Deterministic Lindenmayer languages, nonterminals and homomorphisms

Abstract

Lindenmayer systems are a class of parallel rewriting systems originally introduced to model the growth and development of filamentous organisms. Families of languages generated by deterministic Lindenmayer systems (i.e., those in which each string has a unique successor) are investigated. In particular, the use of nonterminals, homomorphisms, and the combination of these are studied for deterministic Lindenmayer systems using one-sided context (D1Ls) and two-sided context (D2Ls). Languages obtained from Lindenmayer systems by the use of nonterminals are called extensions. Typical results are: The closure under letter-to-letter homomorphism of the family of extensions of D1L languages is equal to the family of recursively enumerable languages, although the family of extensions of D1L languages does not even contain all regular languages. Let P denote the restriction that the system does not rewrite a letter as the empty word. The family of extensions of PD2L languages is equal to the family of languages accepted by deterministic linear bounded automata. The closure under nonerasing homomorphism of the family of extensions of PD1L languages does not even contain languages like { a 1, a 2,⋯, a n } ∗--{λ}, n⩾2 . The closure of the family of PD1L languages under homomorphisms which map a letter either to itself or to the empty word is equal to the family of recursively enumerable languages. Strict inclusion results follow from necessary conditions for a language to be in one of the considered families. By stating the results in their strongest form, the paper contains a systematic classification of the effect of nonterminals, letter-to-letter homomorphisms, nonerasing homomorphisms and homomorphisms for all the basic types of deterministic Lindenmayer systems using context.