A note on one-sided context-sensitive grammars

In this note it is proved that so called one-sided context-sensitive grammars can generate languages which cannot be generated by any context-free grammar. This fact is not quite new. It has been proved in [3], [4] and [5] (as far as the author knows). In [3] it is proved that a special one-sided context-sensitive grammar suggested by Dr. Fris ([1]) generates a language {a m b'"c"; Hn^ m} which is not context-free. In [5] an example of a one-sided context-sensitive grammar is given and in [4] there is proved, concerning this grammar, that it generates a well-known language {a"b"c"; n £ 1}. The proofs given in [3] and [4] are rather complicated though the grammars in question contain about 20 rules only. The aim of the present note is to give a simple proof of the above-mentioned statement. Let us define a one-sided context-sensitive grammar G = (V T , V N , R, S> as follows: