6.841 Advanced Complexity Theory
Apr 13, 2009
Lecture 18
Lecturer: Madhu Sudan
Scribe: Jinwoo Shin
In this lecture, we will discuss about
PCP
(
probabilistically checkable proof
). We first overview the history
followed by the formal definition of
PCP
and then study its connection to inapproximability results.
1
History of
PCP
Interactive proofs were first independently investigated by Goldwasser-Micali-Rackoff and Babai. Goldwasser-
Micali-Wigderson were focused on the cryptographic implications of this technique; they are interested in
zero-knowledge
proofs, which prove that one has a solution of a hard problem without revealing the solution
itself. Their approach is based on the existence of one-way functions.
Another development on this line was made by BenOr-Goldwasser-Kilian-Wigderson who introduced
several provers. 2
IP
has two provers who may have agreed on their strategies before the interactive proof
starts. However, during the interactive proof, no information-exchange is allowed between two provers. The
verifier can them questions in any order. The authors showed that with 2 provers, zero-knowledge proofs
are possible without assuming the existence of one-way functions.
Clearly, 2
IP
is at least as powerful as
IP
since the verifier could just ignore the second prover. Further-
more, one can define 3
IP
, 4
IP
,
. . . , MIP
and naturally ask whether those classes get progressively more
powerful. It turns out that the answer is no by Fortnow-Rompel-Sipser. They proved that 2
IP
= 3
IP
=
. . .
=
MIP
=
OIP
, where
OIP
is called
oracle interactive proof
.
OIP
is an interactive proof system with a
memoryless oracle; all answers are committed to the oracle before the interactive proof starts and the oracle
just answer them (without considering the previous query-history) when the verifier query them.
Now we consider the difference between
OIP
and
NEXPT IME
. They both have an input
x
of
n
bits
and a proof(certificate)
π
of length 2
n
c
for some constant
c
.
The difference relies on the verifier
V
.
If
L
∈
NEXPT IME
, there exists a deterministic-exptime
V
such that
•
If
x
∈
L
, then there exists a
π
such that
V
(
x, π
) accepts,
•
If
x /
∈
L
, then
∀
π
,
V
(
x, π
) rejects.
Similarly, if
L
∈
OIP
, there exists a randomized-polytime
V
such that
•
If
x
∈
L
, then there exists a
π
such that
V
(
x, π
) accepts with high probability,
•
If
x /
∈
L
, then
∀
π
,
V
(
x, π
) accepts with low probability.
Hence the relation
OIP
⊆
NEXPT IME
is obvious since one can build the verifier for
NEXPT IME
by simulating all random coins of
V
in
OIP
.
More interestingly, Babai-Fortnow-Lund showed the other
direction
OIP
⊇
NEXPT IME
.
Using this relation
MIP
=
OIP
=
NEXPT IME
, Feige-Goldwasser-
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- Spring '09
- MadhuSudan
- Computational complexity theory, Interactive proof system, P CP, EXP T IM
-
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