Generalized tag systems

Abstract

Tag systems and cyclic tag systems are forms of rewriting systems which, due to the simplicity of their rewrite rules, have become popular targets for reductions when proving universality/undecidability results. They have been used to prove such results for the smallest universal Turing machines, the elementary cellular automata Rule 110, for simple instances of the Post correspondence problem and related problems on simple matrix semi-groups, and many other simple systems. In this work we compare the computational power of tag systems, cyclic tag systems and a straightforward generalization of these two types of rewriting system. We explore the relationships between the various systems by showing that some variants simulate each other in linear time via simple encodings, and that linear time simulations between other variants are not possible using such simple encodings. We also give a cyclic tag system that has only four instructions and simulates repeated iteration of the Collatz function.