Closure properties of some families of languages associated with biological systems

Abstract

We discuss some families of languages which have originally arisen from the study of mathematical models for the development of some biological organisms. We shall, therefore, call them families of developmental languages. From the computer scientist's point of view, they are all families consisting of languages which are generated by context-free grammars, with the difference that at each step of a derivation every symbol in the sentential form is rewritten. Thus, the behavior of these systems is similar to the behavior of other grammars in which context-free type rules are applied simultaneously at several points in a sentential form. Such grammars have been under active investigation in recent years. Subfamilies (128 of them) of our largest family of development languages are determined by various biologically and mathematically meaningful restrictions. Due to the parallelism in their definition, each of the families will contain languages which are not context free. However, they are all subfamilies of the context-sensitive languages. We investigate the closure properties of these families of languages, and we find that, in contrast to other recently studied families with parallelism, they are closed under only a few operations. In fact, none of them is an AFL or a pre-AFL. We also give a number of examples of how to prove that these families are or are not closed under various operations. The significance of our results is discussed from the point of view of both formal language theory and developmental biology.