CONTEXTUAL GRAMMARS 
- Solomon Marcus - 
InsUtutul de Matematica 
Str, Mihal Eminescuo 47 
BuchareSt 9, ROMANIA 
In the following, we shall introduce a type of generative 
grammars, called contextual grammars. They are not comparable 
with regular grammars- But every language generated by a con- 
textual grammar is a context-£ree language. Generalized con- 
textual grammars are introduced, which may generate non-cox,- 
text-free languages. 
Let V be a finite non-void set ; V 
lary. Every finite sequence of elements in 
ia called a vocnbu- 
V is said to be a 
string on V. Given a string x = ala2...an , the number n is 
called the length of x. The string of length zero is called 
the n tring and is denoted by r~J . Any set of strings on V 
is called a language on V. The set of all strings on V (the 
null-string inclusively)is called the universal language on V. 
By a n- we denote the string a...a, where a is iterated n 
times. 
Any ordered pair (u,v~ of strings on V_ is said to be a 
contex~ on V. The string x is admitted by the context 
<u,v> With respect to the language L if u~ G L. 
Let .~ be a finite set of strings on the vocabulary V~ 
and let@be a finite se@ of contexts on V. The triple 
(v,~, ~)) (1) 
is said to be a contextual l~rammar ; V is the vocabulary of 
the grammar, ~ is the ba_s_e_ of the grammar and ~is the co m~,- 
-2- 
textual ccmoonent of the grammar. 
Let us denote by ~ the contextual grammar defined by 
(1). Oonsider the smallest language L on Vj fulfilling the 
following two conditiom8 
(~J Iz ~ and <u,v>,(~), th-- ~=,L. 
The language L is said to be the lsmguage generated by 
the contextual grammar G. This means that the language gene- 
rated by G is the intersection of all languages L fulfill- 
ing the conai~ions (~) and (pj . 
A language ~L is said to be a eonteF~ual language if 
there exists a contextual grammar G which generates L. 
Proposition i. Eyer~ finite language is a cont~ual lan- 
Proo__f. Let V be a vocabulary and let ~ be a finite lan. 
guage on V. It is obvious that the contextual grammar (V,L4jO), 
where 9 dauotes the void set of contexts, gauerates the lan- 
guage L I. The same language may be gamerated by means of the .g ® 
contextual grammar (V,I~ , where is formed by the nu~ 
cont ex~ only. 
Two contextual grammars are called e~uivalemt if they gems- 
same language. The grammars CV,LI, O ) and (V,~,~ rate the are 
equivalent, since they both generate the language ~ 
The converse of Proposition 1 is not true. Indeed, we have 
Proposition 2. The universal language is a contextual lan- 
guage. 
ProOf. Let V = ~alLa2,...~ ~. De~ote by I~ the umiver. 
Sal language on V.'T.et us put ~" S~.~.~ and t <~''i" " 
C 
-3- 
~,a~, ,...s(~,ian> ~ It is easy to see that thegrammar 
. 
(V,~ generates the universal language on V. 
Remarks. If we put, in the proof Of Proposition 2, LI-V 
instead of h =~' then the grammar (V_,h,@) does not ge- 
nerate the universal language on V, since the language it ge- 
nerates dDes not contain the nu/l-strlng. 
In order to illustrate the activity of the grammar (V,~, 
~defined in the of let consider the proof proposition 2, US 
particular case when the vocabulary iS formed by two elements 
only : V =(a.b~. The general form of a string x on V is 
x = a ~b ~a-~b~...a ~b~ , where il, Jl , i2,j2,...,~,j ~ are 
arbitra~non-negative integers. In order to generate the string 
x, we start with the null.string @@ and we apply il times the 
context ~ ,a~ . The result of this operation is the string 
a 11 ,to which we apply Jl times the context <~,b> and obtain 
the string al~ ~I . Now we apply i~ times the context <~,a> , 
than J2 ti~es the context ~,b_> and we continue so al- 
ternatively. ~hen, ~dter 2p-2 steps, we have obtained the 
J = ai bJlai b 2ooo LP" ib -i , it is 
ply .~ ti~es ~ne context ~@,~ and, to the string so ob- 
tained, jp times ~:e contex~ ~gb> , in order to generate 
completely Uhe string Xo 
Haskell Curry considered Ghe larlg~age L = {abn~ (n=l,2~..o) 
as a model of ~he set of natural numbers \[5~ o We call L the 
language of Curry. 
Prooosiuion 3. The language of Curry is a contextual lan- 
gu~eo 
proof. The considered language is generated by the grammar 
(V,LI,~) , where V= ~a.b~ , I~ =~a 3 -nd~ ~<~,b~. 
We recall that a language is Said to be regular if it may 
be generated by means of a finite aatoma$on (or, equivalently, 
by means of a finite state grammar in the sense of Ohomsky). 
Proposition 4. There exis$~ a contextual language which is 
Proof. Let us consider the language L = ~a-nb n} (n=l,2,...) 
If we put V = {a°b} , L 1 = ~ab~ and ~ ~<a.b>} , then it is 
easy to see that L is generated by the con~extuai grammar (V, 
LI, ~. On the other hand, L is n~t a regular language. This 
fact was assel~ed by Ghomsky in \[3~ and\[~\], but the proof he 
gives is wrong. A correct proof of this assertion and a.discus- 
sion of Chomsky' s proof were given in \[~\]. and ~. 
Propositions 2,3 and # show that there are many infinite 
languages w~ioh are oontextual. This fact may be explained by 
means of 
P~posi~ion 5. If the set ~ is non-void and if the set~ 
contains at least one non-nu/1 contex~ I ~hen the contextual gram- 
ma___r_r (V.Ll, ~ ~enerate s an infinite language. 
Proof. Since L A is non-void, we may find a string x be- 
@ longing to ~i o Since contains, at least one non-nu/\] context, 
@ • let ~u,v~ be a non-null context belonging to . l~rom these 
assumptions, we infer that the strings 
, u2xv 2, . •. ,un~, ,.. 
are mutually distinc~ and belong all to the language generated 
by the grammar (V,I~,). Thus, ~his language is infinite. 
The converse of Proposition 5 is true. Indeed, we have 
/ / 
-5- 
Proposition 6. If the contextual 6rammar (V, LI~ gau~ 
rates an infillite language, then ~Ll. is non-void, whereas@ c0n- 
rains a no~-nult context. 
proof. Let L be the language generated by ~V)LI~. If 
is void, L is void too, hence it cannot be infinite. If 
contains no non-null context, we have L = L I. But ~d is in any 
ease flni~e ; ~hus, L is finite, in contradictiom With the h vpo- 
thesis. 
Since there are contextual language which are not regular 
(see Proposition 4 above), it would be interesting to establish 
whether all contextual languages are context-free ls~guages. The 
amswer is affirmative : 
Proposition 7. _Every contextua~ lan?.ua~e is a context-free 
PrP~oof. Let b be a contextual language. If L is finite, 
it is a regular language. But i~ is well knowm that every regu- 
lar language is a context-free language. Therefore, L is a 
context-free language. Nowe let us suppose that L is infinite. 
Deao~e by G = (V,,L l, a contextual grammar which generates 
the language L, In view of Proposition 6, L I is non-void,whe- 
ream there exists an integer i, l~ i~p , such that the con- 
~ext ~ui,vi~ is non-nu~ Joe. at least one of the equalities 
ui =co , v i =~ is false. Let us make a choice a~d suppose 
tha@ ~.i ~ ~ Let L~ = {xcA,xp_, ... 9~a} and (~) ={<ultVl> , 
~.,U,.~,V."y} . We define a context-free grammar ~)..@| 
as follows. The terminal vocabulary of ~ is V. The non- 
terminal vocabulary of ~ contains one element only- denoted by 
S - which is, of course, the axiom of the grammar ~ . The ter- 
-6- 
minal ~ rul es o f ~' are , 
S -->_x 1 , 
g--~x a , 
S.--* _Xn 
whereas uhe non-terminal rules are 
S ---> u~5% v-i ' 
S ---) uqS v~ , 
It is obvious tha~ the number of terminal rules is equal to the 
number of strings in ~ , whereas the number of non-terminal 
rules is precisely the number of conuexts in ~. Among the non- 
terminal rules, there is one at least which is non-trivial : it 
is the rule S ---> UiS v~. , where '*4 {~ " 
It is not difficult to nrove that the grammar ~ generates 
the given language L. Indeed, the general form of a string in 
L__ is 
where yG V and 
<~i , V~ >E~ for s = 1,2,...,p. 
In order to generate the considered string we begin by apply- 
ing --Jl $imes She rule 
In this way, we obtain the expression 
h 
The next step consists in applying J~2 times the rule 
v 
-7- 
-a ~ ,~2 s v~.~. , 
which yields the expression 
J~ Ja -Ja -J~ s Ha 
N 1 )t~... , -7! l 
Oontinuing in this way, we arrive, after 
pression 
,~z "ia ~-i s J~-\]. J2 ~a 
us-1 ~,a .... .'$1 - "-%'i "'" ~12 ".% " 
Vie now apply j.p times the rule 
and thus we obtain the expression 
p-1 steps, to the ex- 
--Jp.~ .~!2-1 . Ja ':ll .J.l. ,.i2. u jp-1 ~ S - - 
"Ull '~i2 .... ~-l . V~p ~-l "" vi2-V-il 
where, by applying the terminal rule 
S---@ Z, 
the considered string is completely generated, Thus, we have 
proved that L is contained in the language generated by ~ , 
Conversely, let z be a string generated by ~ . The ge- 
neral form of this generation involves sev(ral consecutive ap- 
plications of non-terminal rules (the number of these applica- 
tions may be eventually equal to zero) followed by one and only 
one application of a terminal rule. It is easy to see that the 
result of this generation • is always a string of the form (2). 
Thus we have proved that the language generated by ~ is con- 
taiued in L, In view of the precedim~ eonsiderations, L iS 
precisely the language generated by ~ o 
Proposition 7 easily permits to obtain simple examples of 
..... J 
-8- 
languages which are not contextual l~guages. For instance, ~he l~- 
~uage of Kleene~an~ (m=~,2,...), the first example of an infinite 
language which is not regular, is a very simple example of ~ 
contextual language. It is enough to remark that the sequence 
~n2} (~ = 1,2,...) contains nO subsequaucewhioh is an infinite 
ari~hmebio progression ~ (We have (n+l)2-n22~+l and lira (~+i)=~, 
therefore for every subsequence of ~n2} the difi'erance of two conse- 
cutive terms has the limiu equal to +oo wh~ n-@ ~ ). But a result 
of\[4\] asserts, among others, that given am infinite contex~-f1~ee lan~ 
guage L, the set of integers which represent the len~hs of the 
strings in L contains an infinite arithmetic progression. It follows 
uhab b~Je language of Kleene is not context-free and, in view of Pro- 
oosition 7, it is not a conbex~ual language. T~ sa-~ ~a~T ~@l{ow* ~,~ ~h~-,~ :3.A,~. ®~ \[g'J, ~,,#¢. 
A natural question now arrises : Do there exist non-contextual 
languages a~ong context-free languages 7 The affirmative answer 
follows fro~ the following remark : 
The converse of Proposition 7 is not true. Indeed, we have 
Prooositiou 8. There exists a. cont e.~-free language which is 
not a contextual language. 
Proof. Let V = ~a,b~. In view of a theorem of Gru~Lkl ~ 
~__.-----~there exists, for every positive integer _n~ a context-free 
language I~ on V, such that every context-free grammar of I~ 
contains at least n non-terminal symbols. But, as we can see in 
the proof of Proposition 7, every contextual language may be gene- 
rated with a context-free grammar containing only one non-~erminal 
symbol. Therefore, if _n ~ 2, ~ is not a contex~usl l~guage. 
Proposition 8 suggests the natural question whe~bsr ~bere 
exist regular languages which are not contextual lan~ages. The 
•I 
-9- 
answer is affirmative : 
Pronosition 9- There exists a regular language which is not a 
context ual language. 
Proof~ Let us consider the laugaage L = {abm-~c~a.~ n,) ~,n= 
=1,2,...), which was used b~ H.B.Curry \[5\], in order to descrlbe 
the set of mathematical (true or not) propositions. This language 
is regular, since it can be generated by the rules S--> Abj Ac->Ab, 
A.--~ Ba, B--> CC. , G_--~ ~ , C--~ Db, .D_--~ a, We shall show that .L 
is not a contextual language, Tndeed, let us admit that the contra- 
ry holds and let G = <V,~, ~> be a contextual grammar of L_ 2 
Here, the gene_.-al form of a string in L is 
~ ""-us ~I x~ -~ ... vi= (3) 
, wh eas (t = 1,2,...,=) where 
"'',Pn are arbitrary positive integers. This means that ul,~2,... 
.°- 
---,_Un , Vl,Y2,-..,v ~ in the expression (3) are formed only by 
those elements of V whnse number of occurences in the strings of 
L is unlimited. Only h satisfies this requirement. It follows 
that in any string of .L both occurrences of sand the occurence 
of ~ are terms of the string x in (3). But this implies that 
the intermediate terms between the occurrences of a are terms 
of x, hence we can find two strings y and 
, m l 
The string y is obvioasly .the null-string ~o 
the form 1~. , hence " 
z such that 
,whereas z is of 
But m may be here an arbitrary positive integer. Therefore, since 
-lo- 
X6~ , it follows that ~ is an infinite se~ of mtrimgs. This 
fact contradicts the assus~tion concern_tug G ! v, is ~t a con- 
textual language and Proposition 9 is proved. 
The contextual grammars may be generalized in order to ge- 
nerate some lauguages which are not context-free. 
A generalized contextual ~r~mmar is a quadruple G =~ , 
,L2, ~ , where V, L I and ~have the same meal~g as in 
bhe definition of a contextual grammar, whereas J'2 is a finite 
set of strings on the vocabulary V. We define the language L G 
generabed by G in the following way : Y~ is a language on V 
a~d xe~ if and only if we may e~press x in the form 
.- 
where z~, y~Le , <ui,Yi>~for i : 1,2,...,n and 
pl,P2,...,pn , p are positive integers such that pl+P2..,~n=p. 
Every language generated by a generalized contextual grammar 
is said to be a generalized contextual lsnguage. 
I~, in the delini~ion of G, we take L~ =~c~}, G is equi- 
valent ~o a contextual grammar ! the lang,.% is then precisely 
the language generated by the contextual grammar ~V,LI~.In_ 
deed, the general form of a string in the contextual language ge- 
nerated by ~Y~LA, ~ is 
l 
P~a Pn Pa P2 Pl 
p roy ed 
~roposition lo. \]~ery oontex~ual language ~s a ~eneralized 
context ual lan~uaKe, 
- ll- 
We may consider a conte~ual grammar as a parbicular case 
of generalized contextual grammar, .by ideatifyimg the contextual 
grammar ~¥'~1~ with the generalized contextual grammam~,V,,~, 
" 
It is interesting to point out that somet~imes a cont~ual 
language may be easy generated by a generalized contextual gram- 
mar which is not a contextual grammar. For instance, let. us con- 
sider the l~.~e L= (~=} (~X,2,...) . ~ ~is, or the 
proof of ProDosition A, L is a contextual language. We map ge- 
nerate L by the generalized contextual grammar (which is not 
a co=textual ~r~a~) <v, h~> , where v : {~,b}, .Li_- 
\[c~}, ~ = Ibm, ~= \[a,~ . It is known that ,~_ is not re- 
S~L%ag. We ma~ give a similar example, wi~h a language which is 
regular. In this respect let us consider the language of G~x~V~.~. 
In view of Proposition 5, it is a contextual language. It is a 
regular language too~ since it may be generated by the regular 
gramm~r contain~ ~he following two rules : q--~ Sb and S--> a. 
Now let us consider the generalized contextual grammar < ~i' 
This grammar generates the language of Curlew, but if'is not a 
cent ext,~ al gran~nar. 
~ow let us show that generalized contextual languages are 
an effective generalization of contextual languages. 
Propo .sit ion ii. Th ere _ exist s a_g en=e=~ iaed_gA~nt ext ua! 
language which is ~IQ~ a eon~ext~sl language, 
~, Let us consider the language T, = £an_b.n~.. n} (n:=-l,2,. 4 
It is known that this language is not context-free (see,£or 
instance,66\] ,p.~). 7n view of Proposition 7, every contextual 
- 12- 
language is a context-free language ; hence~ ~ is not a con. 
textual language. Now let us consider the generalized contextual 
gr~m~ G = <V,~,~2,~>, .here v = £~,~ , ~ ~, ,~{~ 
and~ ~(~a>~ . It is easy to see that G generate§ the iea- 
guage L. 
Yrom the proof of Proposition ll it follows immediately; 
Proposition 12. There exists a ~eneralized contextual lan- 
g u_~e which is not a ~.nnteYt-f~ee language. 
We may now ask whether the converse of Proposition 12 is 
true. The answer is given by 
Proposition 13. Th ere exists a cont ext-free~a~e~ 
even a regular language,~ which is.,not a generalized contextual 
!~ua~e.-~ 
P#oof. We may consider the language L = ~sbmc_abn-} (~,n= 
=1,2,...) used in the proof of Proposition 9. It was showed in 
the proof of Proposition 9 that L is regular. Let us admit 
that ~ is a generalized contextual language. Given a string x 
in L, its representation is of the form 
Pl P2 Pn P P~ P2Pl ~m ~ : ui u~ ....~n..~. y'v." ... v2v i 
where ~ui,vi~ ~ (i = 1 .... ,n),ZG~, y~L2,pl+...+pn = p end 
G = ~V, L1,L~, ~ is the grsmmar of L. By a reasoning similar 
to "that used in the proof of Proposition 9, we find that for 
every positive integer m there exists a string z in \]i I such 
that 
z = abmcab s~, 
where s is a non.negative integer, depending mf m. But thls 
means that ~ eontain~ infinitely ma~ strips. This fact con... 
tradicts the definition of a generalized contextu~ grammar. It 
/ 
L 
- 13 - 
follows that L is not a generalized contextual language, 
It is to be expected ~hat every generalized contextual lan- 
guage is a contex~-s~itive language. But the construction of 
the corresponding context-sensitive grammar seems to be very 
complicated, if we thin~ to the generation of the language 
~u.A.~reider has introduced a new type of grammars, called 
gralamatlkl) and defined i~ neighborhood ~ira~.L~ars (okrestnostnye ' 
the following way (\[4o); see ~4\]. Our presentation is some 
what different). Given a finite set V called vocabulary, two 
strings x and y on V, and a context <u,v> on V, We say 
that the pair ~u.v> ,y) is a neighborhood of y with respect 
to x if we can find two strings z and w, such that x=zu~vw. 
Every pair of the i or~ ~<u,v> , ~\] , where ~u,v> is a context on 
~, Whereas y is a string on V, is called a neighborhood on 
V. Let us consider an element e which does not belong to V ; 
G will be called the bo~3dary element. A neighborhood grammar is 
a triple of the form ~ V, e ,~, where V is a vocabulary, 
is the boundary element and ~is a finite set of neighborhoods 
on the vocabulary VU(e} . Let L be a l~aguage on V. 2e say 
that L is generated by the considered neighborhood grammar if 
~i every string x of the form x =~ye (with ymL).and only 
in such strings - there ~ists in ~, far every tera a i of 
X=~la2...a s , a neighborhood of a i with respect to x. 
Neighborhood gray, mrs are closely related to the notion of 
context, since this notion occurs in the definition of a neigh- 
borhood. There is another notion, due to Ja.p.L.Vasilevski~ and 
- 14 - 
~.V.Ghom~ak6v (see ~he refermnce in~2\],p,~o), which e~lains 
this fact. Following these authors, a grammar of contexts (this 
name is imp_roper, since no context occurs among its objects) is 
a triple <V, e ,9> , where .V and @ have the s-me meaning as 
in the definition of a neighborhood grammar, whereas Q is a 
finite set of strings on the vocabulary Vt3{e~ • This grammar 
generates the language _L on V in the following way : x6 
if and only if for every string y and a~y strings z and w 
for which there exist strings u and v such that @ x@ = 
= uzyuv we have either 
l) y = rasp , where sE Q, whereas the strings m and p 
may be eventually 
or 
2) (~x@ = urynt, where qr = z , n t = w mad ryn is a 
string belong~g to Q. 
A string belonging to Q is said to be closed from t~ 
le~ (from the right) if its first (last) term is @ . A string 
belonging to Q is said to be ~ if it is closed bosh from 
the left and f~m the right. 
A grammar of contexts is said ~o be k-bounded if every 
non-closed string of _~ is of length _k, whereas every Clesed 
string of ~ is of length not greater than _kj 
An important theorem of Bor§~ev asserts the equivalance 
between languages generated by neighborhood grammars and lan- 
guages generated by k-bounded grammars of con~s (~£3,p.4o). 
Since grammars of contexts and contextual grammars have some 
similarities in their definitions, it is Interesting to esta- 
blish more ~xac~ly the relation b~een them. 
v 
- 15 - 
Proposition 14. There exists a contextual language ~hioh 
is regular, but which is not a neighborhood language. 
Proof. Let us consider the language L = ~a~n~ (n=l,2,...). 
This language is regular, since it is generated by the regular 
grammar consisting in the rules S ~ ~a, T--->Ua , U--->Ta, 
--->a, where ~ is the start symbol, La~ is the terminal vo- 
cabulary, whereas {S,T,U} is the non-terlainal vocabulary. Let 
us consider the contextual gramnu~r G =~ {a} ,{CO}, {~a,a>~. 
I@ is easy to see that G generates the language ~ $ therefore 
L is a contextual language. 
We shall show that L is not a neighborhood language. In 
this respect, our method will be the following. We shall consider 
all systems of possible neighborhoods of the terms of ~he string 
0aae and we shall show t~}at every such sysbem is either a 
system of aeighborhoods of the ~erms of every string Cane (n= 
= 2,3,4,...) or it is not a system of nei@\]borhoods of the terms 
0t the string ea@e . It is easy to see that the first ~erm of 
the string @aa@ admits ~he following neighborhoods : 1)e , 
2) Ca, 3) @aa, ~) eaa~ . The second term has the neigh- 
borhooas : l) G_a,~)-a~) aa , 4) ~e ., 5) e_a_a , 6) e_~ae. 
The neighborhoods of the third term are : i) e@a , . 2) aa, - 
~) a, ~) _ae , 5) eaa8 , 6)_aaE) . The lass term has the 
neighborhoods - 1) 8 , 2) _a~_ , 3) a a@_ , 4) @aa~ . The no- 
ration _u_xv. represents hier the neighborh~d {<u,v> ,x} . 
It is easy to see that the fourth neighborhood of the firs~ 
and of the lass term c~t b'e a neighborhood of e with respect 
@o @g48 . On She other hand, a is a neighborhood of .aa with 
respect to ea~@ for every n = 1,2, .... It follows that no 
- 16 - 
neighborhood grammar of L = ~a2ZX 3 may contain one of She neigh. 
borhoods _0a2@ , Q a2~ and a. Thus, if a neighborhood gram- 
mar of \]~ exists, it contains at leas~ one neighborhood from 
every group of the following four groups of neighborhoods : 
~) _0 , _~a, _ea 2 . 
~) e_a, _aa, _aae , G~a , ~.aaO. 
b') 6~-~, aa, ..aO , ea_a~ , agO. 
We shall consider all possible combinations betweau a neigh- 
borhood of the group ~ and a neighborhood of the group E . By 
mn we shall denote the combination formed by the m.th neigh- 
borhood of ~ and the n-th neighborhood of ~ . It is easy to see 
Chat every neighborhood grammar containing one of the combina- 
tions 12, 22, 23, 25, 42 generates a language whioh eontain~ 
every string a n with n $ 2. On the other hand~ every neigh- 
borhood grammar containing one of the combinations ll, 13, 14, 
15, 21, 24, 31, 32, 33, ~, 35, 41, 43, 44, 45, 51, 52, 53, 5~, 
55 generates a language which either does not contain the string 
a 4 or contains every string a n with n~ 2 • (This depends on 
the fact if the neighborhoods aa or aa belong or not to the 
considered neighborhood grammar). Thus, there exists no neigh- 
borhood grammar which generates the language ~2n 3. 
But the definition of (generalized) contextual grammars, 
though adequate to the investigation of the generative power of 
purely contextual operations, does not correspond to ~he situa- 
tion existing in real (natural or artificial) l~guages, where 
every string is admired only by some contexts and every o~u~ 
/ 
- 17- 
admits only some strings. Let us try to obtain a type of grammar 
corresponding to this more complex situation. We define a con___y - 
textual grammar with choice as a system G_ =<V,L,~ ,~o>, where 
V, L1 and~are the objects of a contextual grammar, whereas 
is a mappi~ defined on the universal language on V and havi~ 
the values in the set of subsets of~. We define the language 
generated by G as the smallest language L having the follow- 
1 ° ~ L l x ~ L 2 ° ing properties : If x , ! If ye L, <u,y>6 ~(y) 
and Z&~l, then u~L, z v~L and ~L. Thus, every strin~ 
chooses some contexts and every context chooses some strings. We 
define a contextual language with choice a language which is ge- 
nerated by a contextual grammar wit~oioe. The investigation 
of these grammars and languages would better show the generative 
power of contextual operations, in a manner which corresponds to 
the situation existing in real languages. 
--~$ - 

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