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, NLOGSPACE, coNP and EXP. Whether LOGSPACE = NLOGSPACE 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 and thus, LOGSPACE is equal to NLOGSPACE. Furthermore, 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 P under the assumption of P = NP. 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. Indeed, since SUCCINCT-MAXIMUM is a succinct version of a P-complete problem under the complexity of properties of succinctly representable graphs, then this might be a good candidate to be in EXP-complete and therefore, this would imply the complexity class P is not equal to NP as a consequence of the Hierarchy Theorem