An Efficient Generation Algorithm for Lexicalist MT 
Victor Poznafiski, John L. Beaven &: Pete Whitelock * 
SHARP Laboratories of Europe Ltd. 
Oxford Science Park, Oxford OX4 4GA 
United Kingdom 
{vp ~i lb,pete } @sharp. co.uk 
Abstract 
The lexicalist approach to Machine Trans- 
lation offers significant advantages in 
the development of linguistic descriptions. 
However, the Shake-and-Bake generation 
algorithm of (Whitelock, 1992) is NP- 
complete. We present a polynomial time 
algorithm for lexicalist MT generation pro- 
vided that sufficient information can be 
transferred to ensure more determinism. 
1 Introduction 
Lexicalist approaches to MT, particularly those in- 
corporating the technique of Shake-and-Bake gen- 
eration (Beaven, 1992a; Beaven, 1992b; Whitelock, 
1994), combine the linguistic advantages of transfer 
(Arnold et al., 1988; Allegranza et al., 1991) and 
interlingual (Nirenburg et al., 1992; Dorr, 1993) ap- 
proaches. Unfortunately, the generation algorithms 
described to date have been intractable. In this pa- 
per, we describe an alternative generation compo- 
nent which has polynomial time complexity. 
Shake-and-Bake translation assumes a source 
grammar, a target grammar and a bilingual dictio- 
nary which relates translationally equivalent sets of 
lexical signs, carrying across the semantic dependen- 
cies established by the source language analysis stage 
into the target language generation stage. 
The translation process consists of three phases: 
1. A parsing phase, which outputs a multiset, 
or bag, of source language signs instantiated 
with sufficiently rich linguistic information es- 
tablished by the parse to ensure adequate trans- 
lations. 
2. A lexical-semantic transfer phase which em- 
ploys the bilingual dictionary to map the bag 
*We wish to thank our colleagues Kerima Benkerimi, 
David Elworthy, Peter Gibbins, Inn Johnson, Andrew 
Kay and Antonio Sanfilippo at SLE, and our anonymous 
reviewers for useful feedback and discussions on the re- 
search reported here and on earlier drafts of this paper. 
of instantiated source signs onto a bag of target 
language signs. 
3. A generation phase which imposes an order on 
the bag of target signs which is guaranteed 
grammatical according to the monolingual tar- 
get grammar. This ordering must respect the 
linguistic constraints which have been trans- 
ferred into the target signs. 
The Shake-an&Bake generation algorithm of 
(Whitelock, 1992) combines target language signs 
using the technique known as generate-and-test. In 
effect, an arbitrary permutation of signs is input to a 
shift-reduce parser which tests them for grammatical 
well-formedness. If they are well-formed, the system 
halts indicating success. If not, another permutation 
is tried and the process repeated. The complexity of 
this algorithm is O(n!) because all permutations (n! 
for an input of size n) may have to be explored to 
find the correct answer, and indeed must be explored 
in order to verify that there is no answer. 
Proponents of the Shake-and-Bake approach have 
employed various techniques to improve generation 
efficiency. For example, (Beaven, 1992a) employs 
a chart to avoid recalculating the same combina- 
tions of signs more than once during testing, and 
(Popowich, 1994) proposes a more general technique 
for storing which rule applications have been at- 
tempted; (Brew, 1992) avoids certain pathological 
cases by employing global constraints on the solu- 
tion space; researchers such as (Brown et al., 1990) 
and (Chen and Lee, 1994) provide a system for bag 
generation that is heuristically guided by probabil- 
ities. However, none of these approaches is guar- 
anteed to avoid protracted search times if an exact 
answer is required, because bag generation is NP- 
complete (Brew, 1992). 
Our novel generation algorithm has polynomial 
complexity (O(n4)). The reduction in theoretical 
complexity is achieved by placing constraints on 
the power of the target grammar when operating 
on instantiated signs, and by using a more restric- 
tive data structure than a bag, which we call a 
target language normalised commutative bracketing 
261 
(TNCB). A TNCB records dominance information 
from derivations and is amenable to incremental up- 
dates. This allows us to employ a greedy algorithm 
to refine the structure progressively until either a 
target constituent is found and generation has suc- 
ceeded or no more changes can be made and gener- 
ation has failed. 
In the following sections, we will sketch the basic 
algorithm, consider how to provide it with an initial 
guess, and provide an informal proof of its efficiency. 
2 A Greedy Incremental Generation 
Algorithm 
We begin by describing the fundamentals of a greedy 
incremental generation algorithm. The cruciM data 
structure that it employs is the TNCB. We give some 
definitions, state some key assumptions about suit- 
able TNCBs for generation, and then describe the 
algorithm itself. 
2.1 TNCBs 
We assume a sign-based grammar with binary rules, 
each of which may be used to combine two signs 
by unifying them with the daughter categories and 
returning the mother. Combination is the commuta- 
tive equivalent of rule application; the linear order- 
ing of the daughters that leads to successful rule ap- 
plication determines the orthography of the mother. 
Whitelock's Shake-and-Bake generation algorithm 
attempts to arrange the bag of target signs until 
a grammatical ordering (an ordering which allows 
all of the signs to combine to yield a single sign) is 
found. However, the target derivation information 
itself is not used to assist the algorithm. Even in 
(Beaven, 1992a), the derivation information is used 
simply to cache previous results to avoid exact re- 
computation at a later stage, not to improve on pre- 
vious guesses. The reason why we believe such im- 
provement is possible is that, given adequate infor- 
mation from the previous stages, two target signs 
cannot combine by accident; they must do so be- 
cause the underlying semantics within the signs li- 
censes it. 
If the linguistic data that two signs contain allows 
them to combine, it is because they are providing 
a semantics which might later become more spec- 
ified. For example, consider the bag of signs that 
have been derived through the Shake-and-Bake pro- 
cess which represent the phrase: 
(1) The big brown dog 
Now, since the determiner and adjectives all mod- 
ify the same noun, most grammars will allow us to 
construct the phrases: 
(2) The dog 
(3) The big dog 
(4) The brown dog 
as well as the 'correct' one. Generation will fail if 
all signs in the bag are not eventually incorporated 
in tile final result, but in the naive algorithm, the 
intervening computation may be intractable. 
In the algorithm presented here, we start from ob- 
servation that the phrases (2) to (4) are not incorrect 
semantically; they are simply under-specifications of 
(1). We take advantage of this by recording the 
constituents that have combined within the TNCB, 
which is designed to allow further constituents to be 
incorporated with minimal recomputation. 
A TNCB is composed of a sign, and a history of 
how it was derived from its children. The structure 
is essentially a binary derivation tree whose children 
are unordered. Concretely, it is either NIL, or a 
triple: 
TNCB = NILlValue × TNCB x TNCB 
Value = Sign I 
INCONSISTENT I 
UNDETERMINED 
The second and third items of the TNCB triple 
are the child TNCBs. The value of a TNCB is 
the sign that is formed from the combination of its 
children, or INCONSISTENT, representing the fact 
that they cannot grammatically combine, or UN- 
DETERMINED, i.e. it has not yet been established 
whether the signs combine. 
Undetermined TNCBs are commutative, e.g. they 
do not distinguish between the structures shown in 
Figure 1. 
Figure 1: Equivalent TNCBs 
In section 3 we will see that this property is im- 
portant when starting up the generation process. 
Let us introduce some terminology. 
A TNCB is 
• well-formed iff its value is a sign, 
• ill-formed iff its value is INCONSISTENT, 
• undetermined (and its value is UNDETER- 
MINED) iff it has not been demonstrated 
whether it is well-formed or ill-formed. 
• maximal iff it is well-formed and its parent (if it 
has one) is ill-formed. In other words, a maxi- 
mal TNCB is a largest well-formed component 
of a TNCB. 
262 
Since TNCBs are tree-like structures, if a 
TNCB is undetermined or ill-formed then so are 
all of its ancestors (the TNCBs that contain it). 
We define five operations on a TNCB. The first 
three are used to define the fourth transformation (move) 
which improves ill-formed TNCBs. The fifth 
is used to establish the well-formedness of undeter- 
mined nodes. In the diagrams, we use a cross to 
represent ill-formed nodes and a black circle to rep- 
resent undetermined ones. 
Deletion: A maximal TNCB can be deleted 
from its current position. The structure above 
it must be adjusted in order to maintain binary 
branching. In figure 2, we see that when node 
4 is deleted, so is its parent node 3. The new 
node 6, representing the combination of 2 and 
5, is marked undetermined. 
t* 
5 2 5 
I..---- J 
Figure 2:4 is deleted, raising 5 
Conjunction: A maximal TNCB can be con- 
joined with another maximal TNCB if they may 
be combined by rule. In figure 3, it can be seen 
how the maximal TNCB composed of nodes 1, 
2, and 3 is conjoined with the maximal TNCB 
composed of nodes 4, 5 and 6 giving the TNCB 
made up of nodes 1 to 7. The new node, 7, is 
well-formed. 
1 4 7 
2 3 5 6 2 35 6 
Figure 3:1 is conjoined with 4 giving 7 
Adjunction: A maximal TNCB can be in- 
serted inside a maximal TNCB, i.e. conjoined 
with a non-maximal TNCB, where the combina- 
tion is licensed by rule. In figure 4, the TNCB 
composed of nodes 1, 2, and 3 is inserted in- 
side the TNCB composed of nodes 4, 5 and 6. 
All nodes (only 8 in figure 4) which dominate 
the node corresponding to the new combination 
(node 7) must be marked undetermined -- such 
nodes are said to be disrupted. 
1 
2 3 
4 
8 
5 2 3 6 
Figure 4:1 is adjoined next to 6 inside 4 
Movement: This is a combination of a deletion 
with a subsequent conjunction or adjunction. In 
figure 5, we illustrate a move via conjunction. 
In the left-hand figure, we assume we wish to 
move the maximal TNCB 4 next to the maximal 
TNCB 7. This first involves deleting TNCB 4 
(noting it), and raising node 3 to replace node 
2. We then introduce node 8 above node 7, and 
make both nodes 7 and 4 its children. Note 
that during deletion, we remove a surplus node 
(node 2 in this case) and during conjunction or 
adjunction we introduce a new one (node 8 in 
this case) thus maintaining the same number of 
nodes in the tree. 
9 /L 
3 7 
Figure 5: A conjoining move from 4 to 7 
Evaluation: After a movement, the TNCB 
is undetermined as demonstrated in figure 5. 
The signs of the affected parts must be recal- 
culated by combining the recursively evaluated 
child TNCBs. 
2.2 Suitable Grammars 
The Shake-and-Bake system of (Whitelock, 1992) 
employs a bag generation algorithm because it is as- 
sumed that the input to the generator is no more 
than a collection of instantiated signs. Full-scale bag 
generation is not necessary because sufficient infor- 
mation can be transferred from the source language 
to severely constrain the subsequent search during 
generation. 
The two properties required of TNCBs (and hence 
the target grammars with instantiated lexicM signs) 
are: 
1. Precedence Monotonicity. The order of the 
263 
orthographies of two combining signs in the or- 
thography of the result must be determinate -- 
it must not depend on any subsequent combi- 
nation that the result may undergo. This con- 
straint says that if one constituent fails to com- 
bine with another, no permutation of the ele- 
ments making up either would render the com- 
bination possible. This allows bottom-up eval- 
uation to occur in linear time. In practice, this 
restriction requires that sufficiently rich infor- 
mation be transferred from the previous trans- 
lation stages to ensure that sign combination is 
deterministic. 
2. Dominance Monotonicity. If a maximal 
TNCB is adjoined at the highest possible place 
inside another TNCB, the result will be well- 
formed after it is re-evaluated. Adjunction is 
only attempted if conjunction fails (in fact con- 
junction is merely a special case of adjunction 
in which no nodes are disrupted); an adjunction 
which disrupts i nodes is attempted before one 
which disrupts i + 1 nodes. Dominance mono- 
tonicity merely requires all nodes that are dis- 
rupted under this top-down control regime to 
be well-formed when re-evaluated. We will see 
that this will ensure the termination of the gen- 
eration algorithm within n- 1 steps, where n is 
the number of lexical signs input to the process. 
We are currently investigating the mathematical 
characterisation of grammars and instantiated signs 
that obey these constraints. So far, we have not 
found these restrictions particularly problematic. 
2.3 The Generation Algorithm 
The generator cycles through two phases: a test 
phase and a rewrite phase. Imagine a bag of signs, 
corresponding to "the big brown dog barked", has 
been passed to the generation phase. The first step 
in the generation process is to convert it into some 
arbitrary TNCB structure, say the one in figure 6. 
In order to verify whether this structure is valid, 
we evaluate the TNCB. This is the test phase. If 
the TNCB evaluates successfully, the orthography 
of its value is the desired result. If not, we enter the 
rewrite phase. 
If we were continuing in the spirit of the origi- 
nal Shake-and-Bake generation process, we would 
now form some arbitrary mutation of the TNCB and 
retest, repeating this test-rewrite cycle until we ei- 
ther found a well-formed TNCB or failed. However, 
this would also be intractable due to the undirected- 
ness of the search through the vast number of possi- 
bilities. Given the added derivation information con- 
tained within TNCBs and the properties mentioned 
above, we can direct this search by incrementally 
improving on previously evaluated results. 
We enter the rewrite phase, then, with an ill- 
formed TNCB. Each move operation must improve 
p lg 
Figure 6: An arbitrary right-branching TNCB struc- 
ture 
it. Let us see why this is so. 
The move operation maintains the same number 
of nodes in the tree. The deletion of a maximal 
TNCB removes two ill-formed nodes (figure 2). At 
the deletion site, a new undetermined node is cre- 
ated, which may or may not be ill-formed. At the 
destination site of the movement (whether conjunc- 
tion or adjunction), a new well-formed node is cre- 
ated. 
The ancestors of the new well-formed node will 
be at least as well-formed as they were prior to the 
movement. We can verify this by case: 
1. When two maximal TNCBs are conjoined, 
nodes dominating the new node, which were 
previously ill-formed, become undetermined. 
When re-evaluated, they may remain ill-formed 
or some may now become well-formed. 
2. When we adjoin a maximal TNCB within an- 
other TNCB, nodes dominating the new well- 
formed node are disrupted. By dominance 
monotonicity, all nodes which were disrupted 
by the adjunction must become well-formed af- 
ter re-evaluation. And nodes dominating the 
maximal disrupted node, which were previously 
ill-formed, may become well-formed after re- 
evaluation. 
We thus see that rewriting and re-evaluating must 
improve the TNCB. 
Let us further consider the contrived worst-case 
starting point provided in figure 6. After the test 
phase, we discover that every single interior node is 
ill-formed. We then scan the TNCB, say top-down 
from left to right, looking for a maximal TNCB to 
move. In this case, the first move will be PAST to 
bark, by conjunction (figure 7). 
Once again, the test phase fails to provide a well- 
formed TNCB, so we repeat the rewrite phase, this 
time finding dog to conjoin with the (figure 8 shows 
the state just after the second pass through the test 
phase). 
After further testing, we again re-enter the rewrite 
phase and this time note that brown can be inserted 
in the maximal TNCB the dog barked adjoined with 
dog (figure 9). Note how, after combining dog and 
the, the parent sign reflects the correct orthography 
264 
Figure 7: The initial guess 
L___t/ \ 
PAST bark ~ brown .tg 
Figure 8: The TNCB after "PAST" is moved to 
"bark" 
even though they did not have the correct linear 
precedence. 
PAST bark the = browm 
t-___-J 
big 
Figure 9: The TNCB after "dog" is moved to "the" 
After finding that big may not be conjoined with 
the brown dog, we try to adjoin it within the latter. 
Since it will combine with brown dog, no adjunction 
to a lower TNCB is attempted. 
The final result is the TNCB in figure 11, whose 
orthography is "the big brown dog barked". 
We thus see that during generation, we formed a 
basic constituent, the dog, and incrementally refined 
it by adjoining the modifiers in place. At the heart of 
this approach is that, once well-formed, constituents 
can only grow; they can never be dismantled. 
Even if generation ultimately fails, maximal well- 
formed fragments will have been built; the latter 
may be presented to the user, allowing graceful 
degradation of output quality. 
the b~ 
PAST bXark d'og b~o.n ~he ~'bfg, 
Figure 10: The TNCB after "brown" is moved to 
"dog" 
the big brown dog barked 
PA k he 
Figure 11: The final TNCB after "big" is moved to 
"brown dog" 
3 Initialising the Generator 
Considering the algorithm described above, we note 
that the number of rewrites necessary to repair the 
initial guess is no more than the number of ill-formed 
TNCBs. This can never exceed the number of inte- 
rior nodes of the TNCB formed from n lexical signs 
(i.e. n-2). Consequently, the better formed the ini- 
tial TNCB used by the generator, the fewer the num- 
ber of rewrites required to complete generation. In 
the last section, we deliberately illustrated an initial 
guess which was as bad as possible. In this section, 
we consider a heuristic for producing a motivated 
guess for the initial TNCB. 
Consider the TNCBs in figure 1. If we interpret 
the S, O and V as Subject, Object and Verb we can 
observe an equivalence between the structures with 
the bracketings: (S (V O)), (S (O V)), ((V O) S), 
and ((O V) S). The implication of this equivalence 
is that if, say, we are translating into a (S (V O)) 
language from a head-finM language and have iso- 
morphic dominance structures between the source 
and target parses, then simply mirroring the source 
parse structure in the initial target TNCB will pro- 
vide a correct initiM guess. For example, the English 
sentence (5): 
(5) the book is red 
265 
has a corresponding Japanese equivalent (6): 
(6) ((hon wa) (akai desu)) 
((book TOP) (red is)) 
If we mirror the Japanese bracketing structure in 
English to form the initial TNCB, we obtain: ((book 
the) (red is)). This will produce the correct answer 
in the test phase of generation without the need to 
rewrite at all. 
Even if there is not an exact isomorphism between 
the source and target commutative bracketings, the 
first guess is still reasonable as long as the majority 
of child commutative bracketings in the target lan- 
guage are isomorphic with their equivalents in the 
source language. Consider the French sentence: 
(7) ((le ((grandchien) brun)) aboya) 
(8) ((the ((big dog) brown)) barked) 
The TNCB implied by the bracketing in (8) is 
equivalent to that in figure 10 and requires just one 
rewrite in order to make it well-formed. We thus 
see how the TNCBs can mirror the dominance in- 
formation in the source language parse in order to 
furnish the generator with a good initial guess. On 
the other hand, no matter how the SL and TL struc- 
tures differ, the algorithm will still operate correctly 
with polynomial complexity. Structural transfer can 
be incorporated to improve the efficiency of genera- 
tion, but it is never necessary for correctness or even 
tractability. 
4 The Complexity of the Generator 
The theoretical complexity of the generator is O (n4), 
where n is the size of the input. We give an informal 
argument for this. The complexity of the test phase 
is the number of evaluations that have to be made. 
Each node must be tested no more than twice in the 
worst case (due to precedence monotonicity), as one 
might have to try to combine its children in either 
direction according to the grammar rules. There are 
always exactly n - 1 non-leaf nodes, so the complex- 
ity of the test phase is O(n). The complexity of 
the rewrite phase is that of locating the two TNCBs 
to be combined. In the worst case, we can imagine 
picking an arbitrary child TNCB (O(n)) and then 
trying to find another one with which it combines 
(O(n)). The complexity of this phase is therefore 
the product of the picking and combining complex- 
ities, i.e. O(n2). The combined complexity of the 
test-rewrite cycle is thus O(n3). Now, in section 3, 
we argued that no more than n - 1 rewrites would 
ever be necessary, thus the overall complexity of gen- 
eration (even when no solution is found) is O(n4). 
Average case complexity is dependent on the qual- 
ity of the first guess, how rapidly the TNCB struc- 
ture is actually improved, and to what extent the 
TNCB must be re-evaluated after rewriting. In the 
SLEMaT system (Poznarlski et al., 1993), we have 
tried to form a good initial guess by mirroring the 
source structure in the target TNCB, and allowing 
some local structural modifications in the bilingual 
equivalences. 
Structural transfer operations only affect the ef- 
ficiency and not the functionality of generation. 
Transfer specifications may be incrementally refined 
and empirically tested for efficiency. Since complete 
specification of transfer operations is not required 
for correct generation of grammatical target text, 
the version of Shake-and-Bake translation presented 
here maintains its advantage over traditional trans- 
fer models, in this respect. 
The monotonicity constraints, on the other hand, 
might constitute a dilution of the Shake-and-Bake 
ideal of independent grammars. For instance, prece- 
dence monotonicity requires that the status of a 
clause (strictly, its lexical head) as main or sub- 
ordinate has to be transferred into German. It is 
not that the transfer of information per se compro- 
mises the ideal -- such information must often ap- 
pear in transfer entries to avoid grammatical but 
incorrect translation (e.g. a great man translated 
as un homme grand). The problem is justifying 
the main/subordinate distinction in every language 
that we might wish to translate into German. This 
distinction can be justified monolingually for the 
other languages that we treat (English, French, and 
Japanese). Whether the constraints will ultimately 
require monolingual grammars to be enriched with 
entirely unmotivated features will only become clear 
as translation coverage is extended and new lan- 
guage pairs are added. 
5 Conclusion 
We have presented a polynomial complexity gener- 
ation algorithm which can form part of any Shake- 
and-Bake style MT system with suitable grammars 
and information transfer. The transfer module is 
free to attempt structural transfer in order to pro- 
duce the best possible first guess. We tested a 
TNCB-based generator in the SLEMaT MT sys- 
tem with the pathological cases described in (Brew, 
1992) against Whitelock's original generation algo- 
rithm, and have obtained speed improvements of 
several orders of magnitude. Somewhat more sur- 
prisingly, even for short sentences which were not 
problematic for Whitelock's system, the generation 
component has performed consistently better. 

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