Global characterization of Boolean functions with small DREs

Characterize all general Boolean functions that admit small, in particular linear-size, decomposable randomized encodings, and determine how local embeddings of OR, AND, or nontrivial symmetric functions constrain such global characterizations.

Background

The paper completely characterizes symmetric functions with linear DRE complexity: a symmetric function has linear DRE complexity exactly when it has constant period. It does not obtain an analogous characterization for arbitrary Boolean functions.

The authors observe that every nondegenerate monotone Boolean function contains an OR or AND of superconstant size, and every nondegenerate Boolean function contains a nontrivial symmetric function of superconstant size. These local structural facts yield lower bounds for broad classes, but they do not combine into a global classification.