On the parity complexity measures of Boolean functions

TL;DR

This paper explores the complexity of Boolean functions within the parity decision tree model, establishing polynomial relationships with other complexity measures and connecting it to communication complexity.

Contribution

It introduces polynomial relations between deterministic and non-deterministic parity decision tree complexities and relates them to block sensitivity and communication complexity.

Findings

Deterministic parity decision tree complexity is polynomially related to non-deterministic complexity.

Parity decision trees are polynomially related to an analogue of block sensitivity.

The study links parity decision trees with communication complexity.

Abstract

The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any Boolean function is polynomially related to the non-deterministic complexity of the function or its complement. We also show that they are polynomially related to an analogue of the block sensitivity. We further study parity decision trees in their relations with an intermediate variant of the decision trees, as well as with communication complexity.