VP Definition: Why polynomial bound on the degree? The complexity class 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.

Alfredo Cooley

Alfredo Cooley

Answered question

2022-11-05

VP Definition: Why polynomial bound on the degree?
The complexity class 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?

Answer & Explanation

zastenjkcy

zastenjkcy

Beginner2022-11-06Added 14 answers

Step 1
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.
Step 2
2. A bigger class, called V P n b in Interpolation in Valiant's theory (where it is attributed to a 2003 PhD thesis by Guillaume Malod).

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