of 
incorrect translations. 
\[CAT=S 1 ~ \[oKr=NP l \[O~T=VP 1) 
\[SEM=E~\]J LS~M:A~G1 :=~\] L~:M=IEL.,,~o'~=IIljj 
\[CAT=NP\] \[CAT=Det 1 rcA~'=Na \] 2) L~.=I ~ j ~ Ls,~:.,,,~ ~ =l~l L~E~=@\[,,~<----V1\]\] 
\[CAT=N1\] 3) \[~,_~ j 
\[oa,,=m\] 
\[OAT=Eli 
\[cKr=PP\] 
\[CAT=vPl 
Figure 
CAT= A \[CAT---- N1 
SEM:ARG1 :~\] \] 
CAT= N1 \[OAT = PP 
SI,'M: ARG1 ~\[~\]\] \] 
\[ CAT = N 1 
\[ OA'I'= P \[SEM__@~^lm3~\]\[ --=~J\] \[CAT=NP \] 
CAT= Vtra rCAT=N p \] 
1: Simple unification grammar. 
It will be shown that it is possible to exploit 
the connectivity Assumption 2 above in order to 
achieve a reduction in the number of redundant 
wfss constructed by both types of generator de- 
scribed in section 2. 
3.1 Using Connectivity for Pruning 
Take the following bag: 
Ex. 2 {dogl,thcl,brown:,big:} 
(corresponding to 'the big brown dog'). Assume 
that the next wfss to be constructed by the gen- 
erator is the NP 'the dog'. Given the grammar 
in Figure 1, it is possible to deduce that 'brown' 
can never be part of a complete NP constructed 
from such a substring. This can be determined 
as follows. If this adjective were part of such a 
sentence, 'brown' would have to appear as a leaf 
in some constituent that combines with 'the dog' 
or with a constituent containing 'the dog'. From 
the grammar, the only constituents that can com- 
bine with 'dog' are VP, Vtra and P. However, 
none of these constituents can have 'brownl' as 
a leaf: ill the case of P and Vtra this is trivial, 
since they are both categories of a ditferent lexi- 
cal type. In the case of the VP, 'brownl' cannot 
appear as a leaf either because expansions of the 
VP are restricted to NP complements with 2 as 
their semantic index, which in turn would also re- 
quire adjectives within them to }lave this index. 
l,'urthermore, 'brown1' cannot OCCUr as a loaf in 
a deel)er constituent in the VP t)ecause such an 
occurrence would be associated with a different 
index. In such cases 'brown' would modify a dif- 
ferent noun with a different index: 
Ex. a { the\], dog\] , withl ,2 , the~ , lnvwn2 , collar2} 
A naive implementation of this deduction would 
attempt to expand the VP depth-ill'st, left to 
right, ill order to accommodate 'brown' in a com- 
plete derivation. Since this would not be possible, 
the NP 'the dog' would be discarded. This ap- 
proach is grossly inefficient however. What is re- 
quired is a more tractable algorithm which, given 
a wfss and its associated sign, will be able to deter- 
mine whether all remaining lexical elements can 
ever form part of a complete sentence which in- 
cludes that wfss. 
Note that deciding whether a lexical sign can 
appear outside a phrase is determined purely by 
the grammar, and not by whether the lexical ele- 
ments share the same index or not. Thus, a more 
complex grammar would allow 'the man' from the 
bag 
Ex. 4 {thel,manl,shaves<l,\],himselfl} 
even though 'himself' has the same index as 'the 
IIlan'. 
3.2 Outer Domains 
The approach introduced here compiles the rel- 
evant information of\[line fi'om the grammar and 
uses it to check for connectivity during bag gener- 
ation. The compilation process results in a set of 
(Sign,Lex,Bindings) triples called outer domains. 
'l'his set is based on a unification-based phrase 
structure grammar defined as follows: 
Definition 2 d grammar is a tuple (N, 7;P,S), 
where P is a sct of productions ce ~ /3, a is a 
sign, /3 is a list of signs, N is the set of all ee, T 
is the set of all signs appearing as elements of \[3 
which unify with lexical entries, and S is the start 
sign. 
Outer domains are defined as follow: 
Definition 3 {(Sign, Lcx, Binds) I Sign C N tO 
T, Lcx ~ T and there exists a derivation 
Oe ~ /31Signt /32 LeJ /33 or a ~ f11Lez\] /32,S'iqnl /33, 
and Sign' a unifier for Sign, Lez j a unifier 
for Lcx, and Binds the set of all path pairs 
<SignPath, LexPalh> such thai Sign':SignPath is 
Ioken identical with LezS :LexPath} 
Intuitively, the outer domains indicate that 
preterminal category Lex ('an appear in a com- 
plete sentence with subconstituent Sign, such that 
l,cx is not a leaf of Sign. Using ideas from data 
flow analysis (Kennedy, 1981), predictive parser 
constructions (Aho et al., 1986) and feature gram- 
mar compilation (Trujillo, 1994) it is possible to 
construct such a set of triples. Outer domains 
thus represent elements whi(:h may lie outside a 
subtree of category Sign in a complete sentential 
103 
they would be indicated through paths such as 
SYNSEM :LOCAL:CONTI,INT:INI) EX. 
'\[b ensure that only connected lexical signs are 
generated and analysed, the following assumt)tion 
must also be made: 
Assumption 3 A grammar will only generate or 
analyse connected lexical signs. 
2 Bag Generation Algorithms 
Two main tyl)es of rule-based bag generators have 
been proposed. The first type consists of a parser 
suitably relaxed to take into account the un- 
ordered character of tile input (Whitelock, 1994; 
Popowich, 1995; Trujillo, 1995). For example, in 
generators based on a chart 1)arser, the hm(tanmn- 
tal rule is applie(1 only when the edges to be ('om- 
bined share no \]exical leaves, in contrast to re- 
quiring that the two edges have source and target 
nodes in common. The other type of generator ap- 
plies a greedy algorithm to an initial solution in 
order Co find a grammatical sentence (1)oznafiski 
et al., 1.995). 
2.1 Redundancy in Bag Generation 
One disadvantage with the above generators is 
that they construct a nnnd)er of strnctures which 
need Dot have been computed at all. In buihl-- 
ing these structures, the generator is e\[fcctively 
searching branches of the search space which never 
lead to a COml)lete sentence. Consider the the tbl- 
lowing input bag: 
{ dog, barked, the, brown, big} 
Previous rest,archers (Ih:ew, 1992; l)hillil)s, 1993) 
have noted that from such a lx~g, tile following 
strings ;u:e generated but none can fi)rtn part of 
a (;omplete sentence (note that indices are omit- 
ted when there is no possibility of conrnsion; # 
indicates that the subs|ring will never be part of 
~ complete sentence): 
Ex. 1 # the dog 
the dog barked 
# the brown dog 
For simph'~ cases in chart based generators such 
unnecessary strings do not create many problems, 
but for k)nger sentences, each additional su bstring 
implies a further branclt in the search tree to be 
considered. 
Since tile (;Oml)Utational ('Oml)lexity of the 
greedy bag generator (Poznafiski (% al., 1995) is 
polynolni&l (i.e. O(?,.d)), the cf\]'ect of ,'(~(hlnda,lt 
sul)structnres is not as detrimentM as for parser 
based generators. Neverthelc'ss, a (:ert~in am(rant 
of mmccesm~ry work is t)erformed. 'lk) show this, 
consider the test-rewrite sequence for l!'~xaml)h'. I: 
Test: (log barked the brown big 
R.ewrite: __ barked the dog brown big 
Test: barked (the dog) brown big 
Rewrite: __ (the dog) barked brown big 
Test: ((the (log) barked) brown big 
Rewrite: the brown dog barked __ big 
Test: ((the (brown (log)) harked)big 
Rewrite: tile big (brown dog) barked _. 
Test: ((the (big (brown clog))) barked) ('ter- 
minate) 
In this scqnence donble und(,rscorc (__.) indi- 
cates the starting position of a moved constituent; 
the moved constituent itself is given in bold t~ce; 
the bracketing indicates analysed constituents (for 
expository purposes the algorithm has been over- 
simplified, but the general idea remains the salne). 
Now consider the step where 'brown' is inserte(l 
1)etwe(;n '|tie' and 'dog'. This action causes the 
complete structure for 'the dog barked' to be dis- 
carded and replaced with that for %he brown (tog 
barked', which in turn is discarded and replaced 
by 'the big brown dog barked'. 
2.2 Previous Work 
A number of prnning techniqtms have I)een sug- 
gested to re(hwe the mnom,t of redundancy in bag 
generators. Brew (19921 proposed a constraint 
propagation technique which eliminates branches 
during I)ag generation by considering the nec- 
essary lh,~ctor-argument relationships that exist 
between the component basic signs of categorial 
signs. These relationships form a graph indic:Lt- 
ing the necessary conditions for a lexical item to 
form part of a comt/h'.te sentence. Such graphs can 
1)e use(l to elinlinate 1;he substrings in l'3xaml)le 1. 
Unh)rtunately the technique exploits specilic as- 
l)ects of categorial grammars an(l it is not <:lear 
how the.y may he used with other formalisms. 
Trujillo (1995) adapts some of Brew's ideas 
1,o phrase structure grammars by ('emil|ling l!'of 
low functions and constructing adjacency graphs. 
While this al)l)roach reduces the size of the search 
sl)ace , it; does not prune it; sulllciently for cert,|in 
classes of rood|tiers. 
Phillips (199'.{) proposes handling ine\[ticiency ~1; 
the expense of completeness. Ills idea is to main- 
l.a.il~ a queue, of rood|liable constituents (e.g. N Is) 
in order to delay their combination with other 
constituents until rood|tiers (e.g. Pl's) have been 
ana.lysed. While practical, this approach can lead 
to alternative wdid sentences not being gen(;r;(.t(~(I. 
3 Connectivity Restrictions 
In scm('hing ILr a~ nmchanisIn that el i.li~,al.es un 
tteCCssitry WISS, it will I)e l)ossible to use indices in 
lexical signs. As lnenl;ione(\[ earlier, these indices 
derivation. The following definition specifies how 
outer domains are used: 
Definition4 A lexical sign Lea/ is in the 
outer domain of Sign' if\[ there is a triple 
(Sign,Lex, Binds) in outer domains such that Sign 
and Lex unify with Sign I and Lez j respectively, and 
there is at least one pair <PathS, PathL> E Binds 
such that Sign':PathS unifies with LezQPathL. 
In compiling outer domains, inner domains are 
used to facilitate computation. Inner domains are 
defined as follows: 
Definition 5 {(Sign, Lex, Binds) I Sign C N U T, 
Lex 6 7' and there exists a derivation (~ :~ 
/31LezS f12, with Sign I a unifier for Sign, Le~ s a uni- 
fier for Lex, and Binds the set of all path pairs 
<SignPath, LexPath> such that Sign':SignPath is 
token identical with LezS :LexPath} 
The inner domains thus express all the possible 
terminal categories which may be derived from 
each nonterminal in the grammar. 
To be able to exploit connectivity during gen- 
eration, inner and outer domains contain only 
triples in which Binds has at least one element. 
In this way, only those lexical categories which are 
directly connected to the sign are taken into ac- 
count; the implication of this will become clearer 
later. 
As an example, the outer domain of NP as de- 
rived from the above grammar is: 
(N P\[sem:arg l:X\],Vtra\[sem:arg2:Y\], 
{ <sem:argl.,sem:arg2 > } ) (NP\[~em:arg~:X\],Vt~a\[sem :arga:Y\], 
{ <sem:argl,sem:arga > } ) (NP\[~o,n:~rgl:X\],P\[~em:~rga:Y\], 
{ <sem:argl,sem:arg3> }) 
This set indicates that for any NIP, the only ter- 
minal categories not contained in the subtree with 
root NP, and with which the NP shares a seman- 
tic index, are Vtra and P. For instance, the first 
triple arises from the following tree: 
S 
NP/sem:argl:X\] VP\[sem:arg2:X\] 
Vtra\[sem:arg2:X\] NP 
a.a Pruning through Outer Domains and 
Connectivity 
The pruning technique developed here operates 
on grammars whose analyses result in connected 
leaves. 
Consider SOllle wfss W constructed from a bag B 
and with category C; this category, in the form of 
a sign, will include syntactic and lexical-semantic 
information. Such a wfss will have been con- 
structed during the bag generation process. Now, 
either W includes all the input elements as leaves, 
in which case W constitutes a complete sentence, 
or there are elements in the input bag which are 
not part of W. In the latter case, for bags obeying 
Assmnption 2, the following condition holds for 
any W that can form part of a complete sentence: 
Condition 1 Let L be the set of leaves appearing 
in W, let a be the .graph (V, Fd, where V : {C3 
U B- L, and E-- { {x,y} \] x,y 6 Vand y is in 
the outer domain of x}. Then G is connected. 
'lb show that; this condition indeed holds, con- 
sider a grammatical ordering of some input bag 
B, represented as the string W: 
ce.. T&.w 
By Assumption 2, the lexical elements in the bag, 
and therefore in any grammaticM ordering of it, 
are connected. Now consider reducing this string 
using the production rule: 
D~75 
to give the string W': 
O~,. D..o2 
In this case, the signs in W' will also be connected. 
This can be shown by contradiction: 
Proof 1 Assume that there is some sign ~ in W' 
to which D is not connected. Then grammar G 
would allow disconnected strings to be generated, 
contrary to Assumption 3. 7'his is because D 
would not be able to rewrite 7161 in such a way 
that both daughters were connected to ~, leading 
to a disconnected string. 
The situation in string W' is analogous to that 
in Condition 1. By identifying signs which are 
directly connected in E, it is possible to determine 
whether g is connected and consequently whether 
C can form part of a complete derivation, instead 
of simply comparing the value of index paths, it is 
more restrictive to use outer domains since they 
give us precisely those elements which are directly 
connected to a sign and are in its outer domain. 
3.4 Example 
Consider P~xample 2. 'Ib eliminate the wfss 
'the dog' from further consideration, a connected 
graph of lexical signs is constructed before gen- 
eration is started (Figure 2). This graph is built 
by nsing the outer domain of each lexical element 
to decide which of the remaining elements could 
possibly share an index with it in a complete sen- 
tence. 
1@4 
dog1 
big1 
1LhcI 
brown i 
Figure 2: Initial commcted graph. 
When a new wfss is constructed during genera- 
|ion, say by application of the modified fimdame.n- 
tal rule or during the rewrite phase in a greedy al- 
gorithm, this initial graph is updated and tested 
for connectivity. If the updated graph is not con- 
neeted then the proposed wfss cannot form part of 
a complete sentence. Updating the graph involves 
three steps, l"irstly every node in the graph which 
is a leaf' of tit(' new wfss is deleted, toge.t.lmr with 
its associated ares. Secondly, a new node corre- 
sponding to tit(: new wNs is added to the graph. 
Finally, a new arc is added to the graph between 
the uew node and every other node lying in its 
outer domain. The updated (disconnected) graph 
that ensnes after constructing 'the clog' is shown 
in Figure 3; this NP is therefore rejected. 
%|it dog'l 
big1 4 ~ brown1 
Figure 3: Updated disconnected graph after the 
wfss 'the dog' is constructed. 
4 Compiling Connectivity 
Domains 
For reasons of space, the computation of outer do- 
mains cannot be described fully here. The broad 
outline, however, is as follows. First, the inner 
domains of the grammar are calculated. This in- 
volves the calculation of the fixed point of set 
equations, analogous to those used in the con- 
struction of First sets for predictive parsers (Aho 
et al., 1986; Trujillo, 1994). Given the inner do- 
mains of each category in the grammar, the con- 
struction of the outer domains involves the com- 
putation of the lixed point of set equations relat- 
ing the outer domain of a category to the inner 
domain of its sisters and to the outer domain of 
its mother, in a manner analogous to the eoinpu- 
tation of Follow sets. 
I)uring computation, the set of Binds is mono- 
tonically increased as difDreut ways of directly 
connecting sign and lexeme arc found. 
5 Results 
The abow~' pruning tcchnique has been tested on 
bags of different sizes including different combina- 
tkms of modifiers. Sentences were generated using 
two versions of a modified chart parser. In one, 
ew'.ry inactive edge constructed was added to the 
chart. In the. other, every inactive edge was tested 
to see if it led to a disconnected graph; if it did, 
then the edge was discarded. The results of the 
experiment are shown in Table 1. The implemen- 
tation was in Prolog on a Sun SpareS|at|on 10; the 
generation timings do not include garbage collec- 
tion time. The grammar used for the experiment 
consisted of simplified, feature-based versions of 
the 11) rules in GPSG; there were 18 rules and 
50 lexical entries. Compilation of the outer do- 
mains for these rules took apt)roximately 37 min- 
utes, and the resulting set occupies 40K of men> 
ory. In the general case, however, tile size of the 
outer domains is O(n2), where n is the number 
of distinct signs; this number can be controlled 
by employing equivalence classes of different lev- 
els of specificity for pre-terminal and non-terminal 
signs. 
Chart Gen. + Pruning 
Hag size Time Edgcs Time Edges 
2 0.1 15 8.1 15 
4 0.3 37 (1.4 36 
7 1.5 103 2.0 99 
7 0.9 72 11.0 67 
I\] 5.1 213 3.9 138 
12 2.6 133 3.4 123 
15 9.0 294 7.2 186 
15 117.6 448 11.1 253 
17 2.3 126 2.6 1105 
'l'al~le 1: Effect of pruning (times in secs). 
Only one reading was generated for each bag, 
corresponding to one attachment site for PPs. 
'l'he tMJe shows that the technique ctm yieht re- 
ductions in the number of edges (both active aud 
inactive) and time taken, especially for longer sen- 
tences, while retaining the overheads at an accept- 
able level. 
6 Conclusion 
A technique fl)r pruning the search space of a bag 
generator has been implemented and its usefulness 
shown in Lhe geueration of different types of con- 
structions. The technique relies on a connectivity 
constraint imposed on the semantic relationships 
105 
expressed in the input bag. In order to apply the 
algorithm, outer domains needed to be compiled 
from the grammar; these are used to discard wfss 
by ensuring l'exical signs outside a wfss can indeed 
appear outside that string. 
Exploratory work employing adjacency con- 
straints during generation has yielded further im- 
provements in execution time when applied in con- 
junction with the pruner. If extended appropri- 
ately, these constraints could prune the search 
space even further. This work will be reported 
at a later date. 
Acknowledgments 
Two anonymous reviewers provided very useful 
comments; we regret not being able to do justice 
to all their suggestions. 
References 
A. V. Aho, R. Sethi, and J. D. Ullman. 1986. 
Compilers - Principles, Techniques, and Tools. 
Addison Wesley, Reading, MA. 
J. L. Beaven. 1992. Lexicalist Unification Based 
Machine Translation. Ph.D. thesis, Depart- 
ment of Artificial Intelligence, University of Ed- 
inburgh, Edinburgh, UK. 
C. Brew. 1992. Letting the cat out of the bag: 
Generation for Shake-and-Bake MT. In Pro- 
ceedings of the 14th COLING, pages 610-16, 
Nantes, France, August. 
J. Calder, M. Reape, and H. Zeevat. 1989. An 
algorithm for generation in unification catego- 
rial grammar. In Proceedings of the Fourth Eu- 
ropean Conference of the ACL, pages 233-40, 
Manchester, England, April. 
It.H. Chen and Y.S. Lee. 1994. A corrective train- 
ing algorithm for adaptive learning in bag gen- 
eration. In New Methods in Language Process- 
ing, Manchester, UK. 
A. Copestake, D. Flickinger, R. Malouf, S. Riehe- 
mann, and I. Sag. 1995. Translation using 
minimal recursion semantics. In Proceedings of 
the 6th International Conference on Theoretical 
and Methodological Issues in Machine Transla- 
tion, Leuven, Belgium, July. 
II. Kamp and U. Reyle. 1993. From Discourse 
to Logic - Introduction to Modeltheoretic Se- 
mantics of Natural Language, Formal Logic and 
Discourse Representation Theory, volume 42 of 
Studies in Linguistics and Philosophy. Kluwer 
Academic, Dordrecht, The Netherlands. 
Ken Kennedy. 1981. A survey of data flow analy- 
sis techniques. In Muchnick and Jones (1981), 
chapter 1, pages 5-54. 
Steven S. Muchnick and Neil D. Jones, editors. 
1981. Program Flow Analysis: Theory and Ap- 
plications. Software. Prentice-Hall, Englewood 
Cliffs, NJ. 
J. D. Phillips. 1993. Generation of text from log- 
ical formulae. Machine Translation, 8(4):209- 
35. 
C. Pollard and I. Sag. 1994. Head Driven Phrase 
Structure Grammar. Chicago University Press, 
IL. 
Fred Popowich. 1995. Improving the efficiency 
of a generation algorithm for Shake and Bake 
machine translation using Head-Driven Phrase 
Structure Grammar. In Proceedings of Natural 
Language Understanding and Logic Program- 
ming V, Lisbon, Portugal, May. 
V. Poznafiski, J. L. Beaven, and P. Whitelock. 
1995. An efficient generation algorithm for lex- 
iealist MT. In Proceedings of the 33rd Annual 
Meeting of the Association for Computational 
Linguistics, Boston, MA, June. 
Uwe Reyle. 1995. On reasoning with ambigui- 
ties. In Proceedings of the Seventh Conference 
of the European Chapter of the Association for 
Computational Linguistics, pages 1-15, Dublin, 
Ireland, March. 
C. J. Rupp, M. A. Rosner, and R. L. Johnson, ed- 
itors. 1994. Constraints, Language and Com- 
putation. Academic Press, London. 
S. M. Shieber. 1986. An Introduction to 
Unification-based Approaches to Grammar, vol- 
ume 4 of CSLI Lecture Notes. CSLI, Stanford, 
CA. 
A. Trujillo. 1994. Computing FIRST and FOL- 
LOW functions for Feature-Theoretic gram- 
mars. In Proceedings of the 15th COLING, 
pages 875-80, Kyoto, Japan, August. 
A. Trujillo. 1995. Lexicalist Machine Translation 
of Spatial Prepositions. Ph.D. thesis, Computer 
Laboratory, University of Cambridge, April. 
Pete Whitelock. 1994. Shake-and-bake transla- 
tion. In Rupp et al. (1994), pages 339-59. 
106 
