Symbolic dynamics and rotation symmetric Boolean functions

TL;DR

This paper connects the weights of rotation symmetric Boolean functions to the periodic points of finite-type shifts, using algebraic geometry and Weil's Riemann hypothesis to analyze their properties and recurrence relations.

Contribution

It establishes a novel link between Boolean function weights and dynamical systems, and extends previous results on quadratic functions using algebraic geometry techniques.

Findings

Weights satisfy a linear recurrence relation.

Weil's Riemann hypothesis provides bounds and additional properties.

Extension of previous results on quadratic functions.

Abstract

We identify the weights $wt(f_n)$ of a family $\{f_n\}$ of rotation symmetric Boolean functions with the cardinalities of the sets of $n$-periodic points of a finite-type shift, recovering the second author's result that said weights satisfy a linear recurrence. Similarly, the weights of idempotent functions $f_n$ defined on finite fields can be recovered as the cardinalities of curves over those fields and hence satisfy a linear recurrence as a consequence of the rationality of curves' zeta functions. Weil's Riemann hypothesis for curves then provides additional information about $wt(f_n)$. We apply our results to the case of quadratic functions and considerably extend the results in an earlier paper of ours.