P != NP Proof

对Sergey Gubin设计的多项式时间SAT求解器的批评

A Critique of a Polynomial-time SAT Solver Devised by Sergey Gubin

本文反驳了Sergey Gubin提出的求解可满足性的多项式时间算法的有效性。Gubin介绍了使用3-SAT的算法，并最终将其扩展到接受各种形式的布尔可满足性问题。因为3-SAT是NP-完全的，所以如果算法是正确的，它将意味着P = NP。此外，本文还反驳了他的多项式时间约化SAT为2-SAT的正确性。

The threshold for SDP-refutation of random regular NAE-3SAT

Unlike its cousin 3SAT, the NAE-3SAT (not-all-equal-3SAT) problem has the property that spectral/SDP algorithms can efficiently refute random instances when the constraint density is a large constant (with high probability). But do these methods work immediately above the "satisfiability threshold", or is there still a range of constraint densities for which random NAE-3SAT instances are unsatisfiable but hard to refute? We show that the latter situation prevails, at least in the context of random regular instances and SDP-based refutation. More precisely, whereas a random $d$-regular instance…

Space proof complexity for random $3$-CNFs via a $(2-ε)$-Hall's Theorem

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random $3$-CNF $ϕ$ in $n$ variables requires, with high probability, $Ω(n/\log n)$ distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation $ϕ$ requires, with high probability, $Ω(n/\log n)$ clauses each of width $…

A Critique of Keum-Bae Cho's Proof that $\mathrm{P} \subsetneq \mathrm{NP}$

In this paper we critique Keum-Bae Cho's proof that $\mathrm{P} \subsetneq \mathrm{NP}$. This proof relates instances of 3-SAT to indistinguishable binomial decision trees and claims that no polynomial-time algorithm can solve 3-SAT instances represented by these trees. We argue that their proof fails to justify a crucial step, and so the proof does not establish that $\mathrm{P} \subsetneq \mathrm{NP}$.

On the P vs NP question: a proof of inequality

The analysis discussed in this paper is based on a well-known NP-complete problem which is called satisfiability problem or SAT. From SAT a new NP-complete problem is derived, which is described by a Boolean function called core function. In this paper it is proved that the cost of the minimal implementation of core function increases with n exponentially. Since the synthesis of core function is an NP-complete problem, this result is equivalent to proving that P and NP do not coincide.

Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: * We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting plane…