Formal language theory/Parallel replacement systems

Parallel replacement systems
A foray into the language theoretic aspects of Lindenmayer systems. For D0L systems, we have followed the presentation in Salomaa's book.

A D0L (deterministic, zero context) system $$(\Sigma, h, w)$$ over an alphabet $$\Sigma$$ consists of a start string $$w \in \Sigma^{*}$$ and a single replacement rule given by a homomorphism $$h$$. These systems have perhaps surprising properties.

Question: What happens when $$h(w) = wx$$ for some string $$x$$?

Hierachies
0L: instead of a homomorphism, there is a finite substitution

Question: Find a language that is in 0L but not in D0L. (this is not hard)

DTOL: instead of a single homomorphism, there is a table $$H=\{h_1, \dots, h_k\}$$ of homomorphisms

Question: Find a language that is in DT0L but not in D0L or 0L.