Properties of Carry Value Transformation

TL;DR

This paper explores the properties of Carry Value Transformation (CVT), revealing its relation to addition, iterative behaviors, and equivalence classes, thus contributing to understanding its mathematical structure and potential applications.

Contribution

It introduces new properties of CVT, including its relation to addition, iterative convergence behavior, and the formation of equivalence classes, advancing the theoretical understanding of CVT.

Findings

CVT addition equals sum of CVT and XOR values

Iterative process of CVT and XOR converges within binary string length

Defined equivalence relation partitions CV table into classes

Abstract

The notion of Carry Value Transformation (CVT) is a model of Discrete Deterministic Dynamical System. In this paper, we have studied some interesting properties of CVT and proved that (1) the addition of any two non-negative integers is same as the sum of their CVT and XOR values. (2) While performing the repeated addition of CVT and XOR of two non-negative integers "a" and "b" (where a >= b), the number of iterations required to get either CVT=0 or XOR=0 is at most the length of "a" when both are expressed as binary strings. A similar process of addition of Modified Carry Value Transformation (MCVT) and XOR requires a maximum of two iterations for MCVT to be zero. (3) An equivalence relation is defined in the set (Z x Z) which divides the CV table into disjoint equivalence classes.