A PCP Characterization of AM

TL;DR

This paper introduces a new stochastic CSP and shows its approximation version characterizes the complexity class AM, providing a PCP-based framework that could influence derandomization efforts for AM.

Contribution

It presents a PCP characterization of AM using stochastic CSPs and explores implications for derandomization and average-case hardness in complexity theory.

Findings

Approximation version of the stochastic CSP is AM-complete.

Evidence against strong randomized optimization hypotheses.

Existence of hard-on-average optimization problems if NP-hardness holds against circuits.

Abstract

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a `PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some `Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there exist hard-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.