Computational power of insertion–deletion (P) systems with rules of size two

This article investigates insertion–deletion systems of small size, where at most two symbols can be used in the description of insertion or deletion rules in a context-free or contextual manner. The basic result shows a characterization by context-free grammars of insertion–deletion systems, which insert or delete one symbol in one symbol left context (systems of size (1, 1, 0; 1, 1, 0)). If context-free insertion or deletion rules are considered (systems of size (2, 0, 0; 1, 1, 0) or (1, 1, 0; 2, 0, 0)), then we show that corresponding systems are not computationally complete. However, if the insertion and the deletion operations having same size as above are considered in the distributed framework of P systems, then the computational power strictly increases and the obtained models become computationally complete. The article also shows that if context-free insertion and deletion rules of two symbols (of size (2, 0, 0; 2, 0, 0)) are used in combination with P systems, then the obtained model is still not computationally complete. Finally some open problems are presented.