On the index of grammars and languages

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We prove the following results concerning the index of grammars and languages. (1) Any context-free grammar with the terminal vocabulary containing only one letter has a finite index. (2) Any type-O grammar of finite index such that any rule is of the form αAβ → αγβ, generates a context-free language of finite index. (3) The family of matrix languages of finite index is a full AFL.

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