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

We prove an impossibility result for a randomness-efficient parallel repetition. Our result is motivated by an early result of Feige and Kilian (STOC’95), who 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, thus circumventing the impossibility result of Feige and Kilian. In this paper, we show strong limits on an approach to derandomized parallel repetition which we call “proof by embedding”, which is a very general technique that underlies most proofs of parallel repetition, including Raz’s proof (SICOMP’98). We show that under natural assumptions on the repetition transformation, one cannot prove an almost-linear derandomized parallel repetition theorem via the proof by embedding technique. Unlike the Feige and Kilian result, our result is not limited to games with small degree. ∗[email protected]. †[email protected]. ‡[email protected].