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The Unit Gap: How Sharing Works in Boolean Circuits

Kirill Krinkin

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.

The Unit Gap: How Sharing Works in Boolean Circuits

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.
Paper Structure (44 sections, 6 theorems, 8 equations, 2 figures, 7 tables)

This paper contains 44 sections, 6 theorems, 8 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. And-Inverter Graphs.
  4. Circuits and formulas.
  5. Decomposition.
  6. NPN equivalence.
  7. Essential variables.
  8. Gate elimination.
  9. SAT-based exact synthesis.
  10. Experimental scope.
  11. The Unit Gap Theorem
  12. The tree Bellman operator.
  13. Verification.
  14. The Threshold Theorem and Tree Theorem
  15. Tightness.
  16. ...and 29 more sections

Key Result

Lemma 1

Any Boolean function with $k$ essential variables requires at least $k - 1$ AND gates in the AIG basis.

Figures (2)

  • Figure 1: Dual-polarity sharing. Gate $g_0$ feeds arm $g_1$ in positive polarity and arm $g_2$ in negative polarity. In a tree circuit, $g_0$ must be duplicated, costing exactly 1 gate.
  • Figure 2: Common subexpression (CSE) sharing. Gate $g_0$ feeds both $g_1$ and $g_3$ in the same polarity. This template requires 5 distinct input slots, explaining its absence at $n \le 4$.

Theorems & Definitions (13)

  • Lemma 1: Classical; see Savage Savage1976 for exposition
  • proof
  • Theorem 2: Unit Gap
  • proof
  • Theorem 3: Threshold Theorem
  • proof
  • Theorem 4: Tree Theorem
  • proof
  • Remark 5: Optimal substructure
  • Corollary 6: Decomposition Formula
  • ...and 3 more