Transcript Slide 1

AM With Multiple Merlins
Scott Aaronson (MIT)
Dana Moshkovitz (MIT)
Aaronson
Russell Scott
Impagliazzo
(UCSD)
MIT
Two-Prover Games
(the first slide of, like, half of all complexity talks)
Merlin1
Arthur
yY
xX
b(y)B
a(x)A
 x, y  ~ D
Merlin2
V  x, y, a, b   0,1
“VALUE” OF THE GAME (WHAT THE MERLINS ARE TRYING TO MAXIMIZE):
 G  :
max
a: X  A,b:Y  B
E V  x, y, a x , b y 
 x , y ~ D
The PCP Theorem: Given G=(X,Y,A,B,D,V), it’s NP-hard
even just to decide whether (G)=1 or (G)<0.01
The “Scaled-Up” Version [BFL’91]: MIP = NEXP
This work: What if the challenges to the
Merlins have to be independent?
“Free Games”: G’s for which D is a product distribution
Or for simplicity, let’s say, the uniform distribution
A known concept in PCP. Yet we seem to be the first to
explicitly study the complexity of free games
AM(2): Complexity class based on free games. Twoprover, one-round MIP, but where Arthur’s challenges
to the two non-communicating Merlins have to be
independent, uniform random strings
Obvious Objection: The whole power of MIP comes
from Arthur’s ability to correlate questions—take that
away, and two-prover games should become trivial!
As we’ll see, that’s not entirely true…
Summary of Results
Result #1: There’s an AM(2) protocol by which Arthur
can become convinced that a 3SAT instance of size n is
satisfiable, by sending just Õ(n) random bits to the
Merlins, and getting back Õ(n)-bit answers
Assuming the ETH, both of these
results imply the other’s nearoptimality!
Result #2: Given a free game G of size n, there’s an
O  2 log n  algorithm to approximate (G) within 
n
3SAT instance

Can approximate
(G) (and thereby
decide ) in time
Free game
G of size

n
2

O

n
2
O

log 2O 
n

2
O n 
Which means that, assuming 3SAT requires 2Ω(n) time:
~
 
• AM(2) protocols for 3SAT need  n communication
• Approximating free games requires n
~
 log n 
time
• Approximating dense CSPs with polynomial-size
 log n 
n
alphabets also requires
time
~
[Barak et al. 2011] gave an nO(log n)-time algorithm for such
CSPs, but its running time was never previously explained
Going Further
Our algorithm for free games implies AM(2)  EXP—
improving on the trivial bound AM(2)  MIP = NEXP
But AM  AM(2)  EXP is still quite a gap!
Result #3: AM(2) = AM
(with an inherent quadratic blowup in communication)
And more generally, AM(k) = AM for all k=poly(n)
Proof relies heavily on previous work on dense CSPs:
[Alon et al. 2002], [Barak et al. 2011]
Result #1: 3SAT Protocol
Let  be a 3SAT instance of size n. Can assume w.l.o.g. that
 is a balanced PCP, with only polylog blowup [Dinur 2006]
Standard “Clause/Variable Game”:
Random clause C
Random variable xC
CHECKS SATISFACTION & CONSISTENCY
Assignment to C
Assignment to x
“Birthday Game”:
Clauses C1,…,CK
K, L  n
Variables x1,…,xL
CHECKS SATISFACTION & CONSISTENCY ON BIRTHDAY COLLISIONS
Assignments to C1,…,CK
Assignments to x1,…,xL
Proving The 3SAT Protocol Sound
Suppose the Merlins can cheat in the “birthday game.”
We show how they can also cheat in the original
clause/variable game, thereby giving a contradiction
Clause C
“Smuggles” C among
random clauses
C1,…,CK that he picks
himself
Variable xC
“Smuggles” x among
random variables
x1,…,xL that he picks
himself
Then the Merlins run their birthday strategy on C1,…,CK
and x1,…,xL, and return the results restricted to C and x

O

Key Technical Claim (proved with second-moment method):
The induced distribution over C1,…,CK and x1,…,xL is
n 
-close in variation distance to the uniform distribution
KL 

And then we’re done!
High-Error Case: If we only want a 1 vs. 1- soundness
gap, a different argument gives an AM(2) protocol for
3SAT with O n polylog n communication.
Hence, assuming ETH, deciding whether a free game G
~
satisfies (G)=1 or (G)<1- requires n   1 log n  time


Low-Error Case: If we want a 1 vs.  gap, switching from
[Dinur 2006] to [Moshkovitz-Raz 2008] gives an AM(2)
protocol for 3SAT with n1/ 2o 1poly1 /   communication.
Hence, assuming ETH, deciding whether a free game G
satisfies (G)=1 or (G)< requires n poly  log n 1o 1 time
Result #2: Approximation Algorithm
for Free Games
S
Best responses
xX
yY
Let v be the value of the
Best responses
best pair of strategies that
algorithm finds
Followup Workthis
[Brandão-Harrow]:
algorithm
for
Clearly
v(G)
 log YABdifferent



S  O approximating
free games, with
2




Furthermore, v(G)-
exactly the same running time as
w.h.p.
over
S,
by
union
and
Loop over all possible
ours, but based on LP relaxation
strategies on S
Chernoff bounds

Algorithm’s Running Time: O A
S


X AY n
Can derandomize by looping over all possible S
O  2 log n

Result #3: AM(2) = AM
xX
yY
S
T
Subsampling Theorem: Let G be
any free game, and let GS,T be the
subgame induced by restricting
Merlin1’s challenges to SX and
Merlin2’s to TY, where
|S|=|T|=log(|A||B|)/O(1). Then
 G   E  GS ,T    G   
S ,T
The AM simulation of an
AM(2) protocol is then
Trivial
simply: Arthur chooses S,T,
then Merlin replies with
a:SA, b:TB, then Arthur
verifies that (GS,T) is large
Not Trivial
(but [Alon et al. 2002],
[Barak et al. 2011] already
did most of the work)
Generalizing to k Merlins
Let G be a k-player free game (k3). By applying our twoplayer algorithm recursively, to “peel off Merlins one at a
time,” we can approximate (G) to within  in time

n
O  2 k 2 log n

This implies (1) ~AM(k)  EXP, and (2) any AM(k) protocol
for 3SAT needs n1/ 4  communication assuming the ETH
Can’t we do better, by encoding free games as dense CSPs?
Alas, straightforward encoding fails when k=(log n)
We find a better encoding, which yields: (1) AM(k) = AM
for
all k=poly(n), and (2) any AM(k) protocol for 3SAT needs
~
 n total communication (assuming ETH)
 
Quantum Motivation
QMA(2): Arthur receives two unentangled quantum
proofs, |1 from Merlin1 and|2 from Merlin2
Best current knowledge: QMA  QMA(2)  NEXP.
Pathetic!
Upshot
of This
Work: protocol to
[ABDFS, CCC’2008]:
There’s
a QMA(2)
Everything
we’d
prove that a 3SAT
instance of
sizelike
n istosatisfiable, using
provewith
about
QMA(2),
quantum messages
Õ(n)
qubitswe
only
canTheorem
prove about
Protocol uses PCP
andAM(2)!
Birthday Paradox in
almost exactly the same way as our AM(2) protocol!
Conjectures: QMA(2)  EXP. The square-root savings
of [ABDFS’2008] is optimal, assuming the ETH.
Slide Where I Try to Provoke You
Should one call results like ours
“evidence” for the ETH?
Think about it: we gave an Õ(n)-communication
AM(2) protocol for 3SAT, and an nO(log n) approximation
algorithm for free games. Neither result “knew about
the other.” Yet, if either had been slightly better, their
combination would’ve falsified ETH. So if ETH is false,
how did the two results “coordinate”?
Open Problems
Õ  O? 1/2  1/?
Birthday Repetition Theorem?
Is our 3SAT protocol non-algebrizing?
It’s definitely non-relativizing
Better approximation algorithms for free projection games
and free unique games?
Conjecture: Exists a PTAS but not an FPTAS
AM(2) with entangled provers?
Use our techniques to show nΩ(log n) hardness for
approximate Nash equilibrium, assuming ETH?
[Hazan-Krauthgamer 2009]: assuming planted clique is hard