Probabilistically Checkable Proofs And Codes

NP is the complexity class of problems for which it is easy to check that a solution is correct. In contrast, finding solutions to certain NP problems is widely believed to be hard. The canonical example is the SAT problem: given a Boolean formula, it is notoriously difficult to come up with a satisfying assignment, whereas given a proposed assignment it is trivial to plug in the values and verify its correctness. Such an assignment is an "NP-proof" for the satisfiability of the formula.Although the verification is simple, it is not local, i.e., a verifier must typically read (almost) the entire proof in order to reach the right decision. In contrast, the landmark PCP theorem [4, 3] says that proofs can be encoded into a special "PCP" format, that allows speedy verification. In the new format it is guaranteed that a PCP proof of a false statement will have many many errors. Thus such proofs can be verified by a randomized procedure that is local: it reads only a constant (!) number of bits from the proof and with high probability detects an error if one exists.How are these PCP encodings constructed? First, we describe the related and possibly cleaner problem of constructing locally testable codes. These are essentially error correcting codes that are testable by a randomized local algorithm. We point out some connections between local testing and questions about stability of various mathematical systems. We then sketch two known ways of constructing PCPs.