Towards a classification of Lindenmayer systems
This work addresses a foundational problem in formal language theory for researchers studying L-systems, but it appears incremental as it builds on existing classification frameworks like the Chomsky hierarchy.
The paper tackles the problem of classifying Lindenmayer systems (L-systems) by proposing a parametrization of the L-space based on rule properties and generated strings, analogous to the Chomsky hierarchy for normal grammars, but finds that L-systems cluster into non-translatable kinds.
In this paper we will attempt to classify Lindenmayer systems based on properties of sets of rules and the kind of strings those rules generate. This classification will be referred to as a parametrization of the L-space: the L-space is the phase space in which all possible L-developments are represented. This space is infinite, because there is no halting algorithm for L-grammars; but it is also subjected to hard conditions, because there are grammars and developments which are not possible states of an L-system: a very well-known example is the space of normal grammars. Just as the space of normal grammars is parametrized into Regular, Context-Free, Context-Sensitive, and Unrestricted (with proper containment relations holding among them; see Chomsky, 1959: Theorem 1), we contend here that the L-space is a very rich landscape of grammars which cluster into kinds that are not mutually translatable.