A Critique of Quigley's "A Polynomial Time Algorithm for 3SAT"

TL;DR

The paper critically examines Quigley’s claim of a polynomial-time algorithm for ${3SAT}$ by dissecting the implication rules, expansion, and resolution mechanisms. It presents concrete counterexamples to key lemmas (notably Lemmas 5.11, 5.17, and 5.18) and constructs an infinite family of unsatisfiable 3CNF formulas that are incorrectly classified as satisfiable by the algorithm. It further shows that allowing longer implied clauses or removing the length bound leads to exponential growth in the search space, undermining both correctness and efficiency. Collectively, the results support the conclusion that ${P eq NP}$ remains unresolved and that Quigley’s method does not establish a polynomial-time decision for ${3SAT}$.

Abstract

In this paper, we examine Quigley's "A Polynomial Time Algorithm for 3SAT" [Qui24]. Quigley claims to construct an algorithm that runs in polynomial time and determines whether a boolean formula in 3CNF form is satisfiable. Such a result would prove that 3SAT $\in \text{P}$ and thus $\text{P} = \text{NP}$. We show Quigley's argument is flawed by providing counterexamples to several lemmas he attempts to use to justify the correctness of his algorithm. We also provide an infinite class of 3CNF formulas that are unsatisfiable but are classified as satisfiable by Quigley's algorithm. In doing so, we prove that Quigley's algorithm fails on certain inputs, and thus his claim that $\text{P} = \text{NP}$ is not established by his paper.

Theorems & Definitions (16)

• Definition 1: qui:poly-sat
• Definition 2: qui:poly-sat
• Lemma 1: qui:poly-sat
• Lemma 2: qui:poly-sat
• Lemma 3: qui:poly-sat
• Lemma 4: qui:poly-sat
• Lemma 5: qui:poly-sat
• Lemma 6: qui:poly-sat
• Lemma 7
• proof