Random Context and Semi-Conditional Insertion-Deletion Systems

TL;DR

This paper explores insertion and deletion operations in formal systems, demonstrating how semi-conditional rules achieve computational completeness and revealing an asymmetry in power between insertion and deletion rules under random context conditions.

Contribution

It introduces random-context and semi-conditional insertion-deletion systems, showing their computational capabilities and highlighting an asymmetry in their power not seen with other control mechanisms.

Findings

Semi-conditional insertion-deletion systems with context-free rules are computationally complete.

Systems of size (2,0,0; 1,1,0) are computationally complete.

An asymmetry exists between the computational power of insertion and deletion rules in random context systems.

Abstract

In this article we introduce the operations of insertion and deletion working in a random-context and semi-conditional manner. We show that the conditional use of rules strictly increase the computational power. In the case of semi-conditional insertion-deletion systems context-free insertion and deletion rules of one symbol are sufficient to get the computational completeness. In the random context case our results expose an asymmetry between the computational power of insertion and deletion rules: systems of size $(2,0,0; 1,1,0)$ are computationally complete, while systems of size $(1,1,0;2,0,0)$ (and more generally of size $(1,1,0;p,1,1)$) are not. This is particularly interesting because other control mechanisms like graph-control or matrix control used together with insertion-deletion systems do not present such asymmetry.