The Problem of Computational Complexity

Abstract: This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century. Second, building on the theory of functional relations in mathematics, it provides a theoretical framework where we can rigorously distinguish two pairs of concepts: Between solving a problem and verifying the solution to a problem. Between a deterministic and a non-deterministic model of computation. Third, it presents the theory of computational complexity and the difficulties in solving the P versus NP problem. Finally, it gives a complete proof that a certain decision problem in NP has an algorithmic exponential lower bound thus establishing firmly that P is different from NP. The proof presents a new way of approaching the subject: neither by entering into the unmanageable difficulties of proving this type of lower bound for the known NP-complete problems nor by entering into the difficulties regarding the properties of the many complexity classes established since the mid-1970s.