Comments about "NP (Complexity)" in Wikipedia

This document contains comments about the document NP (Complexity) in Wikipedia

• The text in italics is copied from that url
  
• Immediate followed by some comments
  

In the last paragraph I explain my own opinion.

Index

• 1. Formal definition
• 2. Introduction
• 3. Why some NP problems are hard to solve
• 4. Equivalence of definitions

Introduction

The article starts with the following sentence.

In computational complexity theory, NP is one of the most fundamental complexity classes.

By reading the article I get the impression there 4 complexity classes.
The full list is: P, NP, NP-Complete and NP-Hard ?

The abbreviation NP refers to "nondeterministic polynomial time."

To use the word "nondetermistic" is misleading.

Intuitively, NP is the set of all decision problems for which the 'yes'-answers have efficiently verifiable proofs of the fact that the answer is indeed 'yes'.

Total unclear sentence

More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine. In an equivalent formal definition, NP is the set of decision problems where the 'yes'-instances can be recognized in polynomial time by a non-deterministic Turing machine.

Concepts like "deterministic Turing machine" and "non-deterministic Turing machine" can not be used to explain something clearly
See also my comments in Comments about "Non-deterministic Turing machine" in Wikipedia

The equivalence of the two definitions follows from the fact that an algorithm on such a non-deterministic machine consists of two phases, the first of which consists of a guess about the solution, which is generated in a non-deterministic way, while the second consists of a deterministic algorithm that verifies or rejects the guess as a valid solution to the problem.

Totally unclear. You can not mix deterministic with non-deterministic.

The complexity class P is contained in NP, but NP contains many important problems, the hardest of which are called NP-complete problems, whose solutions are sufficient to deal with any other NP problem in polynomial time.

In this sentence two different complexity classes are mentioned.

The most important open question in complexity theory, the "P = NP problem", asks whether polynomial time algorithms actually exist (1) for NP-complete, and by corollary, (2) all NP problems.

Why all?

It is widely believed that this is not the case.

For (1) or for (2) or both ?
Before invastigating this claim it is important to make a list of all programs which belongs to each set (and inditify the differences)

For comments about Euler Diagram see: Euler diagram for P, NP, NP-Complete and NP-hard set of problems

1. Formal definition

This paragraph starts with the text:

The complexity class NP can be defined in terms of NTIME as follows: NP = some function of NTIME

Totally unclear.

Alternatively, NP can be defined using deterministic Turing machines as verifiers.

Please explain in detail.

2.1 Verifier-based definition

Many natural computer science problems are covered by the class NP. In particular, the decision versions of many interesting search problems and optimization problems are contained in NP.

Why using such unclear language.

2. Introduction

Many natural computer science problems are covered by the class NP. In particular, the decision versions of many interesting search problems and optimization problems are contained in NP.

Why using such unclear language.

3. Why some NP problems are hard to solve

4. Equivalence of definitions

Euler diagram for P, NP, NP-Complete and NP-hard set of problems