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P versus NP

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? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Given a positive integer x and a collection S of positive integers, MAXIMUM is the problem of deciding whether x is the maximum of S. We prove this problem is complete for P. Another major complexity classes are LOGSPACE, LOGTIME and coNP. Whether LOGSPACE = P is a fundamental question that it is as important as it is unresolved. We show the problem MAXIMUM can be decided in logarithmic space. Consequently, we demonstrate the complexity class LOGSPACE is equal to P. Moreover, we define a problem called SUCCINCT-MAXIMUM. SUCCINCT-MAXIMUM contains the instances of MAXIMUM that can be represented by an exponentially more succinct way. We show this succinct version of MAXIMUM is in coNP-complete under logarithmic reductions. Hence, under the assumption of P = NP, we obtain the padded version of SUCCINCT-MAXIMUM is in LOGTIME and P-hard. However, this is not possible according to LOGTIME is strictly contained in LOGSPACE, because that result would imply LOGTIME = LOGSPACE. In this way, we demonstrate the assumption of several computer scientists whom fully expect that P is not equal to NP.

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