Complexity 27-1 Complexity Andrei Bulatov Interactive Proofs (continued)

Complexity 27-2 IP = PSPACE Theorem IP = PSPACE Proof. IP  PSPACE If we consider Prover’s messages as nondeterministic guesses, then we get IP  NPSPACE Then, by Savitch’s theorem IP  NPSPACE = PSPACE

Complexity 27-3 PSPACE  IP It is sufficient to show that some PSPACE-complete problem belongs to IP Instance: A quantified Boolean formula where each is a Boolean variable, is a Boolean expression involving and each is a quantifier (  or  ). Question: Is  logically valid ? Quantified Boolean Formula

Complexity 27-4 Arithmetization Given a formula Let be the arithmetization of  Then define polynomials by setting Clearly,  is true if and only if

Complexity 24-5 Reducing degree Since the degree of may be exponential, we need to reduce it. Replace  with or where and We define as follows:

Complexity 27-6 Properties of the new polynomials If then when is linear in x Therefore if then is a linear polynomial

Complexity 27-7 Protocol Step 0. P  V: Prover sends to Verifier Verifier checks if and reject if not Step i. P  V: Prover sends as a polynomial in z. Here denotes the previously selected random values for variables Verifier computes and. Then it checks the degree of the polynomial and that or If either fails, Verifier rejects V  P: Verifier picks a random value and sends it to Prover

Complexity 27-8 Step k + 1. Verifier checks if If yes then Verifier accept, if not rejects