P != NP question

Question

Not a \'pure\' programming question, but since it is deeply involved in programming theory, I thought it best to ask here.

Regarding the P NP problem, this excerpt from

Answer 1

P is the class of all languages that can be computed in polynomial time by a deterministic Turing machine. A modern computer is very much like a deterministic Turing machine, except that a Turing machine essentially has infinite memory. This distinction is generally ignored for practical purposes.

NP is the class of all languages that can be computed in polynomial time by a non-deterministic Turning machine. A nondeterministic Turing machine does not correspond to any real-world device.

It is a basic fact of computational complexity that NP is equivalent to the class of languages whose verification problems are in P. In fact, NP is sometimes defined as this class; the two definitions are interchangeable, and the verification definition has the benefit of direct relevance to the deterministic-Turing-machine-like computers in the real world.

So NP is the class of problems that are verifiable in poly-time on a "real" machine and solvable in poly-time on a very similar theoretical machine. Thus, the questions of solvability and verifiability are linked.

Now, most computer scientists believe that P and NP are not equivalent; that is, that there exist languages computable in poly-time by a nondeterministic Turing machine but not by a deterministic Turing machine, or equivalently that are not solvable in poly-time by a deterministic Turing machine but whose solutions can be verified in poly-time by a deterministic Turing machine.