Properties of Carry Value Transformation

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.