# Case Study of the Proof of Cook's theorem - Interpretation of A(w)

## TL;DR

This paper critically examines Cook's theorem by analyzing the propositional formula A(w), revealing that it appears to be CNF but may not be a true logical form, raising questions about the theorem's foundational proof.

## Contribution

It provides a detailed case study showing potential issues in the logical interpretation of A(w) used in Cook's theorem.

## Key findings

- A(w) appears to be CNF but may not be a true logical form
- The analysis suggests a possible logical flaw in the proof of Cook's theorem
- Raises questions about the validity of the theorem's foundational assumptions

## Abstract

Cook's theorem is commonly expressed such as any polynomial time-verifiable problem can be reduced to the SAT problem. The proof of Cook's theorem consists in constructing a propositional formula A(w) to simulate a computation of TM, and such A(w) is claimed to be CNF to represent a polynomial time-verifiable problem w. In this paper, we investigate A(w) through a very simple example and show that, A(w) has just an appearance of CNF, but not a true logical form. This case study suggests that there exists the begging the question in Cook's theorem.