Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \mathsf{P}_2 \cap \Pi^ \mathsf{P}_2$. A step towards showing $\mathsf{BPP} = \mathsf{P}$ is to first bring it down to the first level of the polynomial hierarchy, i.e. $\mathsf{BPP} \subseteq \mathsf{NP}$.

The containment would mean that nondeterminism is at least as powerful as randomness for polynomial time.

It also means that if for a problem we can find the answers using efficient (polynomial time) randomized algorithms then we can verify the answers efficiently (in polynomial time) .

Are there any known interesting consequences for $\mathsf{BPP} \subseteq \mathsf{NP}$?

 

Are there any reasons to believe that proving $\mathsf{BPP} \subseteq \mathsf{NP}$ is out of reach right now (e.g. barriers or other arguments)?

Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \mathsf{P}_2 \cap \Pi^ \mathsf{P}_2$. A step towards showing $\mathsf{BPP} = \mathsf{P}$ is to first bring it down to the first level of the polynomial hierarchy, i.e. $\mathsf{BPP} \subseteq \mathsf{NP}$.

The containment would mean that nondeterminism is at least as powerful as randomness for polynomial time.

It also means that if for a problem we can find the answers using efficient (polynomial time) randomized algorithms then we can verify the answers efficiently (in polynomial time) .

Are there any known interesting consequences for $\mathsf{BPP} \subseteq \mathsf{NP}$?

Are there any reasons to believe that proving $\mathsf{BPP} \subseteq \mathsf{NP}$ is out of reach right now (e.g. barriers or other arguments)?

Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \mathsf{P}_2 \cap \Pi^ \mathsf{P}_2$. A step towards showing $\mathsf{BPP} = \mathsf{P}$ is to first bring it down to the first level of the polynomial hierarchy, i.e. $\mathsf{BPP} \subseteq \mathsf{NP}$.

The containment would mean that if for a problem we can find the answers using efficient (polynomial time) randomized time algorithms then we can verify the answers efficiently (in polynomial time) and nondeterminism is at least as powerful as randomness for polynomial time.

Are there any known interesting consequences for $\mathsf{BPP} \subseteq \mathsf{NP}$?

Are there any reasons to believe that proving $\mathsf{BPP} \subseteq \mathsf{NP}$ is out of reach right now (e.g. barriers or other arguments)?

Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \mathsf{P}_2 \cap \Pi^ \mathsf{P}_2$. A step towards showing $\mathsf{BPP} = \mathsf{P}$ is to first bring it down to the first level of the polynomial hierarchy, i.e. $\mathsf{BPP} \subseteq \mathsf{NP}$.

The containment would mean that nondeterminism is at least as powerful as randomness for polynomial time.

It also means that if for a problem we can find the answers using efficient (polynomial time) randomized algorithms then we can verify the answers efficiently (in polynomial time) .

Are there any known interesting consequences for $\mathsf{BPP} \subseteq \mathsf{NP}$?

Are there any reasons to believe that proving $\mathsf{BPP} \subseteq \mathsf{NP}$ is out of reach right now (e.g. barriers or other arguments)?