~o~ ~ ~iw~ R~mATI~ \]1, ,A~AL IAmUAQ~S. 
~,O~IOi~ AWA~U~S 
Adrian Birb~eeu 
4./NTaOOO~ -/'/ ON. 
It is an esmontl~, cha~acteristic of nat~al languages 
that one uox~l oan be concatenated mt~h certain others to form 
a et~Ing tha~ enters in correct phrases of the language, whale 
it cannot be eo~atamated ui~h othe~s. The same holds true for 
e~rlng~ of words. Such eoncatenable elements axe also "mutually 
eompatible elements" in the sense used b 7 KVAL in a paper dos- 
o~ib~n8 an aleori~ for formi~K naxlnum classes of such elements~ 
Mathematically, a set of ordered pa~s of such "s~tvall~ ¢ompa- 
t£ble elements" forms a x~lat£~. Ever 7 string Of ~ords belonging 
to a langua~ can be ~esax~d as being obtained by euoceesive 
oo~oatenatton of ordered pair8 of mttuall~ oompat£ble elements, 
i.e. as formed by eueoessive oonoatenation of elements of binary 
relations be~ to the languase, Some of these string e~e 
the phrases of the la~ua6e. It is thu possible to define a ~a~ 
mar of ~elation- and 6ene~ate by it all phrases of the lan~m~. 
If ue tx~ to describe by a ~ this gene~atlon~ the ~aph will 
be a network desc~Iblz~ the mhole system of lan~uaKe under oonsi- 
dex~t ion, 
The equivalence between a g~am~a~ of relations and an 
lO-~a~ma~ and the equivalence between a g£~m~r of ~ela~ions 
and a oategortal 8x~mmu~ ~ a\]~mst self-evident. 
B~ usi~ the notations for union, interseotion and Oarte- 
siam prod~ct it is imssible to urite one single formula~ however 
- 2 - 
cumbersome, containinK all phrases belonKinK ~o the ~ un- 
der oonsidoratton, This formula wan also bo inte~tareted as dos - 
crlb~m~ an electrical network uhloh will be t~e eleotwlcal ana- 
logus of the language, 
Lot L i and T. 2 be two lan6~ages. Lot N(L I), ~osp N(L2) 
be the elec~Tlcal networks uslg~d to LI, z~sp L2e It is possl- 
ble to devi~e a systel of eleo~ioal oonneotlons between N(~) 
and N(L 2) such as to obtain eleo~a117 the ~latlon of a 
phrase belonging to L 2. Let's aall thls method "analogue ~ans- 
lation". Because of the ~oat numbex- of elononts involve4, the 
oonst1~action of a oomplete ~tom for analogue ~nsla*rlon m~7 be 
impractioal. Howevert the oo~astr~ot£on of partial neLlorks for 
simulating translation of a linitod numb~ of phEases mat l~ovo 
itself useful for delonat~atlonal \];narposos. 
2. ~SFIN~ION5 i. Let V be a Vooab~LL~WYp that is a 
set the elementsof whioh a~o words. Assooiste d with the voaa- 
bula~ is a operation called oono~tonation which oenmists of 
writlug one or more words al, a2.,..a k one afte~ another, The 
resultln~ sequonoe al a2... a k 4- a ~0 By extension we 
shall call string also a sequence oontainlng one sln~le word. 
The empty word ~ is cbsx~ote~tsed b~ ail = ~ a i = a i 
fox' ovory ai~ Ve ~me B1;r~lJ U ~33 be oa33edrBontoD~0Be 
The set of all sentences generated on V is t~ 7 defini- 
tion the ~ Lo By ~ax~ar G we shall undez~tand a set 
of rules by which it is possible' to generate the language L. 
Let the set of rules oo~sist of the followln~ 
- lists elasslf~is~ all words and stria into sets 
called categories ; 
- rules of the forn 
where~,~i ana~/,(~.l...n), are aatesor~s, and ~ o~ared 
s~r~ a£j b£k • We adair that an:v ~i or~/ day contsdn one 
single word, or even only the empty word ~ ! 
- a list of the categories which are sentences. 
~. Z~AMPIJ 1. Let the gzamaar G R be defined by 
- the voeabula~ 
v = ~ poor, dear, Jo~-, Ri~=a, sleeps, re.~, ~} , 
- the eategc~ies 
- the rules 
- ~he llst of sentences c~aAz~Iz~ online) ° . 
In this simple case, the rules are of form (~), with i=l. 
Now, stax~ln~ froa ~, by successive substitutions we obtain 
.-, . (~: 
The sOt ~iS thus oomposod of 12 ordered pai~s and triplets. 
Wrttt~ down ~he strlngs associated ~o these paJ~s an~ triplets 
we enwaerate the sentences of the language L(~) generated by 
OR I 
poor ~k~hn sleeps 
poor Joha reade 
poo~ Richard sleeps 
poo.~ Rioha~d ~cla 
do8.~ ,,Tohn sloops 
(3) 
- 4 - 
dsa~ John reads 
dsa~ RiohArd sleeps 
dear R£ohard rea&s 
John sleeps 
John reads 
Richard sleeps 
RicJ1a~d ~eads 
.~ 1. Asetl~= (~u~)x~ isbydsf~- 
~ion a binary relatinn on V.Sinilarly, if V 2 is the set of all 
strings obtained by the ,on,arena, ion of two words ("etrlnge of 
2"), then ~ is a blna~ relation ~ ViU V. These rela- 
tions have d~eot lAub~istlo in~e~p~e~atlonm, for ~ aa~ be ~e- 
~Waz~d as the ~elat£on be~leen adjective and nom~ 0 whale 
is the relation be~een nmm ~oup and verb. Of course, these 
simple inte~-~tationm a~e valid within the flailed 
exposed above. 
For convenAenee of des,rip, ion, in what follows we shall 
c~ll a g~a~ma~ of the type 4eflned in 2p o.K, (~R in example I, 
a ~A-a~ of relations, 
Sets lAMe ~ and Y in example i are sots of ordered 
pai~s of strlnss whose oonoatenation leads ~o o~her e%Tinss that 
can belon8 to sentences of the language under oonsideEatlone Con- 
c~tenable elements are also oomDatible elements, by oompatibili- 
~y underetandln6 a eimmet~io nontEansitlve relation. P~gardsd as 
such 0 these elements can be olassifled into olasseeo one of whioh 
is aaxiaal, by means of an algo£itha developed by KARI~REN ~1~ . 
We are Interested to classify ooncatenable eleaents by 
laposln~ the restriction, that follows. 
5. DEFINITION 2. Two categories ~, d~ 2 are called 
dlfferent if there is at least one third catesoz~ such that, 
a) either 
- 5 - 
alb is a stx.ing contained in a~ leaat one sentence of 
lan~ap, i.e. alb belongs to a categor7 of the langu~e, fox 
eve~ al~ 1 0 b ~ ~ , while a2b is contained in no sen- 
tence of ~he l~e. which ever would be 
a 2 , b , 
b) me 
a2b Is a stri.~g oontalned in at least one sentence 
of the la~. for evex 7 a2~ ~, b~ ~ . while alb is 
oon~alned in no sentence of the language, which ever would be 
6. EXAMPLB Re Let's oonsider ~ follceing relations 
atlnts, e e e~ 
Ac~rding tO definition 21 ~4and ~ are of different ca- 
teKoriee since them exist in English catesories 
~.t= I sees..ants, coaes, reeds, sleeps. "''3 
~= { see. ,ant. coae. read. sleep.... 
such ~hat 
the Engllsh language, while 
be a x~tlon belon~m~ to the ~%~h gx~mm~. 
7. ~OR~. F~ every ~a~a~ of relatic~s we ~ find 
an e~aivalen~ lC-~a~a~ (i~ediate-constit~en~ ~am~ar) and 
conve~asl~. 
- 6 - 
The proof followl iemediatel~v f~a ~ observation that 
a~ rule of the fona (1), £.o. 
can be substtbuted by the followlag set of ~Krsmns~ rules 
R----~D 1 
R ---~D 2 
Ooeoeoeeo 
Oooooooo 
@oeeOOeO 
R .----~ D£ 
D I ~ AI~ 
D 2 ~ A2B 2 
eeo~veoeoo 
@ooooooo 
O0oeoooo 
D i ~ lIB i 
@ooooeqJ@ 
e • eeoeoO 
eoooeeoe 
and convorsol~. 
in a grammar of relations, eorresponds a termiaal 
of the equivalent I~-~ and oonversoly. 
8. REMARK 2. From %he equivalence beCween a K~amma~ of 
relations and an IC-~-amma~ i~ folloues also the equivalenoe 
between a ~auma~ of relations and a oateKorial Kz~um8~. The . 
nocessax'y proofs can be found in BAR-HILLEL, ~AIFHAN and SHAMIR 
-% 
- ? - 
90 EXAMPLES 30. a) The l~e described t.u example 1 
oan be generated also by the followi~ lO-.graJam~ z 
- teralnal vocabula~ 
= dee:. sle, s.  ade. 
- au~ vo,,a~,.-'y 
V s: ~S, A, B, C, D} 
- ~'ules * 
S ----~AB 
A -----~ CD 
A -----.'- D 
C ~poort dear, 
D ~John I Richard 
B ----'-sisops, reads 
b) In the oqu£valent oategor:Lal grammar t 
- poo~ t dea~ a~e of oategox 7 n/n 
- John~ Rl~ha~d are of cateKor 7 n 
- Sleeps, reade are of catessry ~\~ 
- S is the sentence oateKory. 
Io. GRAPH\]EAL REI~ESENTATION. We shall associate to each 
gEaRma~ of relations a graph, obselwing the following conventions! 
- each path must be followed from the extreme left to the 
extreme ~tght, along the arrows ; 
- a sentence is a sequence of words found along a path. 
Thus, the ~aph correspondlng to g~ammar ~ in example i 
is 
(~ 
- 8 - 
An example given by the author for the ~reneh language i-\[4j ie 
me couz~ 
/~voit ~: 
la ~petite--------~.poule ----iv . ange 
ma ~ belle///~ 
C~ 
Such graphs are called networks. An earlier example is to be 
found in MARCUS ~53 , Taking into @onsideratioa what has already 
been written at ~ (ie. in remark 4 ) , the above graphs oan be re- 
married as networks of binar~relatioms. Th~ desoribe not only sen- 
tence structures, but also the whole system of the language, 
Now we shall simplify the graph (~) without altering i~ 
topologically 
poor~ --~ 
John sleeps 
dear (BJ 
In the same mamaer, the graph (5) becomes 
le 
ul --~ 
@e --v- 
sol --~ 
- 9 - 
~petit ~i ~ cheval ~ 
uae petite ~poule 
ee~e~graade ~ourii~ 
ms ~' ~ belle 
~court ~ ~biea 
~voit ~ici 
~mange 
(7) 
ii. ELECTRICAL ANAIDGUES. If ~ g~aph like (6) or (7) is 
considered to be aa electrical diagram where oath word is substitut~ 
by a contact, and each arrow by an electrical conductor (wire), the 
result is u electrical analogue of the grammar. In this analogue if 
ome cIsses all comtee~8 correspomdlng to a semtemce, a contimuous 
electrical path is established and the curremt flows from ome extre- 
mity to the other. The detectiom of thim eurran@ is a proof of 
"grammaticality" for the word sequence under consideration. 
Electrical analogues can be designed also algebraically. For 
this it is necessary to proceed ~S ~o//Q~/~ 
from the list of sentences ~, ~2, .... ~/~ 8tarttng 
P L=U~ 
J=l 
- replace each ~/ by its coW'responding rule of form (i),that 
is 
- io - 
p 
j--I i=l 
- replace each~/and esch~iby the corresponding rules of 
type (1),and so on until the ~lght member of the formula contains 
only the words of the vocabulary. Such a formula is (2) in example 1' 
We have now at our disposal a formula enumerating the ordered n- 
~ples associated to all sentences of the language L. This formula 
is to be interpreted in terms of switching algebra as follows s 
- an n-#~ple (Sl,a2,..°s r) correspon~to a series connec- 
tion of the elements al,s2,...ar ! 
- the union 6 ~/ correspond~ to s paralel connection of the 
i=l 
elements ~/! 
- the Cartesian product ~c~/× gi corresponds %o the paral~l 
connectio  of all .eri= c eotlo  sibl. where 
% 
In \[#\] is presented an electrical analogue of the grammar 
described by graph (7). 
12. GRAPHS OF TRANSIA TIONS~ This chapter and the following 
are intended as su~ested applications of the above discussion, to 
the understanding of the process of translatlon. No attempt is ma@f 
to start from more rigorous definitions, 85 may be found for e~- 
pie in /6\] or \[7\] • Here the process of translation of a simple 
sentence from language L I , into language ~, is regaz~led as consi% 
ring of the following operations s 
a) seek the given sentence in the d/ctionsry ~-~ | 
i b) if the whole sentence is found in the d/otionary, write 
e~ 
down the translation found there and ~ the process | 
c) if tBe sentence is not found in the dictionary, divide it 
into two subst~i~gs admitted by the gra-..sr (in fact, immediate 
constituents) ! 
d) seek each of the substr/ngs in ~he dictionary | 
- 11 - 
e) if one substrlag is found in the dictionar~, write down 
its translation as given in the ~ctto.ar~ ! 
f) if one substring is not found in the dictionar~, divide it 
again into two further substr~ngs and then proceed again as indicated 
under d) ! 
6) the prooess stop~ when a strin6 is obtained which contains 
only words belonging to the vocabulary of the language L2. 
A difficulty rises currently during the process of transla- 
tion, and this is due to the fact that many words, or strings com- 
posed of more than one wo~d , admit two or mo~e translations.Such 
is the case with homonyms. Special subroutines have been developed 
to solve t~e problem of homonyms in digital translation of language. 
Suoh subroutines are based on successive step, of conditioned decisi~ 
To quote only a few very simple examples, MARCUS , in \[8J ,gives ske~ 
ohes of algorithms for translating into Roumanian "example" and 
"this "and for solving the homonymy of the French "pas" or ~he En- 
glish "this"° The problem is related to that of sequential unde1~- 
standing of s sentence, as described by ZIEREH \[9\] • 
Let us put the DEoblem somewhat differently. To choose the 
writs word (or subst~i.~ between more than one possible variants, 
we need some supplementary condition, or conditions. A first and 
most important condition is that the right word (substT~g) must 
match grammatically the other words (subst~ings) in the string. That 
is to say that the right word (string) must form with another word 
~subet~iag) in the string an ordered pair belonging to a certain rela. 
tion accepted by the grammar of the language ~. In the graphical 
representation suggested above, the right word (subst~,ng) must 
find itself on a continuous path with the other words (substr~gs) 
contained in the translation of the sentence under consideration, 
For example, let L I be the Englis~language and ~ the 
French. Let the sentence in L I be "we see the bo~'. It must be 
- 12 - 
divided into (we see) (the boy). Putting side by side the correa- 
ponding parts of the English and French graphs, and marking by do~ 
ted lines the mapping defined in the dictionary, we can draw for 
the first substr/ng the graph 
ue 
\ we 
\ 
IY 
no~ 
' ~ \ *" vois 
1 \\ 1 ~ ~voyaz 
s ~oiant I 
VOyOnB 
We choose as translation only that subst~hg that closes our diagram. 
Sim/.larly, for the second substring 
the 
:I q 
/I ! 
/! j, / 
/ / le / 
/¢ 
1/18 
/ p 
lea 
boy 
l 
t 
L gargon 
The final translation is given by 
(we see ) - ~ (the boy) 
l i 
(nous voyons) L (ls garcon) 
There are cases when the condition of gra~latioality is. not 
sufficient. Then a human translator,uses supplementary information, 
like general knowledge of the subject treated in the text, style used 
etc. Graphically, such information may be taken into account, for 
example, by assigning different colours to different types of subject 
Then the diagram must close through paths of different colour. 
- 13 - 
13. ELECTRICAL TRANSLATION. Let us assume now that we have 
at our disposal an electrical analogue N(L I) of the language ~, 
and an electrical analogue N(~) of the language L 2. We can 
further immaglne such an electrical connexion between the two ana- 
lo~-ues that when a contact "ai" closes in N(LI) ~ all contacts cor- 
responJding to the different possible translations of "ai" indica- 
ted by the dictionary ~-~2 are closed in N(L2). Then, when a con- 
tinuous path of contacts is closed in N(L I) , its translation can 
be only a continuous path resulted in N(L2). For the selection 
pf ~he type of subject or of the style used, we can devise a switch 
that makes only the corresponding connections between N(L I) and 
N(~), or in N(~) and N(L 2) themselves. Such a mwitch may have, 
for ex~nple, positions marked • literature, mechanics, electricity, 
electronics, chemistry, medicine etc. 
Some texts may contain sufficient information to enable the 
above switch to find automatically its right position. This idea 
~serves an entirely separate discussion. 
14. CONCLUDING R~ARKS. What was suggested under 13 and 1@ 
are in fact examples of cabled logic. The implementation of these 
ideas for an entire language may encounter tremenduous technical 
difficulties. Designing graphs and electrical analogues for limited 
parts of language may prove however useful for demonstrational pur- 
poses. Thus it would be possible to achieve other models of language 
understanding and translation than those provided by digital pro- 
grams and cunputers. The author feels that this is indeed interes- 
ting, far as explained by ZIERE~ \[9~ ~ the process of understanding 
must not be actually as divided into elementary steps as in an algo- 
x~thm -- "Dazu kommt noch dass beim Verstehen der gesprochenen Spra- 
che dutch den Menschen auch der soziale Kontext und das Erlebnis- 
verm~en des Menschen zum Abbau der angebotenen Information bei~gt. 
- 14 - 
Hierin ist der, Computer dem Menschen unterle~en". 
Another author, SAUVAN \[lo~ , disoussing other subjects, be- 
lieves that a Sequential computer cannot treat adequately a combJ~atc 
rial problem for it has no possibilities of global perceptions "L'au- 
teur est persuad6 que lee recherehes doivent s'orienter vers le~ 
logiques o~bl6es semblables aux structures e@~brales". 
- 15 - 

References

i. HANS KARLGREN, An algorithm for forming a maximum class of mu- 
tually compatible elements out of a partially incompati- 
ble set, KVAL PM ~85, 25.1.68, Research group for quanti- 
tative linguistics, Stockholm. 

2. Y.BAR-HILLEL~ H.GAIFMAN and E.SHAMIR, On categorial and phrase 
structure grammars, Bull.Res.Council of Israel,9 F,196o, 
p. 16. 
Also to be found in 
YI~OSHUA BAR-HILLEL, Language and Information , Addison-Wesley 
Publishing Company & The Jerusalem Academic Press, Rea- 
ding / Massachussetts - Jerusalem 196@. 

3. HANS KARLGI~, Categorial grammar analysis of context-sensitive 
languages, KVAL Ref. No.441, 1968, Research group for 
quantitative linguistics, Stockholm. 

@. ADRIAN BIRBANESCU, ModUle graphique et module @lectrique d'une 
grammaire ~ eonstituants imm@diats. Cahiers de linguis- 
tique th@orique et appliqu@e, V, 1968, Bucharest. 

5. SOIA)MON~ ~CUS, Introductlnn math~matique h la linguistique struc. 
turale, Paris, Dunod, 196~. 

6. KAHEL ~LIK, Semantics and translation of grammars and ALGOL- 
like languages, KYbeznetiks (Czechoslovakia), i, nr.l, 
1965, p.~7-49. 

7. JIRI EDPRIVA, Gsnsrallzation of well-translation of formal lan- 
guages. Kybernetlka (Czechoslovakia). 2, at.#, 1966,p .305- 313. 

8. S.MAHCUS, No~iunea de alg~Itm ~I implica~iile ei lingvlstice,Llm- 
ba rumJn~, X, nr.3,1961,p.2o~-217. 

9. ~NES~ Z~, Kybernetlk und Sprachverstehen. Grundlagenstudien 
aus Kybernetik und Geisteslissans~aft, lo,nr.l,mart 1969, 
p Ii-I@. 

Io. JAC4~E8 SAUVAN,Inf0rmatique Combinatoire et m~ta-informatique,Cy- 
bernetica (Namum), nr.@, 1968,p.229-23~. 
