On the effect of the finite index restriction on several families of grammars

TLDR: It is proved that this restriction in all these cases yields the same family of languages: the family of finite index ETOL languages (Rozenberg and Vermeir, 1978), which provides another link between the "sequential" and the "parallel" formal language theories.

Abstract: In earlier papers (Rozenberg and Yermeir, 1977, 1978) the effect of the finite index restriction on ETOL systems was quite throoughly investigated. One can give two main reasons for the interest in ETOL systems of finite index: --biological; there are numerous examples of a biological development with a limited number of "active" cells, --mathematical; the finite index restriction is a classical restriction considered in formal language theory (see, e.g., Salomaa, 1973), and it is certainly worthwhile to investigate its effect on language-generating devices parallel in nature. This paper continues the research by Rozenberg and Vermeir (1975a, 1977) First of all, we investigate the effect of the finite index restriction on the classic.a[ extensions of a context-free grammar, namely, on context-free programmed grammars, ordered grammars, and matrix grammars. We prove that this restriction in all these cases yields the same family of languages: the family of finite index ETOL languages (Rozenberg and Vermeir, 1978). In this way we not only demonstrate the importance of the family of ETOL languages of finite index but we also provide another link between the "sequential" and the "parallel" formal language theories. In this way numerous results from Rozenberg and Vermeir (1978) are carried over to "sequential" families of context-free pro- 284