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.


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?

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

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