# Lecture 22

## State of the Art on BPP

BPP = languages decidable with bounded error by polynomial time probabilistic Turing machine.

RP = languages decidable with $$1$$-sided error by polynomial time probabilistic Turing machine.

ZPP = languages decidable with no error by expected polynomial time probabilistic Turing machine.

P $$\subseteq$$ ZPP $$\subseteq$$ RP $$\subseteq$$ BPP

False Claim BPP $$\subseteq$$ NP because for $$L\in$$BPP and input $$x$$, we can have verifier that takes sequence of coins that leads to accept state as the certificate.

When $$x\in L$$, so there exists valid certificate. But when $$x\not\in L$$, there exists certificates that cause the verifier to accept.

Theorem RP$$\subseteq$$NP

Question 1 Is BPP$$\subseteq$$NP? Unknown.

Theorem BPP$$\subseteq$$EXP

Proof Brute-force simulation of the probabilistic Turing machine and count accepting paths.

Question 2 Are there BPP-complete languages? Unknown.

Question 3 Is $$BPTIME(n^2)\neq BPTIME(n)$$? Unknow.

## Interactive Proofs

Definition The class DIP is the class of languages that can be decided in polynomial time by a verifier that interacts with a prover, such that

1. If $$x\in L$$, then prover can always convince the verifier with the interaction
2. If $$x\in L$$, then no possible interaction convinces the verifier (erroneously)

Theorem DIP$$=$$NP

Proof $$NP\subseteq DIP$$ is obvious.

To show $$DIP\subseteq NP$$, we note that DIP verifier $$V$$ can be simulated by an NP verifier $$V'$$ that receives the whole transcript $$c=(m_1,m_2,...)$$ that causes $$V$$ to accept as its certificate.

Definition $$IP$$ is the class of languages decidable with bounded error by a polynomial time probabilistic Turing machine that interacts with a prover.

1. If $$x\in L$$, the prover has a strategy to answer verifier's questions in a way that causes verifier to accept with probability $$p\geq\frac{2}{3}$$.
2. If $$x\not\in L$$, any strategy of the prover still causes the verifier to reject with probability $$p\geq\frac{2}{3}$$

Theorem IP$$\subseteq$$NEXP

## Graph Isomorphism

$$GI=\{\langle G_1,G_2\rangle: G_1\cong G_2\}$$ or $$GNI=\{\langle G_1,G_2\rangle: G_1$$ is not isomorphic to $$G_2\}$$.

Theorem $$G_1\in$$ NP

Proof The certificate is the mapping between $$G_1$$ and $$G_2$$.

Question Is $$GNI\in NP$$? Unknown for this class.

Theorem $$GNI\in IP$$

Proof The verifier drawn $$i\in\{1,2\}$$ uniformly and create $$H$$ by randomly permuting the labels of the vertices of $$G_1$$. The verifier sends $$H$$ to the prover, expects $$i$$ as the response ($$G_i\cong H$$). If $$G_1\not\cong G_2$$, then prover can always determine $$i$$. If $$G_1\cong G_2$$, then prover guess $$i$$ correctly with probability $$\frac{1}{2}$$. With $$2$$ repetitions, error is $$\leq\frac{1}{4}$$