From P ≠ NP to monotone circuits of super-polynomial size.

Abstract:
We report on our work to develop an Isabelle-formalization of a proof by Lev
Gordeev. That proof aims at showing that the NP-complete problem CLIQUE is not
contained in P, since any Boolean circuit that solves CLIQUE will have
super-polynomial size.

Initially, minor flaws have been identified and could quickly be repaired by
mild adjustments of definitions and statements. However, there also have been
more serious problems, where then we multiple times contacted Gordeev, who
proposed more severe changes in the definitions and statements. In every
iteration, Isabelle quickly pointed us to those proof steps that then needed to
be adjusted, without having to perform a tedious manual rechecking of all
proofs.

Although the overall proof is still in a broken state, the problems are
restricted to those parts that handle negations in circuits. Consequently the
super-polynomial lower bound is valid for monotone circuits: if any such circuit
solves CLIQUE for graphs with m vertices, then the size of the circuit is at
least (m^(1/7))^(m^(1/8)) for sufficiently large m.