Critique of Boyu Sima's Proof that ${ m P} eq{ m NP}$

We review and critique Boyu Sima’s paper, “A solution of the P versus NP problem based on specific property of clique function,” (arXiv:1911.00722) which claims to prove that ${\rm P}\neq{\rm NP}$ by way of removing the gap between the nonmonotone circuit complexity and the monotone circuit complexity of the clique function. We first describe Sima’s argument, and then we describe where and why it fails. Finally, we present a simple example that clearly demonstrates the failure.

💡 Research Summary

The paper provides a thorough critique of Boyu Sima’s claim that P≠NP can be proved by eliminating the gap between the monotone and non‑monotone circuit complexities of the clique function. After a brief introduction that situates the P versus NP problem and the relevance of circuit lower‑bounds, the authors recap the necessary preliminaries: Boolean functions, circuits, monotone circuits, standard circuits (where negations appear only on inputs), and the known exponential lower bound on the monotone circuit complexity of CLIQUE(m,s) due to Razborov, Alon, and Boppana.

Sima’s core argument is then summarized. He first shows that any Boolean circuit can be transformed into a standard circuit with at most a factor‑two increase in size, so it suffices to consider standard circuits for the clique function. He rewrites a standard circuit as a Boolean formula of the form
 f = ¬x_i ∧ TermA ∨ TermB ∨ …
where ¬x_i is a negated input variable and none of the terms contain ¬x_i. He then analyses two cases for the sub‑term TermA:

1. TermA = 0 – the whole conjunct ¬x_i ∧ TermA evaluates to 0 regardless of the value of ¬x_i, so the output of the circuit is unaffected.
2. TermA = 1 – Sima argues that because the overall function is monotone, the output must be 1 whenever TermA = 1, independent of ¬x_i. Consequently, he claims that setting ¬x_i to the constant 1 does not change the computed function.

Repeating this replacement for every negated input, Sima concludes that a non‑monotone circuit for CLIQUE can be turned into a monotone circuit of essentially the same size. Since the monotone circuit complexity of CLIQUE is known to be exponential, the same lower bound would hold for general circuits, establishing P≠NP.

The critique identifies two fatal flaws in this reasoning. First, Sima treats each negated variable ¬x_i as an independent Boolean variable, ignoring the logical constraint that ¬x_i must always be the logical complement of x_i. This oversight invalidates the assumption that the value of ¬x_i can be fixed arbitrarily without affecting the rest of the computation. Second, the claim “if TermA = 1 then the whole function equals 1” is not generally true for the clique function; monotonicity alone does not guarantee that the presence of a particular sub‑term forces the overall output to be 1, because the clique function depends on the global structure of the graph, not merely on the truth of a single conjunct.

To make the error concrete, the authors construct a simple counterexample. Let f(x) be a monotone circuit that correctly computes CLIQUE(m,s). Append the contradictory term ¬x_1 ∧ x_1 using an OR gate, yielding the circuit
 f′ = (¬x_1 ∧ x_1) ∨ f(x).
Since ¬x_1 ∧ x_1 is always false, f′ computes the same function as f. Applying Sima’s extraction process, we obtain a formula where ¬x_1 is pulled out: ¬x_1 ∧ x_1 ∨ … . Replacing ¬x_1 by the constant 1 produces the circuit
 g = x_1 ∨ f(x).
Now g outputs 1 whenever x_1 = 1, regardless of the rest of the inputs. For a graph that contains only the edge corresponding to x_1, the original CLIQUE function (with s ≥ 3) should output 0, but g outputs 1. Hence the transformed circuit no longer computes the clique function, disproving Sima’s claim that the replacement preserves functionality.

The paper concludes that Sima’s method rests on an unjustified generalization that neglects the essential relationship between a variable and its negation. Existing results, such as the super‑polynomial lower bound for circuits with a limited number of negation gates (Amano‑Maruoka, 2005), already demonstrate that the gap between monotone and non‑monotone complexities cannot be closed by the proposed transformation. Therefore, the presented proof does not establish P≠NP, and any future attempt to bridge this gap must address the structural dependencies that Sima’s argument overlooks.