LOGSPACE vs P

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 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 when LOGSPACE = P. Hence, under the assumption of P = NP, we obtain SUCCINCT-MAXIMUM is in P-complete as well. However, working with a problem that is complete for P would mean that it would be difficult (or perhaps impossible) to show that the succinct version is in P-complete. In this way, we show some evidences which support the assumption of several computer scientists whom fully expect that P is not equal to NP.