Characterization Of any Non-linear Boolean function Using A Set of Linear Operators

TL;DR

This paper introduces a novel method to characterize any non-linear Boolean function using a set of linear operators, enhancing understanding of non-linear cellular automata dynamics.

Contribution

It proposes a new algebraic approach to systematically analyze non-linear Boolean functions via binary matrices, distinguishing deviant and non-deviant states.

Findings

Non-linear CA dynamics can be systematized using linear operators.

A sequence of binary matrices can characterize any non-linear Boolean function.

The method differentiates between deviant and non-deviant states in CA evolution.

Abstract

Global dynamics of a non-linear Cellular Automata is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In the past efforts have been made to systematize non-linear CA evolutions in the light of Boolean derivatives and Jacobian Matrices. In this paper two different efforts have been made: first we try to systematize non-linear CA evolution in the light of deviant states and non-deviant states. For all the non-deviant states the nearest linear rule matrix is applicable where as for the deviant states we have a set of other matrices. Second using algebraic manipulation, an efficient algorithm is proposed by which every Non-linear Boolean function can be characterized by a sequence of binary matrices.