On Lev Gordeev's "On P Versus NP"

[Timothy Y. Chow]

FOM readers may recall that Lew Gordeew announced a "New paper on P vs NP" 
back in December 2020.

   https://cs.nyu.edu/pipermail/fom/2020-December/022429.html

David Narváez and Patrick Phillips have recently posted a preprint to the 
arXiv with the following 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 ≠ 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.

   https://arxiv.org/abs/2104.07166

Tim