Parallel Complexity of Iterated Morphisms and the Arithmetic of Small Numbers

We improve several upper bounds to the complexity of the membership problem for languages defined by iterated morphisms (D0L systems). The complexity bounds are expressed in terms of DLOGT IME -uniform circuit families. We prove: 1) For polynomially growing D0L systems the membership problem is contained in AC 0 . 2) For arbitrary D0L systems the membership problem is contained in NC 1 . 3) The latter can be improved to T C 0 if and only if upper bounds to a number of natural arithmetic problems can be improved to T C 0 . 4) The general D0L membership problem (the D0L system is part of the input) is contained in Cook's class DET . 1 Introduction We investigate languages defined by iterated homomorphisms or substitutions, i.e.: context -free Lindenmayer languages. They provide a whole framework of classes which have properties similar to the context-free (Chomsky) languages. The complexity of the membership problem for these languages is in most cases well determined, as it is ..