On Lev Gordeev's "On P Versus NP"

TL;DR

This paper critiques Lev Gordeev's attempt to extend approximation methods to De Morgan Normal circuits for proving P versus NP, highlighting a key mistake and emphasizing the need to include negated inputs in such approaches.

Contribution

The paper analyzes Gordeev's approach to proving P ≠ NP via DMN circuit lower bounds and identifies a critical flaw in his method.

Findings

Gordeev's proof contains a crucial mistake in Lemma 12.

Extending approximation methods to DMN circuits requires approximating negated inputs.

The identified flaw undermines the potential of Gordeev's approach to resolve P vs NP.

Abstract

In the paper "On P versus NP," Lev Gordeev attempts to extend the method of approximation, which successfully proved exponential lower bounds for monotone circuits, to the case of De Morgan Normal (DMN) circuits. As in Razborov's proof of exponential lower bounds for monotone circuits, Gordeev's work is focused on the NP-complete problem CLIQUE. If successful in proving exponential DMN circuit lower bounds for CLIQUE, Gordeev would prove that P $\neq$ NP. However, we show that Gordeev makes a crucial mistake in Lemma 12. This mistake comes from only approximating operations over positive circuit inputs. Furthermore, we argue that efforts to extend the method of approximation to DMN circuits will need to approximate negated inputs as well.