Probabilistic Proof Systems: A Primer

Bibliographic Notes 61 Bibliography 64 II Preface A proof is whatever convinces me. The glory attached to the creativity involved in finding proofs makes us forget that it is the less glorified process of verification that gives proofs their value. Conceptually speaking, proofs are secondary to the verification process; whereas technically speaking, proof systems are defined in terms of their verification procedures. The notion of a verification procedure presumes the notion of computation and furthermore the notion of efficient computation. This implicit stipulation is made explicit in the definition of N P, where efficient computation is associated with deterministic polynomial-time algorithms. However, as argued next, we can gain a lot if we are willing to take a somewhat non-traditional step and allow probabilistic verification procedures. In this primer, we shall survey three types of probabilistic proof systems, called interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs. In each of these three cases, we shall present fascinating results that cannot be obtained when considering the analogous deterministic proof systems. Indeed, the use of probabilistic verification procedures is common to the three aforementioned types of proof systems. We note that the association of efficient procedures with deterministic polynomial-time procedures is the basis for viewing NP-proof systems as the canonical formulation of proof systems (with efficient verification procedures). Now, since the notion of efficient computation has been extended to include probabilistic polynomial-time procedures, it is natural to allow the use of randomization also in the context of proof verification. Furthermore, it is natural to allow also a probability of error, which means that these probabilistic verification procedures may rule by (overwhelming) statistical evidence. Needless to say, this probability of error is explicitly bounded (and can be reduced by successive application of the proof system). Let us briefly review the three aforementioned types of probabilistic proof systems. Interactive Proofs. Randomized and interactive verification procedures, giving rise to interactive proof systems, seem much more powerful than their determinis-tic counterparts. In particular, such interactive proof systems exist for any set in PSPACE ⊇ coN P (e.g., for the set of unsatisfied propositional formulae), whereas 1 it is widely believed that some sets in coN P do not have NP-proof systems (i.e., N P = coN P). We stress that a " proof " in this context is not a fixed and static object, but rather a randomized (and dynamic) process in which the verifier interacts with …