Toward Probabilistic Checking against Non-Signaling Strategies with Constant Locality

TL;DR

This paper demonstrates that a variant of a classical PCP construction remains sound against non-signaling strategies with constant locality, under certain geometric noise robustness assumptions, advancing the understanding of non-signaling proof systems.

Contribution

It introduces a new analysis showing PCPs are sound against non-signaling strategies with constant locality, assuming a geometric noise robustness hypothesis.

Findings

Parallel repetition of PCPs is sound against non-signaling strategies.

Linearity and agreement tests are effective in the non-signaling setting.

Assumes a geometric hypothesis on noise robustness of non-signaling proofs.

Abstract

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proof (PCPs) systems that are sound against non-signaling strategies. In this paper we show, assuming a certain geometrical hypothesis about noise robustness of non-signaling proofs (or, equivalently, about robustness to noise of solutions to the Sherali-Adams linear program), that a slight variant of the parallel repetition of the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies with constant locality. Our proof relies on the analysis of the linearity test and agreement test (also known as the direct product test) in the non-signaling setting.