Universal locally verifiable codes and 3-round interactive proofs of proximity for CSP

Universal locally testable codes (universal-LTCs), recently introduced in our companion paper (CJTCS, 2018), are codes that admit local tests for membership in numerous subcodes, allowing for testing properties of the encoded message. Unfortunately, universal-LTCs suffer strong limitations, which motivate us to initiate, in this work, the study of the “NP analogue” of these codes, wherein the testing procedures are also given free access to a short proof, akin the MA proofs of proximity of Gur and Rothblum (Computational Complexity 2018). We call such codes “universal locally verifiable codes” (universal-LVCs). A universal-LVC C:{0,1}^k→{0,1}^η for a family of functions F={f_i:{0,1}^k→{0,1}}_i∈[M] is a code such that, for every i∈[M], membership in the subcode {C(x):f_i(x)=1} can be verified locally using explicit access to a short (sublinear length) proof. A universal-LVC can be viewed as providing an encoding of inputs under which a large family of properties of the encoded inputs can be locally testable using a short proof. We show universal-LVCs of block length O˜(n²) for the family of all functions expressible by t-ary constraint satisfaction problems (t-CSP) over n constraints and k variables, with proof length and query complexity O˜(n^2/3), where t=O(1) and n≥k. In addition, we prove a lower bound of p⋅q=Ω˜(k) for every polynomial length universal-LVC, having proof complexity p and query complexity q, for such CSP functions. We give an application of universal-LVCs for interactive proofs of proximity (IPP), introduced by Rothblum, Vadhan, and Wigderson (STOC 2013), which are interactive proof systems wherein the verifier queries only a sublinear number of input bits to the end of asserting that, with high probability, the input is close to an accepting input. Specifically, we show a 3-round IPP for the set of assignments that satisfy fixed CSP instances, with sublinear communication and query complexity, which we derive from our universal-LVC for CSP functions.