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

TL;DR

This paper explicitly bounds how much a circuit's size can change when a single truth table entry is altered, extending previous implicit results and confirming tightness at small input sizes.

Contribution

It explicitly states the circuit size change bound under truth table perturbation, extends it to Hamming distance, and verifies tightness at small input sizes using SAT-based methods.

Findings

Maximum size change is at most n for one-point perturbations.

Bound is verified to be tight at n=4 for AIG basis.

Extends the bound to general Hamming distance via telescoping.

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.