Parallel Repetition of
Two Prover Games
Ran RazWeizmann Institute
and IAS
Two Prover Games [BGKW]:Player A gets xPlayer B gets y (x,y) 2R publicly known
distributionPlayer A answers a=A(x)Player B answers b=B(y)They win if V(x,y,a,b)=1(V is a publicly known
function)
Val(G) = MaxA,B Prx,y
[V(x,y,a,b)=1]
Example:Player A gets x 2R {1,2}
Player B gets y 2R {3,4}
A answers a=A(x) 2 {1,2,3,4}B answers b=B(y) 2 {1,2,3,4}They win if a=b=x or a=b=y
Val(G) = ½(protocol: a=x, b 2R {1,2})
(alternatively : b=y, a 2R {3,4})
Projection Games:8x,y, 8a, 9 unique b, such that V(x,y,a,b)=1
Probabilistic Protocols:Player A answers a=A(x)Player B answers b=B(y)Where A(x),B(y) may depend
on acommon random string
However:Probabilistic Value = Deterministic Value
Parallel Repetition:A gets x = (x1,..,xn)
B gets y = (y1,..,yn)
(xi,yi) 2R the original distribution
A answers a=(a1,..,an) =A(x)
B answers b=(b1,..,bn) =B(y)
V(x,y,a,b) =1 iff 8i V(xi,yi,ai,bi)=1
Val(Gn) = MaxA,B Prx,y
[V(x,y,a,b)=1]
Parallel Repetition:A gets x = (x1,..,xn)
B gets y = (y1,..,yn)
(xi,yi) 2R the original distribution
A answers a=(a1,..,an) =A(x)
B answers b=(b1,..,bn) =B(y)
V(x,y,a,b) =1 iff 8i V(xi,yi,ai,bi)=1
Val(Gn) = MaxA,B Prx,y [V(x,y,a,b)=1]
Val(G) ¸ Val(Gn) ¸ Val(G)n
Is Val(Gn) = Val(G)n ?
Example [Feige]:A gets x1,x2 2R {1,2}
B gets y1,y2 2R {3,4}
A answers a1,a2 2 {1,2,3,4}
B answers b1,b2 2 {1,2,3,4}
They win if 8i ai=bi=xi or ai=bi=yi
Val(G2) = ½ = Val(G)By: a1=x1, b1=y2-2, a2=x1+2,
b2=y2
(they win iff x1=y2-2)
Parallel Repetition Theorem [R-95]:
(Conjectured by Feige and Lovasz)
8G Val(G) < 1 ) 9 w < 1
(s = length of answers in G)
Assume that Val(G) = 1-What can we say about w ?
Parallel Repetition Theorem:
Val(G) = 1- , ( < ½) )
[R-95]:
[Hol-06]:
For projection games:
[Rao-07]:
(s = length of answers in G)
Strong Parallel Repetition Problem:
Is the following true ?
Val(G) = 1- , ( < ½) )
(for any game or for interesting special cases)
[R-08]: G s.t.: Val(G) = 1-,
Over Expander Graphs [RR-10]:
Val(G) = 1- , ( < ½) )
For general games:
For projection games:
Applications of Parallel Repetition:
1) PCP & Hardness of Approximation: [BGS],[Has],[Fei],[Kho],...
2) Geometry: understanding foams, tiling the space Rn [FKO],[KORW],[AK]
3) Quantum Information: strong EPR paradoxes [CHTW]
4) Communication Complexity: direct sum/product results [PRW],[BBCR]
5) Cryptography
Parallel Repetition, PCP, and
Hardness of Approximation
[BGS],[Has],[Fei],[Kho],...
PCP Theorem [BFL,FGLSS,AS,ALMSS]:
Any (length n) proof can be rewritten
as a length poly(n) proof that can be
(probabilistically) verified by reading a
constant number of bits.
proof is correct ) P(V accepts) = 1statement has no proof ) P(V¼ accepts) ≤ 1-ε
Two Query PCP:Any (length n) proof can be
rewrittenas a length poly(n) proof that can
beverified by reading only 2 symbols (O(1) bits)
proof is correct ) P(V accepts) = 1statement has no proof ) P(V¼ accepts) ≤ 1-ε
PCP as Hardness of Approximation:
Given a two-prover game G (with constant answer size) It is NP hard to distinguish
between:Val(G) = 1 and Val(G) · 1-(for some constant > 0)
[FGLSS]Using Parallel Repetition:It is NP hard to distinguish
between:Val(G) = 1 and Val(G) · (for any constant > 0)
Hardness of Approximation Results:
[BGS],[Has],[Fei],[Kho],...Optimal hardness results for:3SAT, 3LIN, Set-Cover,…
Example: 3LIN [Has-98]:Given a set of linear equations over
GF[2]It is NP hard to distinguish
between:
9¼ solution that satisfies > 1-ε fraction andevery solution satisfies < ½+ε
fraction
PCP Theorem (2 Queries):Original:
Using Parallel Repetition:
proof is correct ) P(V accepts) = 1statement has no proof ) P(V¼ accepts) ≤ 1-ε(for some constant ε > 0)
proof is correct ) P(V accepts) = 1statement has no proof ) P(V¼ accepts) ≤ ε(for any constant ε > 0)
Tiling Rn, Cubical Foams, and
Parallel Repetition of the
Odd Cycle Game
[FKO],[KORW],[AK],[A]...
Cubical Foams:
The unit cube C = tiles Rn bythe lattice Zn. Rn = C £ Zn
What is the minimalsurface area of a(volume 1) object Dthat tiles Rn by Zn ?(Rn = D £ Zn )
Best Cubical Foams:
Minimal surface area of a (volume 1)
object that tiles Rn by Zn :
Upper bound: 2n : unit cube
Lower bound: : volume 1 ball
Best Cubical Foams:
Minimal surface area of a (volume 1)
object that tiles Rn by Zn :
Upper bound: 2n : unit cube
Lower bound: : volume 1 ball
[KORW-08]: There is an object with surface area
similarto the (volume 1) ball, that tiles Rn as a
cube
Odd Cycle Game [CHTW,FKO]:A gets x 2R {1,..,m} (m is odd)
B gets y 2R {x,x-1,x+1} (mod m)
A answers a=A(x) 2 {0,1}B answers b=B(y) 2 {0,1}They win if x=y , a=b
Odd Cycle Game [CHTW,FKO]:A gets x 2R {1,..,m} (m is odd)
B gets y 2R {x,x-1,x+1} (mod m)
A answers a=A(x) 2 {0,1}B answers b=B(y) 2 {0,1}They win if x=y , a=b
To win with probability 1, the players need
to 2-color the odd cycle consistently1
0
1 1
0
Protocol for the Odd Cycle Game:
A gets x, B gets y. They win if(x,y) e (e breaks the cycle)
xy
e
0
0
0
01
1
1
1
1
Parallel Repetition of OCG:A gets x1,..,xn 2R {1,..,m}
B gets y1,..,yn 2R {1,..,m}
8 i yi 2R {xi,xi-1,xi+1} (mod m)
A answers a1,..,an 2 {0,1}
B answers b1,..,bn 2 {0,1}
They win if 8 i xi=yi , ai=bi
1 2 3 n
Parallel Repetition of OCG:A gets x1,..,xn 2R {1,..,m}
B gets y1,..,yn 2R {1,..,m}
8 i yi 2R {xi,xi-1,xi+1} (mod m)
A answers a1,..,an 2 {0,1}
B answers b1,..,bn 2 {0,1}
They win if 8 i xi=yi , ai=bi
We think of the game as played on
then dimensional torus
1 2 3 n
Parallel Repetition of OCG:A gets x1,..,xn 2R {1,..,m}
B gets y1,..,yn 2R {1,..,m}
8 i yi 2R {xi,xi-1,xi+1} (mod m)
A answers a1,..,an 2 {0,1}
B answers b1,..,bn 2 {0,1}
They win if 8 i xi=yi , ai=bi
A,B answer by a color for each coordinate. They win if the colors are consistent on allcoordinates
Odd Cycle Game and Cubical Foams:
The players can color consistently
everycell. Err on edges that cross the surface
The Shorter Story:
[FKO-07]: Cubical Foams implyprotocols for OCG
[R-08]: (a counterexample to strong par.
rep.)
[KORW-08]: Cubical Foams withsurface area (based on the protocol for
OCG)
Cubical Foams:
Minimal surface area of a (volume 1)
cell that tiles Rn by Zn :
Upper bound: 2n : unit cube
Lower bound: : volume 1 ball
[KORW-08]: There is an object with surface area
similarto the (volume 1) ball, that tiles Rn as a
cube
Parallel Repetition, Bell
Inequalities,
and the EPR Paradox
[CHTW]
Entangled Two Prover Games:A,B share entangled quantum state
|siA,B
A gets x, B gets y A measures A, B measures BA answers a, B answers b They win if V(x,y,a,b)=1ValQ(G) = MaxA,B Prx,y
[V(x,y,a,b)=1]
Bell Inequalities (EPR Paradox):
9 G s.t. ValQ(G) > Val(G)
[CHTW 04]: 9 G s.t. ValQ(G) = 1 and Val(G) · 1-(for some constant > 0)Using Parallel Repetition: 9 G
s.t. ValQ(G) = 1 and Val(G) · (for any constant > 0)
Does God Play Dice ?a,b are the outcome of a
quantummeasurement. Does God play
dice ?Hidden Variables Theory: 9 additional variables H, s.t.a=a(x,y,H), b=b(x,y,H) (deterministically)H = outcome of all possible
measurements (independent of x,y)
a=a(x,y,H), b=b(x,y,H)Assume: x,y are independentA chooses x, B chooses y (at time
t-²)Measurements occur at time tInformation cannot propagate
fasterthan light! Hence,Local Hidden Variables Theory:a=a(x,H), b=b(y,H) Thus, ValQ(G) = Val(G)
Bell Inequalities (EPR Paradox):
9 G s.t. ValQ(G) > Val(G)
[CHTW 04]: 9 G s.t. ValQ(G) = 1 and Val(G) · 1-(for some constant > 0)Using Parallel Repetition: 9 G
s.t. ValQ(G) = 1 and Val(G) · (for any constant > 0)
Thank You!
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