COLING 82, J. Horeek~ (ed. J 
North-Holland Publishing Company 
© AcMemta, 1982 
MACHINE TRANSLATION BASED ON LOGICALLY 
ISOMORPHIC MONTAGUE GRAMMARS 
Jan Landsbergen 
Philips Research Laboratories 
Eindhoven - The Netherlands 
The paper describes a new approach to machine translation, 
based on Montague grammar, and an experimental translation 
system, Rosetta, designed according to this approach. It is 
a multi-lingual system which uses 'logical derivation trees' 
as intermediate expressions. 
i. INTRODUCTION 
Usually two approaches to machine translation are distinguished: the inter- 
lingual approach and the transfer approach (cf. Hutchins \[i\]). In the interlingual 
approach translation is a two-stage process: from source language to interlingua 
and from interlingua to target language. In the transfer approach there are three 
stages: source language analysiS, transfer and target language generation. The 
approach advanced in this paper is a variant of the interlingual one. It requires 
that 'logically isomorphic grammars' are written for the languagesunder considera- 
tion. The syntactic rules of these grammars must correspond with logical opera- 
tions, in accordance withthe compositionality principle of Mentague grammar. 
Moreover, the grammars must be attuned to each other as follows: if one grammar 
contains a rule corresponding with a particular logical operation, the other 
grammars must contain rules corresponding with the same operation. Syntactically, 
these rules may differ considerably. If the grammars are attuned to each other in 
this way, 'logical derivation trees', representations of both the syntactical and 
the logical structure of sentences, can be used as intermediate expressions. 
The paper is organized as follows. In section 2 the relevant concepts of 
Montague grammar and the notion 'logically isomorphic grammars' are introduced. In 
section 3 a version of Montague grammar is described, called M-grammar, which is 
more suitable for computational use than Montague's original proposals. The 
property of logical isomorphy is then defined for M-grammars. In section 4 the 
design of the Rosetta translation system, based on this approach, is outlined, 
followed by a brief discussion in section 5. 
2. LOGICALLY ISOMORPHIC MONTAGUE GRAMMARS 
I will first introduce a few concepts of Montague grammar (cf. \[2\], \[3\]), in 
an informal way. 
A Montague grammar defines a language by specifying (i) a set of basic ex- 
pressions and their syntactic categories, (ii) a set of syntactic rules. Each rule 
specifies the categories of the expressions to which it is applicable, prescribes 
how these expressions must be combined to form a new expression and specifies the 
category of this expression. An expression is 'generated' by a Montague grammar if 
it can be derived by applying syntactic rules, starting from basic expressions. 
Exam Ip~_~. Assume that grammar G 1 defines a fragment of English. 
kmong Gl's basic expressions are 'Italian', of category ADJ, and 'girl', of 
category NOUN. 
~mong Gl'S syntactic rules are: 
175 
176 J. LANDSBERGEN 
R4: if expression ~ is of category ADJ and ~ is of category NOUN, then ~ 
is of category NOM. 
R7: if o< is of category NO~, then 'the'=< is of category NP. 
One of the phrases G 1 generates is the NP 'the Italian girl'. It can be 
derived by applying R 4 to basic expressions 'Italian' and 'girl'~ and then applying 
R 7 to the result. 
The way in which an expression is derived from basic expressions by applica- 
tion of rules can be represented by a tree, called derivation tree, with basic 
expressions labelling the terminal nodes and names of applied rules labelling the 
non-terminal nodes. For example, Figure 1 is the derivation tree of 'the Italian 
Italian girl 
Figure 1 
girl' according to grammar G I. 
A Montague grammar must obey the composi- 
tionality principle, which reads: 'The meaning 
of a compound expression is composed from the 
meanings of its parts'. This is achieved by 
choosing the basic expressions and the rules 
in such a way that the meaning of a phrase can 
be defined by a syntax-directed translation in 
a logical language. For each basic expression 
a corresponding expression of the logical 
language is specified: the representation of 
its meaning. For each syntactic rule a logical 
composition rule is given, which shows how the meaning representation of the 
phrase constructed by the syntactic rule is derived from the meaning representa- 
tions of the constituent phrases. The exact nature of the (intensional) logic that 
Montague uses, is not relevant here. For the present discussion it is important 
that a derivation tree displays not only the syntactic structure of a phrase, but 
its logical structure as well. 
Example 2. Suppose grammar G 2 defines a fragment of Italian. 
Among G2's basic expressions are 'italiano', of category ADJ, and 'ragazza', of 
category NOUN. 
Among G2's syntactic rules are: 
R~: if expression ~ is of category ADJ and ~ is of category NOUN, then /3 ~' 
is of category NOM, where ~' is the adjective ~ adjusted to the number and 
gender of the noun ~. 
R~: if ~ is of category NOM, then ~ ~ is of category NP, where ~ is a definite 
article in accordance with the number and gender of ~. 
G 2 generates the phrase 'la ragazza italiana', with the derivation tree of 
italiano "~ ragagga 
Figure 2 
Figure 2. 
A comparison between the example grammars 
G1 and G 2 shows that there is a correspondence 
between the basic expressions and rules of G 1 
and those of G 2. 'Italian' and 'italiano' 
have the same meaning (in at least one of their 
readings) and the same holds for 'girl' and 
'ragazza'. Rule R 4 and rule R~ correspond with 
the same logical composition rule, the same 
holds for R7 and R~. However, syntactically 
the rules differ considerably. 
It may be possible to write grammars for 
large fragments of English and Italian and 
other languages in such a way that this semantic correspondence between basic 
expressions and rules of one language and those of the other languages is main- 
tained. Dowty \[4\] has also pointed out this possibility (referring to similar 
observations by Curry and Dahl) ~nd has given examples of correspondences 
between English, Japanese, Breton and Latin. I will call grammars that correspond 
with each other in this way logically isomorphic grammars. In the next section 
this notion will be defined precisely. 
MACHINE TRANSLATION BASED ON MONTAGUE GRAMMARS 177 
3. M-GRAMMARS 
In Montague's original proposals \[2\] the syntactic rules operate on strings. 
From a linguistic point of view it is desirable to have rules operating on 
syntactic trees (cf. Partee \[5\]). From a computational point of view it is 
necessary to impose restrictions on the grammars in order to make effective 
parsing procedures possible. In an earlier paper \[6\] I developed a version of 
Montague grammar, called M-grammar, with the desired properties. I will briefly 
recapitulate the relevant definitions here. First, I will describe the kind of 
syntactic tree, called S-tree, on which the rules of an M-grammar operate. 
An S-tree is a labelled ordered tree. The labels of the nodes may be com- 
pound entities of the kind more often met in computational linguistics, consisting 
of a syntactic category and a number of attribute-value pairs. The labels of the 
terminal nodes of an S-tree correspond with words i. The branches may be labelled 
with the names of syntactic relations (subject, head, modifier, etc.). The 
syntactic trees defined by a context-free grammar are a special, simple, kind of 
S-tree 2. Each S-tree defines a phrase s, the sequence of terminal labels of t, 
called LEAVES(t). 
An M-grammar defines a set of S-trees by specifying a set of basic S-trees 
(not necessarily terminal S-trees) and a set of rules, called M-rules. 
An M-rule R i defines a function F i from tuples of S-trees to sets of S-trees. 
So application of R i to tuple tl,...,t n results in a set Fi(tl,...,tn). If this 
set is empty, the rule is said to be not applicable. In order to make effective 
analysis procedures possible, M-rules must obey the following conditions. 
Reversibility condition. Each rule R i defines not only the compositional 
function Fi, but also an 'analytical' function F~, from S-trees to sets of 
tuples of S-trees, in such a way that: 
t ~ Fi(t I ..... tn) ~====> <tl ..... t n> E F~(t) 
I will call F~ the reverse of F i. 
Measure condition. There is a measure function~, from S-trees to natural 
numbers, such that for each rule R i the following holds: 
if t ~ Fi(tl,...,tn) then j~(t>>/~g(tj) for each tj. 
So application of the analytical function F~ results in a tople of smaller' 
S-trees. 
Analogously to section 2, we can define derivation trees, to be called 
D-trees here, which show the way in which an S-tree is derived from basic S-trees 
by application of M-rules. 
For a given M-grammar two functions can be defined: 
(i) M-GENERATOR, a function from D-trees to sets of S-trees. For each D-tree d 
M-GENERATOR(d) is the set of S-trees generated by applying the rules in d. 
M-GENERATOR is defined in terms of the compositional functions F i. 
(ii) M-PARSER, a function from S-trees to sets of D-trees. For each S-tree t 
M-PARSER(t) is the set of D-trees that generate t. M-PARSER is defined in terms 
of the analytical functions F~. 
For both functions effective procedures can be written, thanks to the reversibili- 
ty condition and the measure condition. It can be proved that for each D-tree d 
and S-tree t holds: 
t e "M-GENERATOR(d) <===> d ~ MrPARSER(t) 3 
M-grammars must also obey the following condition. 
Surface grammar condition. There must be a surface grammar GS, such that the 
set of S-trees defined by the M-grammar is a subset of the set of S-trees defined 
by G S. A surface grammar defines a set of S-trees, like an M~grammar, but with 
rules, called surface rules, which are simpler and less powerful than M-rules. 
If a surface rule is applied to a tuple of S-trees tl,...,t n it creates an S-tree 
with a new top and with tl,...,t n as immediate subtrees. A context-free grammar 
is a special case of a surface grammar. 
For a surface grammar a parser can be defined: an effective function proce- 
dure, to be called S-PARSER, which assigns to any phrase s the set of grammatical 
178 J. LANDSBERGEN 
S-trees t such that LEAVES(t) = s. 
Thanks to the surface grammar condition, M-PARSER can be extended to a func- 
tion from sentences to D-trees. First the function S-PARSER, defined by the 
surface grammar, is applied to the sentence and then M-PARSER is applied to each 
of its results. M-GENERATOR can be extended to a function from D-trees to senten- 
ces by applying the function LEAVES to each S-tree in M-GENERATOR(d). 
The basic expressions and the rules of an M-grammar must be chosen in 
accordance with the compositionality principle. For each basic expression b i a 
set of logical expressions L.3 must be specified, representing the meanings of b i. 
For each M-rule R i a logical composition rule Sj must be specified (several 
M-rules may share the same Sj). 
Let us now define a logical D-tree as a D-tree with names of logical compo- 
sition rules Sj at the non-termlnal nodes and names of 'basic' logical expressions 
at the terminal nodes. For each M-grammar a function LOG from (syntactical) 
i 3 
Figure 3 
D-trees to sets of logical D-trees can be 
defined, as well as the reverse function LOG'. 
If the example grammars G 1 and G 2 were to be 
reformulated as M-grammars, Figure 3 might be a 
logical D-tree for the D-tree of Figure i, and 
also for the D-tree of Figure 2. 
A logical D-tree defines exactly one 
logical expression and is in fact a redundant 
intermediate step between syntactical D-tree 
and logical expression. However, as we will see 
in the next section~ it is logical D-trees and 
not logical expressions which act as the pivot 
of the translation process in the Rosetta 
system. 
We are now able to define for each M-grarmnar the function ANALYSIS, which 
maps sentences to sets of logical derivation trees and the reverse function 
GENERATION. 
ANALYSIS(s) =def 4e I ~t, ~d: t ~ S-PARSER(s) ~ d E M-PARSER(t) ~ e ~ LOG(d)} 
GENERATION(e) =defiSl ~t,~d: daLOG'(e) ~ t&M-GENERATOR(d) ~ seLEAVES(t) } 
(s ranges over sentences, t over S-trees, d over D-trees, e over logical D-trees) 
The following theorem can be proved easily. 
Reversibility Theorem. 
~/s, ~/e : e e ANALYSIS(s) ~ s ~ GENERATION(e) 
The given definitions enable the notion of logical isomorphy to be defined 
precisely. Let us assume as given two M-grammars G i and Gj with generation 
functions GENERATION i and GENERATIONj. 
Gi t~,~ Gj (G i logically isomorphic with Gj) iff 
\'e : \[ 3s : s ~ GENERATIONi(e) ~ 3 s' : s' ~ GENERATIONj(e) \] 
(for each logical D-tree assigned to a sentence s by Gi, there is a sentence s' 
to which Gj assigns the same logical D-tree, and vlce-versa) 
Proving that two grarmmars are logically isomorphic may be complicated. It 
is simple for grammars of which the rules are complete (i.e. applicable to all 
expressions of the required categories), if there is a one-to-one correspondence 
between the syntactic categories, the basic expressions and the rules of the two 
grammars. 
Because the relation ~ is an equivalence relation, it makes sense to 
speak of a set of logically isomorphic grammars, as I will do in the next section. 
MACHINE TRANSLATION BASED ON MONTAGUE GRAMMARS 179 
4. THE ROSETTA SYSTEM 
Suppose we have logically isomorphic M-grammars GI, G 2 .... for languages 
LI, L2,... In section 3 we have seen that each grammar G i defines a function 
ANALYSIS i and a function GENERATION i. These functions determine a translation 
function for each pair of languages Li, Lj- 
TRANSiJ (s) =def {s'\[ ~e : e ~ ANALYSISi(s) ~ s' 6" GENERATIONj(e)~ 
(s and s' are sentences, e is a logical D-tree) 
An experimental translation system, Rosetta, has been designed, which trans- 
lates isolated sentences of a source language into sets of sentences of a target 
language, according to the definition of TRANSIj. Given isomorphic grammars for a 
set of languages the source language and the target language can be freely chosen 
from this set. The definition of logical isomorphy and the Reversibility Theorem 
guarantee that - given correct grammars - each sentence of any source language 
will be correctly translated into any target language, for each meaning of that 
sentence. 
In Figure 4 the various stages of the translation process in Rosetta are 
shown. The example expressions are simplified, e.g. the attribute-values have 
been omitted in the S-trees. MORFH and MORPH' are the components that perform 
dictionary look-up and morphological rules during analysis and generation. 
the Italian girl---~ _ . ~.~..la ragazza italiana h"'" I IM°RP"! I 
ART + ADJ + NOUN,..~ ~.-..ART + NOUN + ADJ 
I:.°i 1 
Figure 4 
It should be noted that LOG and LOG' may perform more complex operations than 
Figure 4's example suggests. Basic expressions may be larger units than words. 
Furthermore an adjective of the source language may be translated into a preposi- 
tional phrase or a relative clause of the target language, because these catego- 
ries correspond with the same logical type as the category adjective. Rule S 2 
is not only translated into R~, but also into rules that combine a noun with a 
prepositional phrase or a relative clause. Which of the combinations of these 
local ambiguities is correct is decided in M-GENERATOR. 
The first version of Rosetta was designed and implemented in 1981, for very 
small fragments of Dutch, English and Italian. The system operates in a breadth- 
first manner: at each level it generates all results, in the case of ambiguities, 
and ultimately it gives all possible translations. The program was written in 
PASCAL and runs under UNIX on a VAX 11/780. 
There are relatively few applications for a system delivering all syntacti- 
cally possible translations. As is well-known, in order to choose the 'best' out 
of the possible translations, knowledge about the world and the context is needed. 
A future version of Rosetta will presumably be provided with interactive facili- 
ties that enable the user to contribute this kind of knowledge, as in ITS \[7\]. 
180 J. LANDSBERGEN 
5. DISCUSSION 
There is a more obvious way to base a multi-lingual translation system on 
Montague grammar, or on logic in general, than the one described here: use the 
logic itself as the interlingua. The K-X-context-free language of the SALAT 
system \[8\] is an example of this. Friedman \[9\] reports on work by Godden, who 
uses Montague's Intensional Logic. 4 The success of the Rosetta approach depends 
on the correctness of the hypothesis that logically isomorphic grammars can be 
written for interesting fragments of languages. At first sight it may seem that 
we can avoid the necessity to attune the grammars to each other if we do not use 
logical derivation trees but the logical expressions themselves as intermediate 
expressions. However, the important distinction is not between using logical 
derivation trees and using logical expressions, but between making use of the 
form of logical expressions and not doing so. In a powerful logical language each 
meaning can be represented by an infinite variety of logical forms. If a genera- 
tion component has to be able to generate a sentence not only from a particular 
logical form, but also from all logically equivalent forms (which might be the 
result of the analysis component of some other language), this will cause decida- 
bility problems. These are in practice avoided by ~everely limiting the possible 
forms that analysis components may produce and ensuring that the generation 
components can handle these forms (The same holds for systems that use syntactic 
deep qtructures as intermediate expressions). The point I want to make here is 
that this is just another - less explicit - way to attune the grammars of the 
languages to each other. 
The reversibility of the M-grammars used in Rosetta has the advantage that 
the same grammar can be used for analysis and generation. 5 Futhermore it makes 
testing of the system easier: if the analysis component and the generation com- 
ponent of the same language are coupled, each sentence must be one of its own 
possible translations. Ultimately, for most applications it will be necessary 
to give up this nice symmetry, to make the analysis more tolerant and the genera- 
tion more restrictive. But such modifications should be the result of conscious 
decisions and not be mixed up too soon with the incapability to write correct and 
complete grammars. 
NOTES 
i) The relation between the words and the complex labels of the terminal nodes has 
to be defined by a dictionary in combination with morphological rules. This com- 
ponent is not discussed here. 
2) In \[6\] the restricted - context-free - definition of S-tree is used. 
3) For grammars with syntactic variables, the symmetry between M-GENERATOR and 
M-PARSER holds only for 'canonical' D-trees, in which the indices of syntactic 
variables are chosen in a restricted way. This can be done without loss of 
generality. Cf. \[6\]. 
4) Another translation system based on Montague grammar is described by Nishida 
et al \[i0\]. But here the logic is not used as an inter\]ingua, but as the level 
wh@re the transfer, from English to Japanese, takes place. 
5) In the current implementation of Rosetta the rules are not automatically 
compiled or interpreted from the original notation. The analytical and generative 
versions of the rules are 'hand-compiled' into PASCAL. 
ACKNOWLEDGEMENT 
The Rosetta system was designed and implemented in cooperation with Joep 
Rous. 
MACHINE TRANSLATION BASED ON MONTAGUE GRAMMARS 181 

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