A mathematical definition of "simplify"

TL;DR

This paper proposes a rigorous mathematical definition of 'simplify' based on computational complexity and demonstrates that no deterministic exact algorithm can compute the matrix permanent faster than exponential time.

Contribution

It introduces a formal complexity-based definition of 'simplify' and applies it to prove a lower bound on computing the matrix permanent.

Findings

Defined 'simplify' using computational complexity

Proved no deterministic exact algorithm computes permanent in subexponential time

Established a complexity-theoretic lower bound for matrix permanent computation

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.