25 research outputs found
On Fortification of Projection Games
A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious
method to provide a completely elementary proof of the Parallel Repetition
Theorem for certain projection games via a construction called fortification.
However, the construction used in \cite{Moshkovitz14} to fortify arbitrary
label cover instances using an arbitrary extractor is insufficient to prove
parallel repetition. In this paper, we provide a fix by using a stronger graph
that we call fortifiers. Fortifiers are graphs that have both and
guarantees on induced distributions from large subsets. We then show
that an expander with sufficient spectral gap, or a bi-regular extractor with
stronger parameters (the latter is also the construction used in an independent
update \cite{Moshkovitz15} of \cite{Moshkovitz14} with an alternate argument),
is a good fortifier. We also show that using a fortifier (in particular
guarantees) is necessary for obtaining the robustness required for
fortification.Comment: 19 page
Parallel repetition: simplifications and the no-signaling case
Consider a game where a refereed a referee chooses (x,y) according to a
publicly known distribution P_XY, sends x to Alice, and y to Bob. Without
communicating with each other, Alice responds with a value "a" and Bob responds
with a value "b". Alice and Bob jointly win if a publicly known predicate
Q(x,y,a,b) holds.
Let such a game be given and assume that the maximum probability that Alice
and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998) shows that if the game
is repeated n times in parallel, then the probability that Alice and Bob win
all games simultaneously is at most v'^(n/log(s)), where s is the maximal
number of possible responses from Alice and Bob in the initial game, and v' is
a constant depending only on v.
In this work, we simplify Raz's proof in various ways and thus shorten it
significantly. Further we study the case where Alice and Bob are not restricted
to local computations and can use any strategy which does not imply
communication among them.Comment: 27 pages; v2:PRW97 strengthening added, references added, typos
fixed; v3: fixed error in the proof of the no-signaling theorem, minor
change
Parallel Repetition for the GHZ Game: Exponential Decay
We show that the value of the -fold repeated GHZ game is at most
, improving upon the polynomial bound established by Holmgren
and Raz. Our result is established via a reduction to approximate subgroup type
questions from additive combinatorics
Polynomial Bounds On Parallel Repetition For All 3-Player Games With Binary Inputs
We prove that for every 3-player (3-prover) game with value less
than one, whose query distribution has the support of hamming weight one vectors, the value of the -fold
parallel repetition decays polynomially fast to zero;
that is, there is a constant such that the value of the
game is at most .
Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC
2022), our result is the missing piece that implies a similar bound for a much
more general class of multiplayer games: For 3-player game
over and , with value less than 1, there is a constant
such that the value of the game is at most .
Our proof technique is new and requires many new ideas. For example, we make
use of the Level- inequalities from Boolean Fourier Analysis, which, to the
best of our knowledge, have not been explored in this context prior to our
work
Multiplayer Parallel Repetition for Expanding Games
We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz\u27s Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games.
Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense.
We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question
Parallel Repetition of k-Player Projection Games
We study parallel repetition of k-player games where the constraints satisfy
the projection property. We prove exponential decay in the value of a parallel
repetition of projection games with value less than 1.Comment: 17 page
Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines
We prove that a sufficiently strong parallel repetition theorem for a special
case of multiplayer (multiprover) games implies super-linear lower bounds for
multi-tape Turing machines with advice. To the best of our knowledge, this is
the first connection between parallel repetition and lower bounds for time
complexity and the first major potential implication of a parallel repetition
theorem with more than two players.
Along the way to proving this result, we define and initiate a study of block
rigidity, a weakening of Valiant's notion of rigidity. While rigidity was
originally defined for matrices, or, equivalently, for (multi-output) linear
functions, we extend and study both rigidity and block rigidity for general
(multi-output) functions. Using techniques of Paul, Pippenger, Szemer\'edi and
Trotter, we show that a block-rigid function cannot be computed by multi-tape
Turing machines that run in linear (or slightly super-linear) time, even in the
non-uniform setting, where the machine gets an arbitrary advice tape.
We then describe a class of multiplayer games, such that, a sufficiently
strong parallel repetition theorem for that class of games implies an explicit
block-rigid function. The games in that class have the following property that
may be of independent interest: for every random string for the verifier
(which, in particular, determines the vector of queries to the players), there
is a unique correct answer for each of the players, and the verifier accepts if
and only if all answers are correct. We refer to such games as independent
games. The theorem that we need is that parallel repetition reduces the value
of games in this class from to , where is the number of
repetitions.
As another application of block rigidity, we show conditional size-depth
tradeoffs for boolean circuits, where the gates compute arbitrary functions
over large sets.Comment: 17 pages, ITCS 202
Polynomial Bounds on Parallel Repetition for All 3-Player Games with Binary Inputs
We prove that for every 3-player (3-prover) game G with value less than one, whose query distribution has the support S = {(1,0,0), (0,1,0), (0,0,1)} of Hamming weight one vectors, the value of the n-fold parallel repetition G^{?n} decays polynomially fast to zero; that is, there is a constant c = c(G) > 0 such that the value of the game G^{?n} is at most n^{-c}.
Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For every 3-player game G over binary questions and arbitrary answer lengths, with value less than 1, there is a constant c = c(G) > 0 such that the value of the game G^{?n} is at most n^{-c}.
Our proof technique is new and requires many new ideas. For example, we make use of the Level-k inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work