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arXiv · 2610.34516

Propositional formula minimization is $Σ_2^p$-complete under many-one reductions

Reijo Jaakkola
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Abstract

The Minimum Equivalent Expression problem (MEE) asks, given a propositional formula $ϕ$ and an integer $k$, whether $ϕ$ is equivalent to a formula with at most $k$ occurrences of propositional variables. While the problem is clearly in $Σ_2^p$, its completeness under polynomial-time many-one reductions has remained open since at least 1979. Buchfuhrer and Umans showed in 2008 that it is $Σ_2^p$-complete under polynomial-time Turing reductions. In this note, we present a proof discovered by GPT-Sol-6.1 which establishes that MEE is $Σ_2^p$-complete also under polynomial-time many-one reductions.

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Reijo Jaakkola. 2026-10-25. Propositional formula minimization is $Σ_2^p$-complete under many-one reductions. https://arxiv.org/abs/2610.34516

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