The Unit Gap: How Sharing Works in Boolean Circuits

Kirill Krinkin

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arXiv · 2603.08033

The Unit Gap: How Sharing Works in Boolean Circuits

Kirill Krinkin

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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.

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Kirill Krinkin. 2026-03-09. The Unit Gap: How Sharing Works in Boolean Circuits. https://arxiv.org/abs/2603.08033

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