Smooth and Strong PCPs

Abstract

Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:• A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. • A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in N P have PCPs that are both smooth and strong, are of polynomial length and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora et al. (JACM 45(3):501-555, 1998), providing a stronger analysis of the Hadamard and Reed-Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in N P has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of N P witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.

Relaxed Locally Correctable Codes with Nearly-Linear Block Length and Constant Query Complexity

“…Relying on recent progress on canonical PCPPs [13,29], we provide new, simple constructions of relaxedcorrectable O(1)-query canonical PCPPs of polynomial length.…”

“…In this paper we use the following theorem due to Dinur, Goldreich, and Gur [13]. (Alternatively, we could have used the construction by Paradise [29].) Theorem 5.…”

Relaxed Locally Correctable Codes with Improved Parameters

“…Correctable canonical PCPPs (ccPCPP): These are PCPP systems for some specified language L satisfying the following properties: (i) for each w ∈ L there is a unique proof π(w) that satisfies the verifier with probability 1, (ii) the verifier accepts with high probability only pairs (x, π) that are close to some (w, π(w)) for some w ∈ L, i.e., only the pairs where x is close to some w ∈ L, and π is close to π(w), and (iii) the set {w • π w : w ∈ L} is an RLCC. Canonical proofs of proximity have been studies in [DGG18,Par20]. We elaborate on these constructions in Section 5.…”

“…For soundness, the demand is that the only pairs (x, π) that are accepted by the verifier with high probability are those where x is close to some w ∈ L and π is close to π(w). Such proof system have been studies in [DGG18,Par20], who proved that such proof systems exist for every language in P.…”

Regional Categorization by Income Distribution in Tokyo Metropolitan Area

“…Note that regular verifiers are sometimes called smooth verifiers, e.g.,[Par21]. Since the term "regularity" is compatible with that of (hyper) graphs, we do not use the term "smoothness" but "regularity.…”