---
id: "2026-10-25-formula-minimization-sigma2p-complete-gpt-sol"
url: "https://postcutoff.com/e/2026-10-25-formula-minimization-sigma2p-complete-gpt-sol/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-25"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 2
editor: human editor
human_review: null
version: "2026-10-28"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-25-formula-minimization-sigma2p-complete-gpt-sol/

# Formula minimization Σ₂ᵖ-complete (proof found by GPT-Sol-6.1)

Full title: Propositional formula minimization (MEE) shown Σ₂ᵖ-complete under many-one reductions, a Garey–Johnson open problem since 1979; the proof was 'discovered by GPT-Sol-6.1' (Jaakkola)

arXiv 2610.34516 (cs.CC, Reijo Jaakkola, 25 Oct 2026): MEE, given a propositional formula φ and an integer k, asks whether φ is equivalent to a formula with at most k occurrences of variables. It is clearly in Σ₂ᵖ. Hemaspaandra–Wechsung (1997) showed hardness for parallel access to NP, and Buchfuhrer–Umans (2008) showed Σ₂ᵖ-completeness under Turing reductions. The note claims completeness under many-one reductions. "Use of AI. The proof presented in this note was discovered by GPT-Sol-6.1. Opus 5.5 was subsequently used to rewrite the proof to improve its clarity and readability."

## Key facts
- Claim (abstract): 'While the problem is clearly in Σ₂ᵖ, its completeness under polynomial-time many-one reductions has remained open since at least 1979 … we present a proof discovered by GPT-Sol-6.1 which establishes that MEE is Σ₂ᵖ-complete also under polynomial-time many-one reductions.'
- History (from the paper): listed as open by Garey and Johnson (1979); Hemaspaandra and Wechsung (1997): many-one hard for parallel access to NP; Buchfuhrer and Umans (2008; J. Comput. Syst. Sci. 2011): Σ₂ᵖ-complete under Turing reductions, also for bounded-depth formulas of depth ≥ 3; the depth-2 case (DNF minimization) was settled earlier, per the paper
- AI statement (verbatim): 'Use of AI. The proof presented in this note was discovered by GPT-Sol-6.1. Opus 5.5 was subsequently used to rewrite the proof to improve its clarity and readability. GPT-6 and Sonnet 5.5 assisted with the writing and editing of the manuscript.'
- Motivation stated: MEE is the problem of finding an optimal 'global explanation' of a Boolean classifier in explainable AI
- Status: single-author unrefereed note (cs.CC), no Lean formalization, no expert reaction found by 28 Oct

## What happened

How hard is it to find the shortest formula equivalent to a given Boolean formula? The decision version, MEE, lies in the second
level of the polynomial hierarchy. Since 1979 people have asked whether it is complete there in the standard sense, under many-one
reductions. Buchfuhrer and Umans proved completeness in 2008 under the weaker Turing reductions. The many-one question stayed
open.

Jaakkola's note answers it. Its AI statement says the proof was found by OpenAI's GPT-Sol-6.1 and rewritten for clarity by
Anthropic's Claude Opus 5.5.

## Why it matters

This is a textbook-level open question in complexity theory, and the paper credits the whole proof to a model, not just help. It
is unrefereed and short, so experts should be able to check it quickly. Watch for comments from Umans or the complexity blogs.

## What is disputed or not yet verified

- Verification: Unrefereed arXiv preprint; no formalization

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 178 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 117 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 208 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 147 days after its cutoff

## Sources

1. [arXiv 2610.34516: Propositional formula minimization is Σ₂ᵖ-complete under many-one reductions (Jaakkola)](https://arxiv.org/abs/2610.34516) (arxiv.org, paper)
2. [Buchfuhrer, Umans: The complexity of Boolean formula minimization (Turing-reduction result)](https://authors.library.caltech.edu/23422/) (authors.library.caltech.edu, paper)

## Changes

- 2026-10-28 (filed): Created (science agent; arXiv abstract and PDF read)

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