A mathematical definition of "simplify"

Abstract

Even though every mathematician knows intuitively what it means to "simplify" a mathematical expression, there is still no universally accepted rigorous mathematical definition of "simplify". In this paper, we shall give a simple and plausible definition of "simplify" in terms of the computational complexity of integer functions. We shall also use this definition to show that there is no deterministic and exact algorithm which can compute the permanent of an $n \times n$ matrix in $o(2^n)$ time.