The term experimental must be taken quite seriously here though, if m y view of machine translation were more generally held, it would be redundant.. I believe that all machine translati
Trang 1F U N C T I O N A L U N I F I C A T I O N G R A M M A R :
A F O R M A L I S M F O R M A C H I N E T R A N S L A T I O N
M a r t i n K a y
Xerox Palo Alto Research Center
3333 Coyote Hill Road Palo Alto California 94304 and CSLI, Stanford
A b s t r a c t Functional Unification G r a m m a r provides an opportunity
to e n c o m p a s s within one f o r m a l i s m and c o m p u t a t i o n a l s y s t e m
the p a r t s of m a c h i n e t r a n s l a t i o n s y s t e m s t h a t h a v e u s u a l l y been
treated separately, n a t a b l y analysis, transfer, and s y n t h e s i s
M a n y of t h e a d v a n t a g e s of t h i s f o r m a l i s m come from t h e fact
t h a t it is monotonic allowing d a t a s t r u c t u r e s to grow differently
as different n o n d e t e r m i n i s t i c a l t e r n a t i v e s in a c o m p u t a t i o n a r e
pursued, b u t n e v e r to be modified in a n y way A s t r i k i n g feature
of t h i s s y s t e m is t h a t it is f u n d a m e n t a l reversible, allowing a to
t r a n s l a t e as b only if b could t r a n s l a t e as a
I O v e r v i e w
A M a c h i n e T r a n s l a t i o n
A classical t r a n s l a t i n g m a c h i n e s t a n d s w i t h one foot on t h e
i n p u t text a n d one on the output The i n p u t t e x t is analyzed by
t h e components of t h e m a c h i n e t h a t m a k e u p t h e left leg, each o n e
feeding i n f o r m a t i o n into t h e one above it Information is passed
from component to c o m p o n e n t down the r i g h t leg to c o n s t r u c t
the o u t p u t text The c o m p o n e n t s of each leg correspond to t h e
chapters of a n introductory textbook on l i n g u i s t i c s with phonology
or graphology at the bottom, t h e n s y n t a x , s e m a n t i c s , a n d so on
The legs join where l a n g n a g e s are no longer differentiated a n d
linguistics s h a d e s off into psychology and philosophy The h i g b e r
levels are also the ones whose theoretical u n d e r p i n n i n g s a r e less
well k n o w n and s y s t e m d e s i g n e r s therefore often tie t h e legs
together s o m e w h e r e lower down, c o n s t r u c t i n g a more or less ad
hoe bridge, pivot, or t r a n s f e r component
We connot be s u r e t h a t t h e classical design is t h e r i g h t
design, or the best design, for a t r a n s l a t i n g m a c h i n e B u t it does
h a v e several s t r o n g points Since t h e s t r u c t u r e of t h e c o m p o n e n t s
is grounded in linguistic theory, it is possible to divide each of
t h e s e components into two parts: a formal description of t h e
r e l e v a n t facts a b o u t t h e l a n g u a g e , a n d a n i n t e r p r e t e r of t h e
formalism The formal description is d a t a w h e r e a s t h e i n t e r p r e t e r
is program The formal description should" ideally serve the n e e d s
of s y n t h e s i s a n d a n a l y s i s indifferently On t h e o t h e r h a n d we
would expect different i n t e r p r e t e r s to be required in t h e two legs
of the machine• We expect to be able to u s e identical i n t e r p r e t e r s
in corresponding places in all m a c h i n e s of s i m i l a r design because
the information t h e y embody comes from g e n e r a l lingusitic theory
and not from p a r t i c u l a r l a n g u a g e s The s c h e m e therefore h a s
t h e a d v a n t a g e of modularity The linguistic descriptions are
i n d e p e n d e n t of the leg of t h e m a c h i n e t h e y are used in a n d t h e
p r o g r a m s are i n d e p e n d e n t of t h e l a n g u a g e s to which t h e y are
applied
For all t h e a d v a n t g a g e s of t h e classical design, it is n o t
h a r d to i m a g i n e i m p r o v e m e n t s In t h e best all possible worlds,
t h e r e would only be one f o r m a l i s m in which all t h e facts a b o u t a
l a n g u a g e - - m o r p h o l o g i c a l , syntactic, s e m a n t i c , or w h a t e v e r - - c o u l d
be stated A f o r m a l i s m powerful e n o u g h to accommodate t h e
v a r i o u s different kinds of linguistic p h e n o m e n a with equal facility
m i g h t be u n a p p e a l i n g to theoretical l i n g u i s t s because powerful formal s y s t e m s do not m a k e powerful claims B u t the e n g i n e e r i n g
a d v a n t a g e s are clear to see A single f o r m a l i s m would straightfor- wardly reduce t h e n u m b e r of i n t e r p r e t e r s to two, one for a n a l y s i s
a n d one for s y n t h e s i s F u r t h e r m o r e , t h e e x p l a n a t o r y v a l u e of a theory clearly r e s t s on a g r e a t deal more t h a n the restriciveness of its formal base In particular, the possiblity of e n c o m p a s s i n g w h a t
h a d hitherto been t h o u g h t to require a l t o g e t h e r different k i n d s of
t r e a t m e n t w i t h i n a single f r a m e w o r k could be theoretically inter- esting
A n o t h e r clear i m p r o v e m e n t on t h e classical design would
"result from m e r g i n g 'the two i n t e r p r e t e r s associated with a for-
m a l i s m T h e m o s t obvious a d v a n t a g e to be hoped for with
t h i s move would be t h a t t h e overall s t r u c t u r e of the t r a n s l a t i n g
m a c h i n e would be greatly simplified, t h o u g h this would not neces- sarily happen It is also reasonable to hope t h a t t h e m a c h i n e would
be m o r e robust, easier to modify a n d m a i n t a i n , a n d a l t o g e t h e r more perspicuous T h i s is because a device to which a n a l y s i s a n d
s y n t h e s i s look e s s e n t i a l l y t h e s a m e is one t h a t is f u n d a m e n t a l l y
l e s s t i m e dependent, w i t h fewer i n t e r n a l v a r i a b l e s and states; it
i s a p t to work by m o n i t o r i n g c o n s t r a i n t s laid down in t h e formal description and e n s u r i n g t h a t t h e y are m a i n t a i n e d , r a t h e r t h a n
c a r r y i n g out long a n d complex sequences of steps in a carefully prescribed order
• T h e s e a d v a n t a g e s are available in large m e a s u r e t h r o u g h
a class of formal devices t h a t are slowly g a i n i n g acceptance in linguistics a n d which are based on t h e relations contracted by formal objects r a t h e r t h a n by t r a n s f o r m a t i o n s of one formal object into another These s y s t e m s are all procedurally monotonic in t h e
s e n s e that, while new information m a y be added to e x i s t i n g d a t a
s t r u c t u r e s , possibly different i n f o r m a t i o n on different b r a n c h e s of
a n o n d e t e r m i n i s t i c process, n o t h i n g is ever deleted or changed
A s a result, the p a r t i c u l a r order in which e l e m e n t a r y e v e n t s t a k e place is of little importance Lexical F u n c t i o n a l G r a m m a r a n d Generalized P h r a s e - S t r u c t u r e g r a m m a r s h a r e t h e s e relational a n d monotonic properties T h e y are also c h a r a c t e r i s t i c s of F u n c t i o n a l Unificational G r a m m a r (FUG) w h i c h I believe also h a s additional properties t h a t s u i t it p a r t i c u l a r l y well to t h e needs of e x p e r i m e n - tal machine-translation systems
The term experimental must be taken quite seriously here though, if m y view of machine translation were more generally held, it would be redundant I believe that all machine translation
of natural languages is experimental and that he w h o claims otherwise does his more serious colleagues a serious disservice I should not wish any thing that I say in this paper as a claim to have solved any of the miriad problems that stand between us and working machine translation systems worthy of the name T h e contribution that F U G might m a k e is, I believe, a great deal more
75
Trang 2modest, namely to reformalize more simply and perspicuously
what has been done before and which has come to be regarded, as
1 said at the outset %lassical'
B F u n c t i o n a l U n i f i c a t i o n G r a m m a r
FUG traffics in descriptions and there is essentially only one
kind of description, w h e t h e r for lexical items, phrases, sentences,
or entire languages Descriptions do not distinguish a m o n g levels
in the linguistic hierarchy This is not to say t h a t the distinctions
among the levels are unreal or t h a t a linguist working w i t h
the formalism whould not respect them It m e a n s only t h a t the
notation and its interpretation are always uniform• E i t h e r a pair
of descriptions is incompatible or they are combinable into a single
description
Within FUG, every object h a s infinitely m a n y descriptions,
though a given g r a m m a r partitions the descriptions of the words
and phrases in its language into a finite n u m b e r of equivalence
classes, one for each interpretation t h a t the g r a m m a r assigns to it
The m e m b e r s of an equivalence class differ along dimensions t h a t
are grammatically i r r e l e v a n t - - w h e n they were uttered, w h e t h e r
they a m m u s e d Queen Victoria, or w h e t h e r they contain a prime
n u m b e r of words Each equivalence class constitutes a lattice
with j u s t one m e m b e r t h a t contains none of these grammatically
irrelevant properties, and this canonical member is the only one
a linguist would normally concern himself with However, a
grammatical irrelevancy t h a t acquires relevance in the present
context is the description of possible t r a n s l a t i o n s of a word or
phrase, or of one of its interpretations, in one or more other
languages
A description is an expression over an essentially arbitrary
basic vocabulary The relations a m o n g sets of descriptions there-
fore remain unchanged under one-for-one mappings of their basic
vocabularies It is therefore possible to arrange that different
g r a m m a r s share no terms except for possible quotations from
the languages described Canonical descriptions of a pair of
sentences in different languages according to g r a m m a r s that
shared no terms could always be unified into a single descrip-
tion which would, of course, not be canonical Since all pairs
are unifiable, the relation that they establish between sentences
is entriely arbitrary However, a third g r a m m a r can be written
that unifies with these combined descriptions only if the sentences
they describe in the two langaunges stand in a certain relation
to one another The relation w e are interested in is, of course,
the translation relation which, for the purposes of the kind'of
expcrimantal system I have in mind I take to be definable o':en
for isolated sentences Such a transfer g r a m m a r can readily cap-
ture all the components of the translation relation that have in
fact been built into translation systems: correspondences between
words and continuous or discontinuous phrases, use of selectional
features or local contexts, case frames, reordering rules, lexical
functions, compositional semantics, and so on
I I T h e F o r m a l i s m
A F u n c t i o n a l D e s c r i p t i o n s
I n ' F U G , linguistic objects are represented by functional
descriptions (FDs) The basic constituent of a functional descrip-
tion is a feature consisting of an attribute and an associated value
We write features in the form a ~ v, where a is the attribute and
v, the value Attributes are a r b i t r a r y words with no significant
internal structure Values can be of various types, the simplest of
which is an atomic value, also an arbitrary word So Cat ~- S is
a feature of the most elementary type It appears in the descrip-
tions of sentences, and which declares t h a t their Category is S
The only kinds of non-atomic values t h a t will concern us here are
constituent sets, patterns and FDs themselves
A FD is a Boolean expression over features We distinguish
their members; the conjuncts and disjuncts of a p, b ~-~ q, and
c ~ r are written
b -~ q and b ~ - q
c ~ q c ~ r
respectively The vertical a r r a n g e m e n t of these expressions has proved convenient zind it is of minor importance in t h a t braces
of the ordinary variety are used for a different purpose in FUG, namely to enclose the ]nembers of consituent sets The following
FD describes all sentences whose subject is a s i n g u l a r noun phrase
in the nominative or accusative cases
pu°' = l[case om l I
L LLCase = A c c J J J
It is a crucial property of FDs t h a t no a t t r i b u t e should figure more t h a n once in any conjunct, t h o u g h a given a t t r i b u t e m a y appear in feature lists t h a t are themselves the values of different attributes This being the case, it is ahvays possible to identify
a given conjunct or disjunct in a FD by giving a sequence of
a t t r i b u t e s ( a l a k ) a I is a attribvte in the FD whose value,
el, is a n o t h e r FD The attribute a2 is an a t t r i b u t e in Vl whose value if an FD, and so on Sequences of a t t r i b u t e s of this kind are referred to as paths If the FD contains disjuncts, then the value
identified by the p a t h will n a t u r a l l y also be a disjunct
We sometimes write a path as the value of a n a t t r i b u t e to indicate t h a t t h a t value of t h a t a t t r i b u t e is not only eaqual to the value identified by the p a t h b u t t h a t these values are one and the same, inshort, t h a t they are unified in a sense soon to
be explained Roughly, if more information were acquired about one of the values so t h a t more features were added to it, the same additions would be reflected in the other value This would not automatically h a p p e n because a pair of values happened to be the
• same So, for example, if the topic of the sentence were also its object, we m i g h t write
Object -~ v 1
Topic = (Object)J
where v is some FD
Constituent sets are sets of p a t h s identifying within a given
FD the descriptions of its constituents in the sense of phrase-
s t r u c t u r e g r a m m a r No constituent set is specified in example (l) above and the question of w h e t h e r the subject is a constituent is therefore left open
Example (2), t h o u g h still artificially simple, is more realis- tic It is a syntactic description of the sentence J o h n k n o w s Mary
P e r h a p s the most s t r i k i n g property of this description is t h a t descriptions of constituents are embedded one inside another, even
t h o u g h the constituents themselves are not so embedded The value of the H e a d attribute describes a constituent of the sentence,
a fact which is declared in the value of the CSet attribute We also
see t h a t the sentence has a second a t t r i b u t e whose decription is
to be found as the value of the Subject of the Head of the Head of the sentence The reason for this a r r a n g e m e n t will become clear shortly
In example (2), every conjunct in which the CSet a t t r i b u t e
has a value other t h a n N O N E also has a substantive value for the
attribute Pat The value of this attribute is a regular expression
over p a t h s which restricts the order in which the constituents m u s t appear By convention, if no p a t t e r n is given for a description which nevertheless does have constituents, they m a y occur in a n y order We shall have more to say about p a t t e r n s in due course
76
Trang 3E s s e n t i a l l y t h e only operation used in processing F U G is t h a t
of Unification, the p a r a d i g m e x a m p l e of a monotonic operation
Given a pair of descriptions, the unification process first deter-
m i n e s w h e t h e r t h e y are compatible in t h e s e n s e of allowing the
possibility of t h e r e being some object t h a t is in t h e extension of
both of t h e m T h i s possibility would bc excluded if t h e r e were a
p a t h in one of t h e two descriptions t h a t lead to a n atomic v a l u e
while t h e s a m e p a t h in the other one lead to some other value
T h i s would occur if, for example, one described a s e n t e n c e w i t h a
s i n g u l a r subject and the other a sentence with a plural subject, or
if one described a sentence a n d the other a n o u n phrase T h e r e can
also be incompatibilities in respect of other k i n d s of value T h u s ,
if one h a s a p a t t e r n r e q u i r i n g the subject to precede the m a i n verb
w h e r e a s the other specifies t h e other order, t h e two descriptions
will be incompatible C o n s t i t u e n t sets are incompatible if t h e y
are not t h e s a m e
We h a v e briefly considered how three different types of descrip-
tion b e h a v e u n d e r unification Implicit in w h a t we h a v e said is
t h a t descriptions of different types do not u n i f y with one another
G r a m m a r s , which are t h e descriptions of the infinite s e t s of sen-
tences t h a t m a k e up a l a n g u a g e constitute a type of description
t h a t is s t r u c t u r a l l y identical a n ordinary FD b u t is d i s t i n g u i s h e d
on the grounds t h a t it b e h a v e s slightly differently u n d e r unifica-
tion In particular, it is possible to unify a g r a m m a r with a n o t h e r
g r a m m a r to produce a new g r a m m a r , b u t it is also possible to
unify a g r a m m a r w i t h a FD, in which case t h e r e s u l t is a n e w
FD The rules for u n i f y i n g g r a m m a r s with g r a m m a r s are t h e
s a m e as those for u n i f y i n g F D s w i t h FDs T h e rules for unify-
ing g r a m m a r s with FDs, however, are slightly different a n d in
t h e difference lies t h e ability of F U G to describe s t r u c t u r e s recur-
sively a n d hence to provide for s e n t e n c e s of u n b o u n d e d size The
rule for u n i f y i n g g r a m m a r s w i t h F D s requires t h e g r a m m a r s to
be u n i f i e d ~ f o l l o w i n g the rules for FD u n i f i c a t i o n ~ w i t h each in-
dividual c o n s t i t u e n t of t h e FD
(s)
H e a d ~-~ [tIead = [Cat ~ - V]]
CSet = {(Head H e a d Subj)(Head)}
I
P a t = ((Itead H e a d Subj}(Heed))
I / I O b j = N O N E
Head = |[Obj = [Cat = NP]
LCSet = N O N E [Head = [Cat = N II
By way of illustration, consider t h e g r a m m a r in (3) Like
m o s t g r a m m a r s , it is a disjunction of clauses, one for each (non-
t e r m i n a l ) category or c o n s t i t u e n t type in t h e l a n g u a g e T h e
first of t h e t h r e e c l a u s e s in t h e principle dir.junction describes
s e n t e n c e s as h a v i n g a h e a d whose head is of category V T h i s
characterization is in line with so called X-theory, according to
which a sentenceI belongs to the category ~ In general, a p h r a s e
of category X , for w h a t e v e r X , h a s a head c o n s t i t u e n t of category
X , t h a t is, a category with the s a m e n a m e b u t one less bar X
is built into the very fabric of t h e version of F U G i l l u t r a t e d h e r e
where, for example, a setence is by definition a p h r a s e whose
bead's head is a verb T h e h e a d of a s e n t e n c e is a V, t h a t is,
a p h r a s e whose head is of category V a n d which h a s no h e a d
of its own A p h r a s e with t h i s description c a n n o t u n i f y w i t h
t h e first clause in t h e g r a m m a r because its head h a s t h e f e a t u r e
[Head = NONE]
Of sentences, t h e g r a m m a r s a y s t h a t t h e y h a v e two con-
s t i t u e n t s It is no surprise t h a t t h e second of t h e s e is its head
The first would u s u a l l y be called its subject b u t is h e r e charac-
must be lexical entries not only for all the verbs in the language but that there must be such an entry for each of the subjects that the verb might have W h a t it does m e a n is that the subject must
be unifiable with any description the verb gives of its subject and thus provides automatically both for any selectional restrictions that a verb might place on its subject but also for agreement in person and number between subject and verb Objects are handled
in an analogous manner Thus, the lexical entries for the French
verb forms cm, nait a n d salt m i g h t be as follows:
Lex ~ connaitre /
T e n s e = P r e s I
Subj = |Num = Sing|/
LAnim = + J[
Obj = [Cat = NP] J
[Pers = 3 I I Subj = INure = Sing|I [ A n i m ~ + J/ Obj ~i~ [Cat ~ - S] J Each requires its subject to be third person, singular and animate Taking a rather simplistic view of the difference between these
verbs for the sake of the example, this lexicon states that connatt takes noun phrases as objects, whereas salt takes sentences
III T r a n s l a t i o n
A S y n t a x
Consider now t h e F r e n c h s e n t e n c e J e a n connaft Marie which
is p r e s u m a b l y a reasonable r e n d e r i n g of the E n g l i s h s e n t e n c e
J o h n k n o w s Mary, a possible f u m c t i o n a l description of w h i c h
we w a s given in (2) I t a k e it t h a t t h e F r e n c h s e n t e n c e h a s
a n e s s e n t i a l l y isomorphic s t r u c t u r e In fact, following t h e p l a n laid o u t a t the b e g i n n i n g of t h e paper, let u s a s s u m e t h a t t h e functional description of t h e F r e n c h s e n t e n c e is t h a t given in (2)
w i t h obvious r e p l a c e m e n t s for the v a l u e s of the Lex a t t r i b u t e a n d
w i t h a t t r i b u t e n a m e s z~ in t h e E n g l i s h g r a m m a r s y s t e m a t i c a l l y
replaced by F - z i in t h e French T h u s we h a v e F-Cat, F-Head, etc
Suppose now, t h a t , u s i n g t h e E n g l i s h g r a m m a r a n d a suitable
p a r s i n g a l g o r i t h m , t h e s t r u c t u r e given in (2) is derived from t h e
E n g l i s h sentence, a n d t h a t t h i s description is t h e n unified w i t h
t h e following transfer g r a m m a r :
:F-Lex ~ - JeanJ | [ Lex = M a r y ] / /
" ~ = know l I /
= conna'tre1111
LF-Lex -= savoir J J ) J The first clause of the principal conjunct states a very strong requirement, namely that the description of a phrase in one of the two languages should be a description of a phrase of the same category in the other language The disjunct that follows
is essentially a bilingual lexicon that requires the description of
a lexical item in one language to be a description of that word's counterpart in the other language It allows the English verb
k n o w to be set in correspondence w i t h either connattre or savoir
a n d gives no m e a n s by w h i c h to d i s t i n g u i s h t h e m In t h e simple
e x a m p l e we a r e developing, t h e choice will be d e t e r m i n e d on t h e
b a s i s of criteria expressed only in t h e F r e n c h g r a m m a r , n a m e l y
w h e t h e r the object is a n o u n p h r a s e or a sentence
T h i s is a b o u t as trivial a t r a n s f e r g r a m m a r as one could readily i m a g i n e writing It profits to t h e m i n i m a l possible e x t e n t from t h e power of F U G N e v e r t h e l e s s , it should a l r e a d y do better
t h a n word-for-word t r a n s l a t i o n because t h e t r a n s f e r g r a m m a r s a y s
n o t h i n g at all a b o u t t h e order of t h e words or p h r a s e s If t h e
7 7
Trang 4and the French one says that they precede, the same transfer
grammar, though still without any explicit mention of order,
will cause the appropriate "reordering" to take place Similarly,
nothing more would be required in the transfer g r a m m a r in order
to place adjectives properly with respect to the nouns they modify,
and so forth
B S e m a n t i c s
It m a y be objected to t h e line of a r g u m e n t t h a t I h a v e b e e n
p e r s u i n g t h a t it r e q u i r e s t h e legs of t h e t r a n s l a t i n g m a c h i n e to be
tied t o g e t h e r a t too lower a level, e s s e n t i a l l y a t t h e level of s y n t a x
To be sure, it allows more elaborate t r a n s f e r g r a m m a r s t h a n t h e
one j u s t i l l u s t r a t e d so t h a t t h e t r a n s l a t i o n of a s e n t e n c e would
n o t h a v e to be s t r u c t u r a l l y isomorphic w i t h its source, modulo
ordering B u t t h e device is e s s e n t i a l l y syntactic However, t h e
relations t h a t c a n be characterized by F U G a n d s i m i l a r m o n o t o n i c
devices a r e in fact a g r e a t deal m o r e diverse t h a n this s u g g e s t s In
p a r t i c u l a r , m u c h of w h a t falls u n d e r t h e u m b r e l l a of s e m a n t i c s in
m o d e r n l i n g u i s t i c s also fits c o n v e n i e n t l y w i t h i n t h i s f r a m e w o r k
S o m e t h i n g of t h e flavor of t h i s c a n be c a p t u r e d from t h e following
example Suppose t h a t t h e lexieal e n t r i e s for t h e words all a n d
dogs are a s follows:
"Cat -~ Det
Lex ~ all
N u m ~ P l u r
D e f ~ +
| [Type - - I m p l i e s
Sense = [P op = [P1 = [Arg = (Sense Varl]
L LP2 = [Arg ~ (Sense Var)JJJ
- - T y p e ~ Pred
W h e n t h e first of t h e s e is unified w i t h t h e v a l u e of t h e Art
a t t r i b u t e in t h e second as required by t h e g r a m m a r , t h e r e s u l t is
as follows:
"Cat -~ N
Lex clog
C a t ~ D e t
Lex = All
A r t Def ~ +
N u m ~ P l u r
~ense = (Sense'
/ [Type - Implies I l l
Se~ |Prop = lP1 = |Pred = dog / / / I
[ LP2 [Arg ~ (Sense Var)] JJJ
This, in turn, is readily interpretable as a description of the logical
expression
Vq.dogCq)AP(q)
It r e m a i n s to provide v e r b s w i t h a s e n s e t h a t provides a s u i t a b l e
v a l u e for P , t h a t is, for (Sense Prop P2 Pred) A n e x a m p l e would
be t h e following:
"Cat ~ V Lex ~ barks Tense ~ Pres
r P e r s = 3 1 Subj - - | N u m ~ S i n g | LAnim ~ + J Obj : N O N E
S e n s e = [Prop ='- [P2 = [Pred = bark]]]
I V C o n c l u s i o n
It h a s n o t b e e n possible in t h i s p a p e r to give m o r e t h a n a n
i m p r e s s i o n of how a n e x p e r i m e n t a l m a c h i n e t r a n s l a t i o n s y s t e m
m i g h t be c o n s t r u c t e d b a s e d on F U G I hope, however, t h a t it
h a s b e e n possible to convey s o m e t h i n g of t h e v a l u e of m o n o t o n i c
s y s t e m s for t h i s purpose I m p l e m e n t i n g F U G in a n efficient w a y
r e q u i r e s skill a n d a v a r i e t y of little k n o w n t e c h n i q u e s However,
t h e p r o g r a m s , t h o u g h subtle, a r e n o t l a r g e and, once w r i t t e n ,
t h e y provide t h e g r a m m a r i a n a n d lexicographer w i t h a n e m m e n s e
w e a l t h of e x p r e s s i v e devices A n y s y s t e m i m p l e m e n t e d s t r i c t l y
w i t h i n t h i s f r a m e w o r k will be reversible in t h e s e n s e t h a t , if it
t r a n s l a t e s from l a n g u a g e A to l a n g u a g e B the, to t h e s a m e e x t e n t ,
it t r a n s l a t e s from B to A If t h e set S is a m o n g t h e t r a n s l a t i o n s
it delivers for a, t h e n a will be a m o n g t h e t r a n s l a t i o n s of e a c h
m e m b e r of S I k n o w of no s y s t e m t h a t c o m e s close to p r o v i d i n g
t h e s e a d v a n t a g e s a n d I k n o w of no facility provided for in a n y
system proposed hitherto that it not subsumable under F U G
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