A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian

In this work we show a barrier towards proving a randomness-efficient parallel repetition, a promising avenue for achieving many tight inapproximability results. Feige and Kilian (STOC'95) proved an impossibility result for randomness-efficient parallel repetition for two prover games with small degree, i.e., when each prover has only few possibilities for the question of the other prover. In recent years, there have been indications that randomness-efficient parallel repetition (also called derandomized parallel repetition) might be possible for games with large degree, circumventing the impossibility result of Feige and Kilian. In particular, Dinur and Meir (CCC'11) construct games with large degree whose repetition can be derandomized using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However, obtaining derandomized parallel repetition theorems that would yield optimal inapproximability results has remained elusive. This paper presents an explanation for the current impasse in progress, by proving a limitation on derandomized parallel repetition. We formalize two properties which we call "fortification-friendliness" and "yields robust embeddings." We show that any proof of derandomized parallel repetition achieving almost-linear blow-up cannot both (a) be fortification-friendly and (b) yield robust embeddings. Unlike Feige and Kilian, we do not require the small degree assumption. Given that virtually all existing proofs of parallel repetition, including the derandomized parallel repetition result of Dinur and Meir, share these two properties, our no-go theorem highlights a major barrier to achieving almost-linear derandomized parallel repetition

On the Control of Asynchronous Automata

The decidability of the distributed version of the Ramadge and Wonham controller synthesis problem,where both the plant and the controllers are modeled as asynchronous automataand the controllers have causal memoryis a challenging open problem.There exist three classes of plants for which the existence of a correct controller with causal memory has been shown decidable: when the dependency graph of actions is series-parallel, when the processes are connectedly communicating and when the dependency graph of processes is a tree. We design a class of plants, called decomposable games, with a decidable controller synthesis problem.This provides a unified proof of the three existing decidability results as well as new examples of decidable plants

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 entangled k-player games via fast quantum search

We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a $k$-player free game $G$ with entangled value $\mathrm{val}^*(G) = 1 - \epsilon$, the $n$-fold repetition of $G$ has entangled value $\mathrm{val}^*(G^{\otimes n})$ at most $(1 - \epsilon^{3/2})^{\Omega(n/sk^4)}$, where $s$ is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is $\mathrm{val}(G^{\otimes n}) \leq (1 - \epsilon^2)^{\Omega(n/s)}$, due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games $G$ where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by $(1 - \epsilon^2)^{\Omega(n/sk^2)}$. We also show that the dependence of the exponent on $k$ is necessary: we exhibit a $k$-player free game $G$ and $n \geq 1$ such that $\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}$. Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.Comment: This paper is a significantly revised version of arXiv:1411.1397, which erroneously claimed strong parallel repetition for free entangled games. Fixed author order to alphabetica