Universality and Computational Completeness of Controlled Leftist Insertion-Deletion Systems

In this article, we consider leftist insertion-deletion systems (LIDS), in which all rules have contexts on the same (left) side, and may only insert or delete one symbol at a time. We start by introducing extended rules, in which the contexts may be specified as regular expressions, instead of fixed words. We prove that in this case the computational completeness is achieved when additional control mechanisms are used (graph control with two states, matrix control with binary matrices and random-context control). We then show how rules with regular contexts can be simulated by conventional rules checking one-symbol (resp. two-symbol) left contexts for insertion and two-symbol (resp. one-symbol) left contexts for deletion. This simulation does not generally hold in the controlled case, however. Hence, we provide a construction simulating an arbitrary 2-tag system using extended rules and which can be rewritten in terms of conventional rules of types above, which implies that the latter systems are universal.