Year: 2026
Type: preprint
Abstract: Exact And-Inverter Graph (AIG) size of the NPN classes 0x1669 and 0x166b of 4-variable Boolean functions: opt = 10 for both, the largest value among the 222 classes (two other classes also reach it). Cost model: 2-input AND gates, free inversions, size = number of AND gates. These were the two entries that Krinkin's NPN-4 AIG catalog (arXiv:2603.09379, data at github.com/krinkin/bounds) left at an upper bound of 10 with a solver timeout at 9 gates. Upper bound. An explicit 10-gate circuit for each class, checked by exhaustive simulation over the 16 input rows with a script that shares no code with the encoder. Circuits of this size for these classes were already in public exact-synthesis databases. Lower bound. For every k = 1..9 the CNF asking for a normalized AIG with exactly k gates is unsatisfiable. A normalization lemma (paper proof, included) shows that a minimum-size AIG can always be put in the normal form the CNF searches, so each UNSAT rules out opt = k. The record contains the 18 CNFs, the encoder that writes them, the 16 DRAT proofs for k = 1..8 (zstd-compressed, all s VERIFIED by drat-trim on 2026-10-12, byte-identical to the proofs of July 2026), and the July 2026 drat-trim outputs for k = 9. The k = 9 proofs (4.8 GB and 3.9 GB) are not included; the k = 9 CNFs reproduce the SHA-256 pinned in July, and the README gives the command that regenerates and checks each proof in about 50 minutes on one core. Limits. DRAT certifies unsatisfiability of each CNF, not that the CNF encodes the intended question; that step rests on the normalization lemma (not machine-checked) and on an encoder that is not formally verified (it was validated against an independent exhaustive enumerator for n = 2 and n = 3 with k ≤ 3, and on a known class with opt = 7). The lemma holds at k = opt and must not be reused with extra constraints above it. Two classes, one basis; nothing asymptotic. Related records by the same author, on the n = 4 AIG census: the preprint The Unit Gap Is Base-Dependent (10.5281/zenodo.23328820) and its artefacts (10.5281/zenodo.21630762). Text and data under CC BY 4.0; code under MIT.
Source: arXiv (Cornell University)
Author: Krinkin, Kirill
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7134889837