Take the 2-minute tour ×
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It's 100% free, no registration required.

I'd like your help with the following question:

Assume we proved that $\mbox{BPP}\subseteq \Pi_2$ -What conclusions can you make?

BPP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1/3 for all instances,

$\Pi_2$ is the class of all languages $L$ such that there's a polynomial algorithm $M$ and a polynom $p$ so that $\forall x.x\in L\Leftrightarrow \forall u\in \{ 0,1 \}^*.\exists v \in \{ 0,1 \}^*.M(x,u,v)=1$.

We already know that $\mbox{BPP}\subseteq \Sigma_2$, so $\mbox{BPP}\subseteq \Pi_2\cap \Sigma_2$.

share|improve this question

1 Answer 1

It is known. As your final statement says, $\mbox{BPP} \subseteq \Pi_2 \cap \Sigma_2$. It is called the Sipser–Gács–Lautemann theorem. (All though your "so" is somewhat misleading.)

It can be strengthened to $\mbox{BPP} \subseteq \mbox{MA} \subseteq \mbox{S}^P_2 \subseteq \Pi_2 \cap \Sigma_2 \subseteq \Pi_2$ (see also Arthur-Merlin protocol).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.