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P versus NP under codings

Frank Vega

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? This question was first mentioned in a letter written by John Nash to the National Security Agency in 1955. A precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. We define a coding to be a mapping from symbols of some alphabet (not necessarily one-to-one). NP is closed under codings. However, P is closed under codings if and only if P = NP. We show a coding of a NP language which produces a NEXP-complete problem when the empty string is considered as a symbol. If P = NP, then this NEXP-complete language would be in P, but this is not possible due to the Hierarchy Theorem. In this way, we prove P is not equal to NP when the empty string is taken as a symbol.

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