A finite alternation result for reversible boolean circuits

TL;DR

This paper establishes upper bounds on the alternation depth for even reversible boolean functions, showing they can be decomposed into a limited number of simpler functions acting on subsets of bits.

Contribution

The paper proves that all even reversible boolean functions on four or more bits have bounded alternation depth, specifically at most 9, and at most 13 for even functions.

Findings

Any even reversible boolean function on n ≥ 4 bits has alternation depth ≤ 9.

Any even reversible boolean function on n ≥ 4 bits has even alternation depth ≤ 13.

The results provide bounds on the complexity of decomposing reversible boolean functions.

Abstract

We say that a reversible boolean function on n bits has alternation depth d if it can be written as the sequential composition of d reversible boolean functions, each of which acts only on the top n-1 bits or on the bottom n-1 bits. Moreover, if the functions on n-1 bits are even, we speak of even alternation depth. We show that every even reversible boolean function of n >= 4 bits has alternation depth at most 9 and even alternation depth at most 13.