On ET0L Systems of Finite Index

The theory of L systems (see, e.g., Herman and Rozenberg (1975) and Rozenberg and Salomaa (1974, 1976)) constitutes today one of the most pursued fragments of formal language theory. Its relevance to the formal language theory stems from the facts that it introduced new range of problems and proof techniques and has put some of the old concepts and problems in a better perspective. In this paper we examine the important formal language theoretic concept of a rewriting system of finite index in the framework of L systems. In particular we are concerned with the class of ETOL systems (see Rozenberg (1973b)) which form perhaps the central class in the theory of L systems. After modifying the concept of a finite index (remember that the distinction between nonterminal and terminal symbols in L systems is more subtle than in Chomsky grammars) we investigate properties of ETOL systems of finite index as well as the properties of the class of languages that they generate. _; r We would like to point out that the finite index restriction in ETOL systems is also biologically reasonable. It happens quite often that the development of an organism is such that things are happening in a limited number of places only. The organism may be programmed in such a way that the number of active cells do not exceed a certain threshold. Such a limitation may be also imposed by an outside controlling factor (e.g., nutrition). Since such a behavior is rather typical, we can conclude that if the concept of finite index would not have been introduced