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The complexity class $\mathbf{VP}$ (Valiant P) is defined to be the class of all polynomials of polynomially bounded degree which can be realized by an arithmetic circuit family with polynomially bounded size.

My questions are:

  1. Why do we require the degree to be polynomially bounded?
  2. Which complexity class to we get if we omit this constraint?
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up vote 1 down vote accepted
  1. It rules out sequences like $p_n=X^{2^n}$, which can be computed by a straight-line program using $n$ multiplications, but on a Turing machine these multiplications would take exponentially long. However, VP does allow things like $p_n=X+2^{2^n}$, which is still weird from the Turing machine point of view.

  2. A bigger class, called $VP_{nb}$ in Interpolation in Valiant's theory (where it is attributed to a 2003 PhD thesis by Guillaume Malod).

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