# Lecture 24

## Probabilistically Checkable Proofs

**Theorem** MIP=class of languages that can be verified by polynomial time probabilistic verifiers in the RAM model given access to an exponential-length proof.

**Proof** MIP\(\rightarrow\)Verifier

We can encode strategy of each prover and send it over as the proof. Verifier can use the proof to simulate MIP verification protocol.

- Proof length \(\leq\) exponential
- Verifier only needs to check one path if the interaction tree so polynomial time.

Verifier\(\rightarrow\)MIP

Key idea 1: If we can ask a prover to fix a possible proof that \(x\in L\) of exponential length and not modify it midway through the interaction and allow verifier to query bits of the proof, we can design a protocol that shows \(L\in MIP\).

Does this work?

- When \(x\in L\), yes. Prover can adhere to the condition, verifier accepts.
- When \(x\not\in L\), no. If the provers do not follow the promise, we have no guarantees on probabilities that verifier rejects.

Key idea 2: Use prover \(1\) to query the bits of the proof I need to examine. Use prover \(2\) to detect if prover \(1\) failed to stick to the same initial proof.

Details in book.

**Definition** PCP\((r(n),q(n))\) (probabilistically checkable proof) is the class of languages that can be verified by a polynomial-time randomized RAM algorithm that can access a proof of length \(2^{O(r(n))\) and queries only \(q(n)\) bits of the proof.

**Theorem** PCP(poly(n), poly(n))=MIP=NEXP

**Corollary** NP=PCP(log(n),log(n))

**PCP Theorem** NP=PCP(log(n),1)

## Beyond CS 365

- CS 466: proof of correctness.
- CS 467: challenge the Church-Turing Theorem
- CS 462: Formal Languages and Parsing
- CS 489(?) by Eric
- Grad courses
- Math courses (CO)
- Research

## Final Exam

- Friday, April 21, 9:00-11:30 am, MC 4059
- Lectures, Problems, Exercises.
- One cheat sheet, regular size, double sides.
- Q&A session: 1pm.

完结撒花ヽ(*≧ω≦)/~