Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34516

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

Reijo Jaakkola
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CCcs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A proof of P != NP (New symmetric encryption algorithm against any linear attacks and differential attacks)

P vs NP problem is the most important unresolved problem in the field of computational complexity. Its impact has penetrated into all aspects of algorithm design, especially in the field of cryptography. The security of cryptographic algorithms based on short keys depends on whether P is equal to NP. In fact, the security requirements for cryptographic keys are much stricter than those for P$\neq$NP, the security of the key must ensure not only a sufficiently high computational complexity to crack it, but also consider the security of each bit of the key, while fully avoiding the effectiveness of various attack methods. In this paper, we innovatively propose a new encoding mechanism and develop a novel block symmetric encryption algorithm, which be named Eagle, whose encryption and decryption can be completed in linear time. The key consists of 10 variables, for the attacker, in the case when only the plaintext-ciphertext correspondence is known, the problem of cracking the key is equivalent to solving a system of equations about ten unknown variables together with some implicit variables which are generated by the encryption process. In this paper, we will prove that there is no universal algorithm without brute-force searching at least one variable during the process of verifying four of the ten variables whose computational complexity is not greater than solving the ten variables. After performing elimination the implicit variables, the problem of verifying the four variables will be equivalently transformed into a series of general quintic equations about one variable with coefficient given by other four variables, according to the conclusion that ``there is no algebraic solution to quintic equations", we can't establish a system of algebraic equations only containing the other four variables......

cs.CC↗
arXiv2026

The estimates for Boolean surface area of polynomial threshold functions

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture recently solved by OpenAI, \cite{GLsol}. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the second bound \[ \mathbf{BSA}[f]\le \exp(C(d)\sqrt{\log n}). \] This bound was obtained in \cite{FSVZ} without clarifying the magnitude of $C(d)$, which we do below. For lower moments $\bE |\nabla f|^η \le C(d, η)$ there is no dependence on $n$ for $0<η<1$.

cs.CC↗
arXiv2026

Local Search with Correlated Randomness

The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $ν$ satisfying $ν[w]\le DP[w]^s$ for every finite prefix $w$ of the random tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth, a tree max-flow/min-cut identity that makes worst-case survival probabilities exact and optimizable over the algorithm's choices. Three applications follow. For $n$ tasks on one machine, each retried until the next random symbol lies in its success set, every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-Θ(n)]$; a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment that satisfies a majority of each clause, we determine the sharp source threshold $s_*$. Below $s_*$, on some instances a single admissible source forces nontermination however clauses are selected; at $s_*$, with fixed $D$, the expected number of flips is at most cubic in the initial Hamming distance, against linear under independence, with matching worst-case instances; and any fixed margin above $s_*$ restores linear dependence on the distance. Third, verified backtracking for noisy predecessor search remains correct under a sufficiently strong bound on the comparison errors. In every setting the algorithm may also restart, which supplies no fresh randomness; restarting restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.

cs.CC↗