Abstract:
Let $\tau(k)$ be the minimum number of arithmetic operations required to build the integer $k \in \N$ from the constant 1. A sequence $x_k$ is said to be ``easy to compute'' if there exists a polynomial $p$ such that $\tau(x_k) \leq p(\log k)$ for all $k \geq 1$. It is natural to conjecture that sequences such as $\lfloor 2^n \ln 2 \rfloor$ or $n!$ are not easy to compute. In this paper we show that a proof of this conjecture for the first sequence would imply a superpolynomial lower bound for the arithmetic circuit size of the permanent polynomial. For the second sequence, a proof would imply a superpolynomial lower bound for the permanent or separate P from PSPACE.