# Inquiry of P-reduction in Cook's 1971 Paper -- from Oracle machine to Turing machine

## TL;DR

This paper critically examines the concept of P-reduction in Cook's 1971 theorem, arguing that its definition is flawed due to a disguised shift from Oracle machines to Turing machines, questioning the proof of NP problems' reducibility.

## Contribution

It reveals a fallacy in the original definition of P-reduction and challenges the foundational assumptions in Cook's theorem regarding NP problem reductions.

## Key findings

- Identifies a fallacy in the definition of P-reduction
- Argues that P-reduction is essentially equivalent to Turing computability
- Questions the proof of NP problems being reducible to logical forms or each other

## Abstract

In this paper, we inquire the key concept P-reduction in Cook's theorem and reveal that there exists the fallacy of definition in P-reduction caused by the disguised displacement of NDTM from Oracle machine to Turing machine. The definition or derivation of P-reduction is essentially equivalent to Turing's computability. Whether NP problems might been reduced to logical forms (tautology or SAT) or NP problems might been reduced each other, they have not been really proven in Cook's 1971 paper.