Dynamics and Application Through the Rule Based Transformation of Non-negative Integer Pairs

The relationship between Cellular Automata (CA) rules and Integral Value Transformations (IVTs) have been studied previously. The notion of Carry Value Transformation (CVT) and Exclusive OR Transformation (XORT) were defined earlier over the set of non-negative integers, and are the two fundamental operations in developing the concept of IVTs. This paradigm is highly multi-functional for almost all arithmetic operations and multiple states. In this article a dynamical system is conceived based on CVT and XORT. Consequently a notion of tree (called CVT-XORT tree) comes in through trajectory/orbit of the dynamical system. A comprehensive study has been made based on the CVT-XORT trees and several graph theoretic properties from different perspectives have been investigated. One of the important properties is the height analysis of CVT-XORT tree which is very much needed for the efficient bulk computing environment. Again, an application is presented how the CVT-XORT trees can be used for ´encoding and decoding' of various data. Additionally, some basic operations such as insertion, deletion, and searching a node on CVT-XORT tree are demonstrated and specific CVT-XORT tree properties are revealed to make the operations efficient. These theoretical results will help to study the dynamics of pairwise CA rules that act on a two-dimensional finite lattice.