Valiant’s model and the cost of computing integers

Abstract.

Let τ(n) be the minimum number of arithmetic operations required to build the integer $n \in \mathbb{N}$ from the constants 1 and 2. A sequence x _n is said to be “easy to compute” if there exists a polynomial p such that $\tau (x_n ) \leq p(\log n)$ for all It is natural to conjecture that sequences such as $\left\lfloor {2^n \ln 2} \right\rfloor$ 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 P ≠ PSPACE.