The Unit Gap: How Sharing Works in Boolean Circuits

TL;DR

This paper investigates the relationship between Boolean circuit size and formula size, establishing precise conditions under which sharing reduces circuit complexity and characterizing the structure of optimal sharing.

Contribution

It proves the Unit Gap Theorem, Threshold Theorem, and Tree Theorem, providing exact formulas and structural insights into sharing in Boolean circuits over AIGs.

Findings

The gap between circuit and formula size is always 0 or 1.

Sharing requires at least n essential variables.

No sharing is needed when opt(f) <= 3.

Abstract

We study the gap between the minimum size of a Boolean circuit (DAG) and the minimum size of a formula (tree circuit) over the And-Inverter Graph (AIG) basis {AND, NOT} with free inversions. We prove that this gap is always 0 or 1 (Unit Gap Theorem), that sharing requires opt(f) >= n essential variables (Threshold Theorem), and that no sharing is needed when opt(f) <= 3 (Tree Theorem). Gate counts in optimal circuits satisfy an exact decomposition formula with a binary sharing term. When the gap equals 1, it arises from exactly one gate with fan-out 2, employing either dual-polarity or same-polarity reuse; we prove that no other sharing structure can produce a unit gap.