Solutions to four open problems concerning controlled pure grammar systems

摘要

In this paper, we address several open problems concerning pure grammar systems (pGSs) and their controlled versions. More specifically, we prove the following four results. (Ⅰ) Regular-controlled pGSs having a single component define the family of regular languages. (Ⅱ) pGSs having two components controlled by infinite regular languages define the family of recursively enumerable languages. (Ⅲ) Regular-controlled pGSs without any erasing rules define the family of regular languages not containing the empty string. (Ⅳ) pGSs define a proper subfamily of the family of regular languages.