ON COMPUTATIONAL SENTENCE GENERATION FROM LOGICAL FORM 
Juen-tin Wang 
Institut fur Angewandte Informatik, TU Berlin 
Summary. 
This paper describes some computational 
and linguistical mechanisms in our 
program written in SIMULA to generate 
natural language sentences from their 
underlying logical structures in an 
extended predicate logic. After the 
presentation of the augumented logical 
formalism to deal with illocutionary 
acts,we explain then the basic devices 
used in the generation process:semiotic 
interpretation,orders of quantifications 
or derivational constraints,the referen- 
tial property of variables and the 
Leibniz-Frege idea. Examples from system 
output will be given. 
1.Introduction 
Logical form is one ot the most used 
notions in philosophy,logi c and linguis- 
tics. It goes back at least to Aristote- 
les in his linguistical and logical ana- 
lysis of natural language sentences. 
This direct reference to the immediate 
sentence form which has been characteri- 
stic for the logic of syllogism remains 
unchanged throughout the whole period 
of scholastic logic until the develop- 
ment of the formal predicate logic. Sin- 
ce then,this logical formalism, with or 
without variation and modification, has 
been widely used in the linguistic phi- 
losophy to analyse and study the natural 
language. And it is then the resulted 
representations in logical formalism 
which will be taken as the logical form 
of the analyzed natural language senten- 
ces. This changed notion of logical form 
can be found everywhere in the tractatus 
of Carnap,Quine,Geach,Hintikka and many 
others. And this notion of logical form 
will be now used universally. In recent 
times, a lot of logically minded lingui- 
ts like Lakoff, Harman,Keenan and Kar- 
ttunen have even attempted to put logi- 
cal form into the relationship with the 
notion of deep structure in connection 
with Chomsky~ theory of generative gra- 
mmar. They hold the view that the seman- 
tical representation of natural language 
sentences can be obtained from the for- 
mal logical structures and that these 
semantical representations can be adap- 
ted as a basis for sysntactical genera- 
tion of natural language sentences. 
However,this school of generative gram- 
mar has not given any constructive 
demonstration of their assertions. 
In this paper we do not concern with 
the question whether this theory,the so- 
called generative semantics ,will yield 
a true grammar theory or a genuine the- 
ory of the semantics of natural language. 
We are rather motivated by real needs. 
We have already at our disposal a ques- 
tion-answering data base system which 
uses essentially the language of predi- 
cate logic as the formal query language. 
We need to know how to express these 
logical forms in natural language sen- 
tences. And Since we have to do with a 
question-answering system, we need not 
only to treat logical forms underlying 
indicative sentences but,more important, 
the logical structures which have been 
used by the system as the representa- , 
tions for interrogative sentences. In the 
following we present at first the 
extended logical formalism. We describe 
then the conceptions and principles 
being used in implementation. The pro- 
gram is written in SIMULA. 
2.Logical formalism as semantical rep- 
resentation of natural language senten- 
ces 
The logical formalism which we have used 
to represent the sentence structure of 
a natural language is in its essence a 
many-sorted language of predicate logic. 
In the conception of representation we 
have adopted some ideas from the speech 
act theory of Austin and Searle.Accor- 
ding to this theory, the utterance of 
any sentences in a natural language is 
characteristically performing at least 
three distinct kinds of acts:(1) the 
uttering of words,(2) referring and pre- 
dicating, (3) stating, questioning, 
commanding, promising,etc. The notion 
of referring and predicating should be 
thus detached from the notions of such 
speech acts as asserting,questioning, 
commanding,etc. , since the same refe- 
rence and predication can occur in the 
performance of different complete speech 
acts. In taking account of this distinc- 
tion between proposition and illocutio- 
nary act we make one addition to the 
usual logical formalism. We let the pro- 
--405-- 
positional part be represented by the 
usual logical expression. In addition, 
we have an auxiliary component to repre- 
sent the different illocutionary acts. 
This additional component will be con- 
nected with the left end of the logical 
expression by a convention sign " = ", 
which,by the way, should not be read as 
"equal". A detailed description of this 
extended logical formalism 'is given in 
Habel,Schmidt and Schweppe (1977). Some 
examples can be given as follows: 
Assertions: 
/true/=.all.x2 (.ex.x3 (city(x3) .eq. 
"~okyo ~.and. takeplacecity(x2,x3))) 
(I) 
Requests (Wh-questions) : 
conference(x2)=.ex.x3 (city(x3) .eq. 
"Tokyo. .and. takeplacecity(x2,x3)) 
(2) 
yes-no questions: 
=.all.x2 (.ex.x3 (city(x3) .eq: ~okyo " 
.and. takeplacecity(x2,x3) )) 
(3) 
The illocutionary indicators like "con- 
ference(x2)", which is itself a name 
function, can be compared with the de- 
signator of Woods(1968) in his query 
language formalism. In general, several 
such illocutionary indicators can be 
allowed at the same time; they could 
then lead to the representation of mul- 
tiple questions as discussed by Hinti- 
kka. Here,however, we leave the question 
open, whether this proposed logical 
formalism as a representation symbolism 
is complete and adequate for natural 
language. For example, we do not consi~ 
der whether WHY- and HOW-question can 
also be treated in the same framework. 
It is obvous that this proposal for 
the semantical representation of natural 
language sentences does not follow 
Chomsky "s theory, according to which 
interrogative sentences should be deri- 
ved from non-interrogative ones by the 
application of optional transformations. 
This approach has rather some affinity 
with the suggestion of Ajdukiewicz(1928) 
who has described the logical structure 
of a question as consisting of senten- 
tial matrix(a sentence with one or more 
of its components replaced by variables) 
preceded by an interrogative operator 
"for what x" (or "for what x,y ,z,..." ,if 
the matrix has more than one free vari- 
able). In such cases, we can take illo- 
cutionary indicators as interrogative 
operators in the sense of Ajdukiewicz. 
The proposed way of giving semantical 
representations both to indicative and 
question sentences seems to have some 
advantages. Above all, it enable us to 
deal with question sentences directly 
without using the somehow artificial 
method to paraphrase them as indicative 
sentences or spistemic statements, as 
suggested by Hintikka. Any way, the 
suggested kind of semantical representa- 
tion of question sentences receives a 
quite natural set-theoretical interpre- 
tation. For example, the form (2) used 
for request corresponds to the meaning 
of the set expression: 
{x2 I ex.x3(city(x3) "Tokyo" and. eq. Q O I 
takeplacecity(x2,x3)) 
J (4) 
In such cases, the interrogative opera- 
tors function as quantifiers; they bind 
free variables and thus transform con- 
ditions exhibited in sentential matrix 
into complete closed forms. 
3. Examples 
In order to let the reader have a rough 
impression of what the system can accom- 
plish at the present stage, we give 
below at first some output examples, 
before we step into the scattered des- 
crition of the conceptions and princi- 
ples to be used. The examples taken from 
output consists of pairs of a given 
logical form and its corresponding natu- 
ral language sentence generated. 
/true/=.all.x2(.ex.x3(city(x3).eq. 
"Tokyo'.and. takeplacecity(x2,x3))) 
EVERY MEETING WILL BE HELD IN THE 
CITY'~okyo" 
/true/=.all.x1(.ex.x2((.ex.x3(make- 
journeycity(xl,x3).and.city(x3).eq. 
"tokyo')).imp. (takepart(xl,x2)))) 
EVERYBODY WHO MAKES A JOURNEY TO THE 
CITY'~OKYO" TAKES PART AT SOME MEETING 
person(xl)=.ex.x2((.ex.x4(takeplace- 
country(x2,x4).and.country(x4) .eq. 
"japan')).and. (takepart(xl,x2))) 
WHO TAKES PART AT THE MEETING WHICH 
TAKES PLACE IN THE COUNTRY "JAPAN" 
conference(x2)=.ex.x3(city(x3).eq. 
"tokyo'.and. takeplacecity(x2,x3)) 
WHICH MEETINGS WILL BE HELD IN THE 
CITY "TOKYO" 
person(xl)=.ex.x2(conference(x2).eq. 
"colling-8o'.and. (.ex.x3(takeplace- 
city(x2,x3).and.city(x3).eq.'tokyo')) 
.and.givelectureconf(xl,x2)) 
WHO GIVES A LECTURE AT THE CONFERENCE 
"COLLING-8o" WHICH TAKES PLACE IN THE 
CITY "TOKYO" 
country(x4)=.ex.x2(takeplacecountry 
(x2,x4).and.conference(x2).eq. c¢lling 
-80 ") 
IN WHICH COUNTRIES WILL THE MEETING 
---406-- 
"COLLING-8o'BE HELD 
person(xl)=.ex.x3(city(x3).eq.'tokyo" 
.and. (.ex.x2(takeplacecity(x2,x3).and. 
conference(x2).eq.'colling-8o')).and. 
traveltocity(xl,x3)) 
WHO TRAVELS TO THE CITY "TOKYO'IN 
WHICH THE MEETING "COLLING-8o" TAKES 
PLACE 
person(xl)=.ex.x2((.ex.x4(takeplaceco- 
untry(x2,x4).and.country(x4).eq.'japan" 
)).and.(takepart(xl,x2))) 
WHO TAKES PART AT THE MEETING WHICH 
TAKES PLACE IN THE COUNTRY "JAPAN" 
4. Semiotic interpretation as sentence 
generation basis 
Let us proceed to consider the devices 
for sentence generation from the under- 
lying logical structure. Essentially 
the generation process will be based on 
the semiotic interpretation,called by 
Scholz and Hasenjaeger, of the predica- 
tes and functions used in the logical 
structure. Some of them are listed as 
follows: 
Predicates: 
takepart(xl,x2) =def person xl takes 
part at meeting x2 
takeplacecity(xl,x2) =def meeting xl 
will be held in city x2 
takeplacecountry(xl,x2) =def meeting xl 
takes place in country x2 
makejourneycity(xl,x2) =def person xl 
makes a journey to city x2 
Functions: 
city(x).eq.y =def the name of city x 
is y 
conference(x).eq.y =def the name of 
meeting x is y 
pezson(x).eq.y =def the name of person 
x is y 
The semiotic interpretation strings are 
the building basis for surface senten- 
ces. In this respect the semiotic inter- 
pretation of predicate may be comparable 
with the underlying string in the gene- 
ration tree or phrase-marker which is 
assumed both in the theory of Chomsky 
and in the theory of Montague as well. 
If we look at its actual form more clos- 
ely, the strings given as semiotic inter- 
pretations differ in one essential 
point from the underlying strings adop- 
ted in the school of generative grammar. 
The underlying string in the deep struc- 
ture for grammatical transformation con- 
tains no variable as used in the logic. 
On the ground of this essential differ- 
ence we can make no direct comparison 
between our approach and that of genera- 
tive semantics. 
At the disposal of semiotic interpreta- 
tions of predicates and functions, we 
could already in principle implement a 
program to generate somehow quasi natural 
language sentences from the given logical 
structures. All what we need to do is to 
follow the type of reading the logical 
formula which we have been taught at the 
class room. We have been taught, for ex- 
ample, to read the following logical 
structure 
/true/=.all.x2(.ex.x3(city(x3).eq. 
"tokyo'.and.takeplacecity(x2,x3))) 
as: 
for every meeting x2 it holds: 
there is a city x3,for which it hold: 
the name of city x3 is "tokyo" 
and 
meeting x2 will be held in city x3 
This might be considered as a quasi natu- 
ral language sentence formulation. It 
has above all the advantage of being 
universal to the extent that it can be 
applied to every kind of logical struc- 
tures. And actually a program has worked 
in this style (Habel,Schmidt,Schweppe 
1977). However,this kind of formulation 
is not the usual surface sentence and 
it is also not so intelligible as it 
could. We need therefore to find out an 
alternative which might give us a simple 
and natural formulation . For eample, the 
logical form given above has the meaning 
which can be expressed simply as: 
"Every meeting will be held in the 
city "Tokyo" " 
It contains no formal logical quantifiers 
and no free or bounded variables. We 
describe below some main methods and 
principles which we have used to achieve 
the generation of such surface sentences 
computationally. 
5. Quantification order and derivational 
constraint 
The problem of quantifiers constitutes 
one of major obstacles in the computa- 
tional sentence generation from logical 
structures. As is well known, the order 
of different quantifiers has an influ- 
ence on the meanil~g of the expression 
whether it is in the case of natural 
language or it is in the case of predi- 
--407- 
cate logic. Thus, Peirce has already po- 
inted out that the sentences 
"some woman is adored by whatever 
spaniard may exist" 
and 
"whatever spaniard my exist adores 
some woman" 
have quite different meanings. Hintikka 
and Lakoff have made the same observa- 
tion in their analysis of natural lan- 
guage (but it seems that Chomsky has 
overlooked this fact in his formulation 
of Passive-transformation).This pheno- 
menon that the order in which universal 
and particular quantifier occur is mate- 
rial for the meaning is even more obvi- 
ous in the language of predicate logic. 
Let us consider as example the pre- 
dicate 
personvisitcity(x,y) 
with the assigned semiotic interpreta- 
tion: 
person x visits city y 
The two logical expressions 
.ail.xl(.ex.x2(personvisitcity(xl,x2 ))) 
.ex.x2(.all.xl(personvisitcity(xl,x2 ))) 
which differs from each other just in 
the order of quantification means quite 
differently. In the process of sentence 
generation from logical structure we can 
thus not simply take the semiotic inter- 
pretation string and substitute for its 
variables the corresponding types of 
quantifiers. In other words, the opera- 
tion of "quantifier-lowering", as Lakoff 
has called it, can not be applied in all 
cases without pertinent differentiation. 
In our example, it can be applied in the 
first case and yields the correct sen- 
tence: 
"every person visits some city " 
However,its direct application would 
lead rather to incorrect sentence in 
respect to the secand logical form. It 
has rather the meaning 
"some city will be visited by every 
person " 
The regularity for the possibility of 
substitution can be perceived if we look 
at the semiotic interpretation string 
and consider the patterns of the follow- 
ing logical forms together: 
.all.x1(.ex.x2 
.ex.x1(.all.x2 
personvisitcity(xl,x2) 
personvisitcity(xl,x2) 
II. 
.all.x2(.ex.xl 
.ex.x2(.all.xl 
personvisitcity(xl,x2) 
personvisitcity(xl,x2) 
It is then obvious that only in cases, 
while the order of logical quantifiers 
is in the same sequence in which the cor- 
responding variables occur in the given 
semiotic interpretation, the operation 
of quantifier-lowering can be directly 
carried out. And it yields correct 
sentences. In other cases such as in (II 
),it is without measures not possible. 
This kind of regularity has been also 
observed by Lakoff in his discussion of 
the notion of derivational ;it occurs 
in the transformational derivation of 
surface sentences from the underlying 
deep structures. Without going into the 
details of his final modifications,the 
derivational constraint means roughly 
like this: if one quantifier commands 
another in underlying structure, then 
that quantifier must be leftmost in sur- 
face structure. He uses the derivational 
constraint as a means to rule out certa- 
in kind of transformational generation 
of incorrect surface sentences. Our aim 
is ,however, not to block out but to 
obtain correct and meaningful surface 
sentences from meaningful logical struc- 
tures. We thus try to find out means so 
that the condition of derivational con- 
straint can always,or at least to a 
large part, be fulfilled. For this pur- 
pose we introduce the notion of the 
associated forms of the semiotic inter- 
pretation of the given predicate. We add 
for example to the original semiotic 
interpretation 
"person x visits city y" 
its associated form like 
(5) 
"city y will be visited by person x" 
(6). 
It will be simply stored. In dependence 
on the orders of quantifiers the corres- 
ponding semiotic interpretation string 
will be selected. By this additional 
means, correct sentences could then be 
computationally generated from the lo- 
gical patters mentioned in (II). 
The same problem occurs with the trea- 
tment of logical structures underlying 
Wh-questions (which, who, etc.,). In our 
conception and in accordance also with 
the theory of Hintikka,the interrogative 
operators has the quantification nature. 
They subject thus to the same derivat- 
ional constraints. We use thus the asso- 
ciated semiotic interpretation strings 
in the required cases. By this means, we 
can generate computationally from the 
logical structures 
person(xl)=.all.x2(personvisitcity 
(xl,x2)) 
city(x2)=.all.xl (personvisitcity(xl,x2 
--408 
)) 
the following interrogative sentences 
respectively: 
"Who visits every city" , 
"Which cities will be visited by every 
person" 
It is of interest to note that with this 
device the topic of interrogative sen- 
tences has been treated and solved for 
the simple cases at the same time. In 
general, the device of associated forms 
of the semiotic interpretation,which 
from the linguistical viewpoint relate 
to each other transformationally, will 
be extensively used. Among others, it 
will be applied in the treatment of the 
relative sentences. In other words,asso- 
ciated form like 
"who makes a journey to city y " 
will be stored together with the given 
interpreted predicate; and this asso- 
ciated for~ ~ill be used eventually for 
relative sentence formation.We return 
to this problem below. 
6. Referential property of variable, 
relative sentence generation and 
and property of connectivity 
In computational sentence generation 
from the underlying logical structure 
we make an extensive use of the refe- 
rence nature of the variables. Variables 
have been called by Quine as pronouns 
of logic and mathematics. The referen- 
tial character will be used by us as a 
kind of red thread in building up the 
composed sentences. This feature shows 
clearly in generating sentences with 
relative clauses. Let us consider as 
example the logical structure 
person(xl)=.ex.x2((.ex.x4(takeplace- 
country(x2,x4).and.country(x4).eq. 
"japan')).and. (takepart(xl,x2))) 
The variable xl in the interrogative 
operator, namely person(xl), indicates 
the topic of the question concerned. 
This topic is in general specified by 
the composition of predicates and func- 
tions in a certain way which is expres- 
sed by the logical matrix. The generation 
of the corresponding interrogative sen- 
tence means to express verbally this 
composition of predicates and functions 
after the given prescription in matrix. 
In making use of the referential pro- 
perty of variavles, it is seen that the 
topic will be characterized at first by 
the predicate 
takepart(xl,x2) 
On this ground its associated form of 
semiotic interpretation, namely 
"who takes part at meeting x2" 
will be used as the main building compo- 
nent of the question sentence to be 
generated. By means of the variable, we 
can find that this predicate 
takepart(xl,x2) 
is connected directly with the predicate 
takeplacecountry(x2,x4) . 
In other words, the variable x2 contain- 
ed in the predicate 
takepart(xl,x2) 
is in its turn specified by the predica- 
te takeplacecountry(x2,x4). We use 
thus in consideration of its modifica- 
tion character the corresponding associ- 
ated form of semiotic interpretation, 
namely 
"which takes place in country x4" 
to build up the relative clause. In the 
same way, we find that the variable x4 
contained in the predicate 
takeplacecountry(x2,x4) 
is referred by the name function 
country(x4) , 
whose function value indicates the name 
Japan. This constant will be thus inser- 
ted at the place x4. The termination of 
these connecting and inserting processes 
lead then to the generation of the sen- 
tence 
"Who takes part at the meeting which 
takes place in the country "Japan'" 
In connection with the referential fea- 
ture of variables it is of interest to 
note that all the logical structures 
which we have used in our question-ans- 
wering system shows a remarkable proper- 
ty which we have called the property of 
connectivity. A logical structure is 
called to have the property of connec- 
tivity, if in the case where it contains 
more than one predicate or function each 
of its predicates ~d functions shares 
some argument with others,i.e, has com- 
mon variables with other functions or 
predicates. 
It is on the ground of the property of 
this connectivity that we can even let 
the program processing under certain 
circumstances be driven by variables, 
such as explained just above. On the 
contrary, let us consider the following 
logical structure: 
/true/=.ex.x1(.ex.x2(.ex.x3(city(x3) 
.eq. "tokyo'.or.takepart(xl,x2)))) 
Since the function city(x3) and the 
predicate takepart(xl,x2) do not share 
any common argument, this logical form 
--409-- 
does not have the defined property of 
connectivity. Its corresponding surface 
sentence can therefore not be computed 
by the process driven by variables• Ins- 
tead, a different procedure must be ap- 
plied. At present stage, we let, how- 
ever, such types of sentences out of 
our consideration. 
The usefulness of variables is not ex- 
hausted in relative sentence generation. 
In general, we intend to use it to dif- 
ferentiate the varied patterns of the 
logical forms concerned. And as a result 
of this differentiation, sentences of 
varied patterns will be generated. Let 
us consider the following simple logical 
form: 
person(xl)=.ex.x2(takepart(xl,x2).and. 
givelectureconf(xl,x2)) 
For such pattern, no attempt to generate 
relative sentence will be made. Instead, 
it tries to express the surface sentence 
as follows: 
"Who takes part at some meeting and 
gives a lecture at this meeting " 
Our program is thus in trying to discern 
as much of logical patterns as possible. 
It works after them. 
7. Categorical and hypothetical senten- 
ces, idea of Leibniz and Frege 
In our computational sentence generation 
we have made use of an old idea, which 
goes back at least to an observation 
made by Leibniz in his famous nouveau 
essais sur l~ntendement humain. In the 
classical logic, one is customed namely 
to divide the judgements or assertional 
indicative sentences into three major 
types: 
categorical,hypothetical and dis- 
junctive 
Leibniz has remarked that in some cases 
an actual hypothetical judgement can be 
expressed in a categorial form. This 
regularity is discussed also by Frege 
on the relation between auxiliary sen- 
tences (Beis~tze) and conditional sen- 
tences (Bedingungss~tze)in his essay 
0ber Sinn und Bedeutung. According to 
Frege the cinditional or hypothetical 
sentence 
"Wenn eine Zahl kleiner als I und 
gr~Ber also ist, so ist auch ihr 
Quadrat kleiner als I und gr6Ber als 
O " 
can be expressed in a categorial form: 
"Das Quadrat einer Zahl, die kleiner 
als I and gr~Ber also ist, ist kle- 
iner als I und gr~Ber also " 
In our system design, we have adopted 
this old conception. From the underlying 
logical implication structure its sur- 
face sentence will not be generated in 
hypothetical, but rather in categorial 
form.' This approach has its practical 
and stylistic advantages. It can be seen 
in consideration of the following logi- 
cal form: 
/true/=.all.xl (.ex.x2((.ex.x3 ( 
makejourneycity(xl ,x3) .and.city(x3) 
• eq. "tokyo" ) ) . imp. (takepart (xl ,x2) ) ) ) . 
In following this line of thought,the 
corresponding surface sentence will be 
generated by the system as follows: 
"Everybody who makes a journey to the 
city'~okyo" takes part at some 
meeting " 
It is natural and simple. For its gene- 
ration we need no more additional meth- 
ods than the ones which have been at our 
disposal:the quantifier-lowering and 
formation of relative sentence. The only 
thing which we must take care of is to 
choose the semiotic interpretation str- 
ing of the conclusion rather than that 
of antecedent as the main building 
component. Otherwise, the meaning would 
be distorted. 
The usefulness of this conception of 
Leibniz and Frege consists for our pur- 
pose, above all, in the fact that it 
can be even extended to the treatment of 
logical structures for interrogative 
sentences. Without using this idea,the 
surface sentences to be computationally 
generated would have a cumbersome look. 
This feature may appear clearly,if we 
try to deal with the following simple 
logical structure: 
conference(x2)=.all.xl ((.ex.x3( 
makejourneycity(xl,x3).and.city(x3) 
.eq.'tokyo')).imp. (takepart(xl,x2))) 
It is a logical form underlying an in- 
terrogative sentence; it contains the 
logical form mentioned just above 
almost as component. In combination of 
this Leibniz-Frege idea with the other 
principles like referential property of 
variables, topic handling and formation 
of relative sentence which we have des- 
cribed above the system yields then 
without other detour the interrogative 
sentence: 
"Which meetings will be visited by 
everyone who makes a journey to the 
city "Tokyo" " 
--410-- 
8. General remark and discussion 
We have above described some main con- 
ceptions and principles upon which we 
have built up the program. The system 
works essentially after logical patterns, 
after certain features of logical struc- 
tures such as connectivity, the occu- 
rence of implication sign and so on. It 
is thus properties-oriented and not 
syntax-driven. It is needless to say 
that our program can not deal with all 
kinds of logical structures. This is 
also not our original aim, besides the 
fact that,as Chomsky makes remark about 
the nature of deep structures, not all 
logical structure can underly or have 
a meaningful surface sentence. From the 
right beginning we have confined ourself 
to just a specified set of logical stru- 
ctures used as a formal query language. 
It is remarkable that for such a set of 
logical formscertain regularities and 
patterns can be generally established 
and be used to generate meaningful 
surface sentences computationally. The 
progress will depend to a large extent 
on the careful observation of logical 
patterns and insightful linguistic ana- 
lyses. 

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