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A Simple Constructive Bound on Circuit Size Change Under Truth Table Perturbation

Kirill Krinkin

TL;DR

This note states the bound explicitly for arbitrary fixed finite complete bases with unit-cost gates, extends it to general Hamming distance via a telescoping argument, and verifies it exhaustively at $n = 4$ in the AIG basis using SAT-derived exact circuit sizes for 220 of 222 NPN equivalence classes.

Abstract

The observation that optimum circuit size changes by at most $O(n)$ under a one-point truth table perturbation is implicit in prior work on the Minimum Circuit Size Problem. This note states the bound explicitly for arbitrary fixed finite complete bases with unit-cost gates, extends it to general Hamming distance via a telescoping argument, and verifies it exhaustively at $n = 4$ in the AIG basis using SAT-derived exact circuit sizes for 220 of 222 NPN equivalence classes. Among 987 mutation edges, the maximum observed difference is $4 = n$, confirming the bound is tight at $n = 4$ for AIG.

A Simple Constructive Bound on Circuit Size Change Under Truth Table Perturbation

TL;DR

This note states the bound explicitly for arbitrary fixed finite complete bases with unit-cost gates, extends it to general Hamming distance via a telescoping argument, and verifies it exhaustively at in the AIG basis using SAT-derived exact circuit sizes for 220 of 222 NPN equivalence classes.

Abstract

The observation that optimum circuit size changes by at most under a one-point truth table perturbation is implicit in prior work on the Minimum Circuit Size Problem. This note states the bound explicitly for arbitrary fixed finite complete bases with unit-cost gates, extends it to general Hamming distance via a telescoping argument, and verifies it exhaustively at in the AIG basis using SAT-derived exact circuit sizes for 220 of 222 NPN equivalence classes. Among 987 mutation edges, the maximum observed difference is , confirming the bound is tight at for AIG.
Paper Structure (6 sections, 1 theorem, 4 equations, 1 table)

This paper contains 6 sections, 1 theorem, 4 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result
  4. Tightness
  5. Computational Verification
  6. Discussion

Key Result

Theorem 1

For any fixed finite complete gate basis $B$ with unit-cost gates and any two Boolean functions $f, f'$ of $n$ variables, where $c_B$ is a constant depending only on $B$.

Theorems & Definitions (2)

  • Theorem 1
  • proof