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 GoldwasserMicaliRackoff and Babai. Goldwasser
MicaliWigderson were focused on the cryptographic implications of this technique; they are interested in
zeroknowledge
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 oneway functions.
Another development on this line was made by BenOrGoldwasserKilianWigderson 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 informationexchange is allowed between two provers. The
verifier can them questions in any order. The authors showed that with 2 provers, zeroknowledge proofs
are possible without assuming the existence of oneway 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 FortnowRompelSipser. 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 queryhistory) 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 deterministicexptime
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 randomizedpolytime
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, BabaiFortnowLund showed the other
direction
OIP
⊇
NEXPT IME
.
Using this relation
MIP
=
OIP
=
NEXPT IME
, FeigeGoldwasser
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 Spring '09
 MadhuSudan
 Computational complexity theory, Interactive proof system, P CP, EXP T IM

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