**Abstract** : 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 where S is represented by an array. 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. We define a problem called CIRCUIT-MAXIMUM. CIRCUIT-MAXIMUM is nothing else but the instances of MAXIMUM represented by a positive integer x and a Boolean circuit C which represents the collection S. We show this version of MAXIMUM is in coNP-complete. In addition, CIRCUIT-MAXIMUM contains the instances of MAXIMUM that can be represented by an exponentially more succinct way. In this way, we show the succinct representation of a P-complete problem is indeed in coNP-complete.