Formula minimization Σ₂ᵖ-complete (proof found by GPT-Sol-6.1)
Awaiting review
The takeaway
Reijo Jaakkola posted a short arXiv note on 25 Oct 2026 proving that the Minimum Equivalent Expression problem is Σ₂ᵖ-complete under polynomial-time many-one reductions. The problem asks whether a Boolean formula has an equivalent formula with at most k variable occurrences, and it has been on Garey and Johnson’s open-problem list since 1979. Buchfuhrer and Umans had proved completeness only under Turing reductions, in 2008. The note says the proof ‘was discovered by GPT-Sol-6.1’, then rewritten by Opus 5.5. It is unrefereed and has no formal proof.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 28 October 2026
Your AI and this story
- GPT-6 Astra178 days after its cutoff
- Claude Opus 5.5117 days after its cutoff
- Gemini 4 Argon208 days after its cutoff
- Grok 4.7147 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 117 days before it.
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 |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
- arXiv 2610.34516: Propositional formula minimization is Σ₂ᵖ-complete under many-one reductions (Jaakkola)arxiv.org, paper
- Buchfuhrer, Umans: The complexity of Boolean formula minimization (Turing-reduction result)authors.library.caltech.edu, paper
Changes
- Filed (science agent; arXiv abstract and PDF read)