This paper presents a new characterization of NP in terms of probabilistically checkable proofs (PCPs), improving upon previous results. The authors show that every language in NP has a probabilistic verifier that checks membership proofs using a logarithmic number of random bits and a constant number of bits from the proof. This result has important implications for the hardness of approximation problems, particularly for MAX SNP-hard problems. The authors prove that no MAX SNP-hard problem has a polynomial-time approximation scheme (PTAS) unless NP = P. They also improve upon previous results regarding the hardness of approximating the maximum clique size in a graph. The paper also discusses the connection between PCPs and the hardness of approximation, and provides a new characterization of NP in terms of PCPs. The authors use techniques from interactive proofs and algebraic complexity theory to achieve their results.This paper presents a new characterization of NP in terms of probabilistically checkable proofs (PCPs), improving upon previous results. The authors show that every language in NP has a probabilistic verifier that checks membership proofs using a logarithmic number of random bits and a constant number of bits from the proof. This result has important implications for the hardness of approximation problems, particularly for MAX SNP-hard problems. The authors prove that no MAX SNP-hard problem has a polynomial-time approximation scheme (PTAS) unless NP = P. They also improve upon previous results regarding the hardness of approximating the maximum clique size in a graph. The paper also discusses the connection between PCPs and the hardness of approximation, and provides a new characterization of NP in terms of PCPs. The authors use techniques from interactive proofs and algebraic complexity theory to achieve their results.