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devel / comp.lang.c / ℙ≠ℕℙ proof

ℙ≠ℕℙ proof

<77d5729fa878630614efb969cc98cb23ecf5461d.camel@gmail.com>

https://www.novabbs.com/devel/article-flat.php?id=32478&group=comp.lang.c#32478

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From: wynii...@gmail.com (wij)
Newsgroups: comp.lang.c
Subject: ℙ≠ℕℙ proof
Date: Sun, 04 Feb 2024 11:13:04 +0800
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by: wij - Sun, 4 Feb 2024 03:13 UTC

This file is intended a proof that ℙ≠ℕℙ.
https://sourceforge.net/projects/cscall/files/MisFiles/PNP-proof.txt/download

-------------
ANPC::= Set of decision problems that additional information c must be
provided
to compute the problem in P-time (including processing the
information c
(certificate)).

Since the information c can be generated in O(2^N) time, the upper
bound of
ANPC is O(2^N) (checking c is in P-time).

Corollary: ANPC ⊆ ℕℙ (from ℕℙ definition: "... ℕℙ is the set of
decision
problems verifiable in polynomial time by a deterministic Turing
machine. https://en.wikipedia.org/wiki/NP_(complexity)")

If we try to prove or show that an ANPC problem can be solved in P-
time, we
would also be trying to prove/show that the requirement of the
additional
information c is not necessary. Thus, such a proof will just prove
itself not a valid proof, IOW, ANPC==P is unprovable.

Conclusion: ANPC is not P-time provable/solvable (implied by ANPC
definition)

Corollary: ℕℙ!=ℙ (ANPC ⊆ ℕℙ AND ANPC!=ℙ)

QED. ------------