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ℙ≠ℕℙ proof

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wij

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Feb 3, 2024, 10:11:36 PMFeb 3
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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. ------------



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