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### Paper:

TR04-003 | 22nd December 2003 00:00

#### Valiant's model and the cost of computing integers

TR04-003
Authors: Pascal Koiran
Publication: 12th January 2004 20:28
Keywords:

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.

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