A Randomized Algorithm for BBCSPs in the Prover-Verifier Model

Abstract

In this paper we introduce a Prover-Verifier model for analyzing the computational complexity of a class of Constraint Satisfaction problems termed Binary Boolean Constraint Satisfaction problems (BBCSPs). BBCSPs represent an extremely general class of constraint satisfaction problems and find applications in a wide variety of domains including constraint programming, puzzle solving and program testing. We establish that each instance of a BBCSP admits a coin-flipping Turing Machine that halts in time polynomial in the size of the input. The prover P in the Prover-Verifier model is endowed with very limited powers; in particular, it has no memory and it can only pose restricted queries to the verifier. The verifier on the other hand is both omniscient in that it is cognizant of all the problem details and insincere in that it does not have to decide a priori on the intended proof. However, the verifier must stay consistent in its responses. Inasmuch as our provers will be memoryless and our verifiers will be asked for extremely simple certificates, our work establishes the existence of a simple, randomized algorithm for BBCSPs. Our model itself serves as a basis for the design of zero-knowledge machine learning algorithms in that the prover ends up learning the proof desired by the verifier.