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$\VP = \VNP$ and GRH implies $\Ppoly = \NPpoly$
Arithmetic circuits are generalization for the study of boolean circuits. It is regarded as a weaker model in the sense that proving lower bound should be easier here. For instance, there is a super-linear lower bound for general arithmetic circuits while the state-of-the-art lower bound for general boolean circtuis is $(3+O(1))n$.
In addition to the state-of-the-art lower bounds, there is also an evidence that resolving the analogous $\P$ versus $\NP$ problem in the arithmetic setting is not more difficult than in the boolean setting. Formally, we are going to prove the following theorem.
If $\VP=\VNP$ over $\mathbb{C}$ and the Generalized Riemann Hypothesis (GRH) holds, then $\Ppoly=\NPpoly$.
Namely, to show $\Ppoly\neq\NPpoly$, it is necessary to show $\VP\neq\VNP$ over $\mathbb{C}$ if you believe GRH is correct.
Proof Sketch
Let us sketch the proof for the main theorem in this section. The goal is basically show that $\NPpoly\subseteq\Ppoly$ under (i) $\VP=\VNP$ and (ii) GRH. The proof can be divided into four steps. Before stating them, let us define $\class{BP(VP)}$ (resp. $\class{BP(VNP)}$) as the restriceted version of $\VP$ ($\VNP$) on boolean input.
- Non-uniform Toda’s theorem: $\NPpoly\subseteq\ShP\class{/poly}$
- Arithmetization: $\ShP\class{/poly}\subseteq\class{BP(VNP)}$
- Under $\VP=\VNP$: $\class{BP(VNP)}=\class{BP(VP)}$
- Under GRH: $\class{BP(VP)}\subseteq\Ppoly$
The above four steps give us $\NPpoly\subseteq\Ppoly$ and thus complete the proof. One can see that the first two steps of the proof basically follow from classic results from complexity theory. The third step is also trvially true. Thus, this notes will focus more on the last step. Nevertheless, for the completeness of the proof, we will briefly explain the first two stesp in this section.
Step 1: $\NPpoly\subseteq\ShP\class{/poly}$
Recall that Toda showed that $\PH\subseteq\P^{\ShP}$ while here we only need to go to the first level of polynomial time hierarchy, though in the non-uniform setting. Nevertheless, we actually don’t have to go through the whole proof of Toda’s theorem. $\NPpoly\in\P^{\ShP}\class{/poly}$ can be easily shown as follows.
Let $L$ be a language in $\NPpoly$, there exists a polynomial size circuit family ${C_n}$, such that $x\in L$ iff there exists $y$ such that $C_{\card{x}}(x,y)=1$. Now, the difficulty here is constructing a new polynomial size circuit family ${C_n’}$ such that $\sum_yC_{\card{x}}(x,y)=\mathbf{1}_{x\in L}$.
The main idea is derandomizing the Valiant-Vazirani randomized reduction for Unique-SAT ($\USAT$). Concretely, Valiant and Vazirani showed that SAT ($\SAT$) can be randomized reduced to $\USAT$ in the sense that there is a polynomil time algorithm given an $n$-variate predicate $\phi(x,y)$ outputs $\phi’(x,z)$ such that $\phi\in\SAT\Leftrightarrow\phi’\in\USAT$ with constant probability.
In uniform complexity classes, it is not clear how to derandomize Valiant-Vazirani, however, in the non-uniform setting, this can be trivially done by error amplification and guess the certificate $z^*$ that works for every input $x$. Thus, we conclude that $\NPpoly\subseteq\ShP\class{/poly}$.
Step 2: $\P^{\ShP}\class{/poly}\subseteq\class{BP(VNP)}$
This step is a simple arithmetization by replacing boolean gates to arithmetic operation over $\mathbb{C}$. Concretely,
- for an AND gate $x\wedge y$, replace it with $x\cdot y$,
- for an OR gate $x\vee y$, replace it with $1-(1-x)\cdot(1-y)$, and
- for a NOT gate $\neg x$, replace it with $1-x$.
Note that this should also work for larget enough finite field.
Generalized Riemann Hypothesis (GRH)
In this application, we actually don’t need the full power of GRH. What we need is its consequence on guaranteeing the existence of some desired prime numbers.
Once we focus on this specific application on the number of prime numbers, then the Riemann Hypothesis (RH) implies the following.
\begin{equation}
\card{\pi(x)-li(x)}=O(\sqrt{x}\log x)
\end{equation}
for integer $x$ large enough where $\pi(x)$ is the number of prime numbers less than $x$ and $li(x)=\Theta(\frac{x}{\log x})$.
The Generalized Riemann Hypothesis (ERH) basically extends RH to number fields or other mathematical objects.
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