Complexity

1
Complexity
27-1
Interactive Proofs (continued)
Complexity Andrei Bulatov
2
Complexity
27-2
IP PSPACE
Proof.
IP ? PSPACE
If we consider Provers messages as
nondeterministic guesses, then we get IP ? NPSPACE
Then, by Savitchs theorem
IP ? NPSPACE PSPACE
3
Complexity
27-3
PSPACE ? IP
It is sufficient to show that some
PSPACE-complete problem belongs to IP
4
Complexity
27-4
Arithmetization
Given a formula
Let be the
arithmetization of ?
Then define polynomials
by setting
5
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
6
Complexity
27-6
Properties of the new polynomials
7
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
8
Complexity
27-8
Step k 1.
Verifier checks if If yes then Verifier accept,
if not rejects