What is Cook's theorem?

TL;DR

This paper offers a reinterpretation of Cook's theorem, highlighting cognitive biases that obscure the understanding of nondeterminism and its role in NP-completeness, suggesting foundational issues in P versus NP comprehension.

Contribution

It identifies cognitive biases in the proof of Cook's theorem as a source of confusion about nondeterminism, impacting the understanding of NP's definitions and the P versus NP problem.

Findings

Cognitive biases affect the interpretation of Cook's theorem

Loss of nondeterminism is linked to misunderstandings in NP definitions

Fundamental difficulties in P vs NP stem from cognition and logic issues

Abstract

In this paper, we make a preliminary interpretation of Cook's theorem presented in [1]. This interpretation reveals cognitive biases in the proof of Cook's theorem that arise from the attempt of constructing a formula in CNF to represent a computation of a nondeterministic Turing machine. Such cognitive biases are due to the lack of understanding about the essence of nondeterminism, and lead to the confusion between different levels of nondeterminism and determinism, thus cause the loss of nondeterminism from the NP-completeness theory. The work shows that Cook's theorem is the origin of the loss of nondeterminism in terms of the equivalence of the two definitions of NP, the one defining NP as the class of problems solvable by a nondeterministic Turing machine in polynomial time, and the other defining NP as the class of problems verifiable by a deterministic Turing machine in polynomial time. Therefore, we argue that fundamental difficulties in understanding P versus NP lie firstly at cognition level, then logic level.