Proof. The considered language is generated by the grammar (V.I^,^)), where V = {a.b\ , 1^ ={«} and{<w,b>} .

We recall that a language is said to be regular if it may be generated by means of a finite automaton (or, equivaleatly, by means of a finite state grammar in the sense of Ohomsky).

Proposition 4. There exists a contextual language which is nftt regular^

Propositions 2,3 and 4 show that there are many infinite languages which are contextual. This fact may be explained by means of

Proposition 5« If the set is non-void and if the set(1g) contains at least one nnn-nnjl context, then the contextual gram­mar^(V.Lj_«(jg)) generates an infinite language.

Proof. Since lq_ is non-void, we may find a string x be­longing to L^. Since contains, at least one non-null context, let <"u,v> be a non-null context belonging to(^) « From these assumptions, we infer, that the strings

uxy , u»?,...,u^xvÄ,.., are mutually distinct' and belong all to the language generated by the grammar (V,;L,,(^). Thus, this language is infinite.

The converse of Proposition 5 is true. Indeed', we have

Proposition 6. If the contextual grammar (V.I/^feJ!) gene­rates an infinite language, then 1^ is non-void, whereas con­tains a non-null context.

Proof. Let L be the language generated by (V,L, If Lj_ is void, L is void too, hence it cannot be infinite. If contains no non-null context, we have L = L-^. But is in any case finite ; thus, L is finite, in contradiotion with the hypo­thesis.

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 languages. The answer is affirmative :

Pro-no sit ion 7. Every contextual language is a context-free language^,

j>roof. Let L_ be a contextual language. If _L is finite, it is a regular language. But it is well known that every regu­lar language is a context-free language. Therefore, L is a context-free language. Howe let us suppose that L is infinite. Denote by G = (V.Lj^^j)) a contextual grammar which generates the language L, In view of Proposition 6, L^ is non-void,whe­reas there exists an integer i, 1-4 ié.p , such that the con­text <u±,vi> is non-null, i.e. at least one of the equalities uA =co , v^ = co is false. Let us make a choice and suppose

«^»▼e?- »•••« <.V_vp>} * We define a w)ltext-fr8e grammar ' P as follows. The terminal vocabulary of V is V. The non­terminal vocabulary of P contains one element only- denoted by S - which is, of course, the axiom of the grammar P « The ter-

minai rules of P are

whereas che non-terminal rules are s -Blä-V-l ,

It is obvious thab 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 contexts in Among the non­terminal rules, there is one at least which is non-trivial : it is the rule S —* u^S.v^ , where n. 4w •

It is not difficult to prove that"the grammar P generates the given language L. Indeed, the general form of a string in L_ is <üi f v± >é(^) ' for s = 1,2,...,p.

In order to generate the considered string we begin by apply­ing times the rule >

In this way, we obtain the expression

The next step consists in applying times.the rule which yields the expression

V uu.s- v; v.

Continuing in this way, we arrive, after p-1 steps, to the ex­pression

We now apply times the rule and thus we obtain the expression

J2. jp-i M s lv àp-i h 4i

the considered string is completely generated. Thus, we have proved that X is contained in the langua-ge generated by P ,

Conversely, let z be a string generated by P . The ge­neral form of this generation involves several consecutive ap­plications of non-terminal rules (the number of these applica­tions 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 P is con­tained in Ii. In view of the preceding considerations, 1 is precisely the language generated by f ,

Proposition 7 easily permits to obtain simple examples of languages which are not contextual languages. For instance, the lan­guage of Kleene^a—] (n=i,2,...), the first example of an infinite langaage which is not regular, is a very simple example of non-contextual language. It is enough to remark that the sequence {n} (a = 1,2,...) contains no subsequence*hich is an infinite therefore for every subsequence of fnJ the difference of two conse­cutive terms has the limit equal to +oo when n—>oo). But a result of[A ] asserts, among others, that given an infinite context-free lan­guage L, the set of integers which represent the lengths of the strings in 1 contains an infinite arithmetic progression. It follows ■chat the language of Kleene is not context-free and, in view of Pro­position 7, it is not a contextual language. Tfc« *a«< fact £o|f«vj

A natural question now arrises : Do there exist non-contextual languages among context-free languages V The affirmative answer follows trou toe following remark :

The converse of Proposition 7 is not true. Indeed, we have

Proposition 8. There exists a context-free language which is not a contextual language.

Proof, let V = In view of a theorem of GruskB C ¥ J _^there exists, for every positive integer n_, a context-free language 1^ on V, such that every context-free grammar of contains at least n non-terminal symbols. But, as we can see in the proof of Proposition 7, every contextual language may be gene­rated with a context-free grammar containing only one non-terminalsymbol. Therefore, if n > 2, .1^ is not a contextual language.

Proposition 8 suggests the natural question whether there exist regular languages which are not contextual languages. The answer is affirmative :

Proposition 9« There exists a regular language which is not a contextual language.

Proof. Let us consider the language L = •Ç.ab -le lalfr'-j (ffl;>Ç= =1,2,...), which was used by H.B.Curry [5], in order to describe the set of mathematical (true or not) propositions. This language is regular, since it can be generated by the rules 8 —> Ab, A_—>Ab, A:—^ Ba, B—> Cc, C_—> Cb , C—> Db, D_—* a. We shall show that L_ is not a contextual language. Indeed, let us admit that the contra­ry holds and let G = < V,!^, De a contextual grammar of L. Here, the gensral form of a string in L is where ytl^ , whereas <a^,y^> (i = l,2,...,n) and .Px»^**** ...,Pn are arbitrary positive integers. This means that .u^,u7,... ...,11^ , v1,y2, ...,va in the expression (3) are formed only by those elements of V whose number of occurences in the strings of L is unlimited. Only ï> satisfies this requirement. It follows that in any string of L both occurrences of a and the occurence of _c_ are terms of the string jl 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 z such that x = y a|b icaz.

x m ab-- (c ab.

But ja. may bë here an arbitrary positive integer. Therefore,since x6.% » it follows that 1^ is an infinite set of strings. This fact contradicts the assumption concerning G ; L is not a con­textual language and Proposition 9 is proved.

The contextual grammars may be generalized, in order to ge­nerate some languages which are not oontext-free.

A generalized contextual grammar is a quadruple G =^V , .I^fLgt t where V, In and (^)have the same meaning as in bhe definition of a contextual grammar, whereas is a finite set of strings on the vocabulary _V. We define the language X &generated by G_ in the following way : is a language on V and x e 2g if and only if we may express x in the form pl».?-2,••• *Pn » P are Posifcive integers such that Vi+p^.t+p^p.

Every language generated by a generalized contextual grammar is said to be a generalized contextual language.

I£» i-Q the definition of G, we take Xg =£coj-, G is equi­valent to a contextual grammar ; the language T^, is then precisely the language generated by the contextual grammar <V,X1^^>.In­deed, the general form of a string in the contextual language ge­nerated by <VJXl,^> is

Proposition lo. Every contextual language is a generalized contextual language.