Valiant's model and the cost of computing integers.

Abstract : Let tau(k) be the minimum number of arithmetic operations required to build the integer k from the constants 1 and 2. A sequence (x_k) is said to be ``easy to compute'' if tau(x_k) is bounded by a polynomial function of log(k). 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 P different from PSPACE.
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Pascal Koiran. Valiant's model and the cost of computing integers.. [Research Report] LP RR-2004-01, Laboratoire de l'informatique du parallélisme. 2004, 2+13p. ⟨hal-02102106⟩

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