On the Knowledge Complexity of NP

We show that if a language has an interactive proof of logarithmic statistical knowledge-complexity, then it belongs to the class AM \ co?AM. Thus, if the polynomial time hierarchy does not collapse, then NP-complete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict NP languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The rst asserts that statistical zero knowledge is contained in AM\co?AM AH-87], while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in BPP NP GOP-94]. Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the error probability (n) is less than 2 ?3k(n) (where k(n) is the knowledge complexity) then the language proven has to be in the third level of the polynomial time hierarchy. In the standard setting of negligible error probability, there exist PSPACE-complete languages which have linear knowledge complexity. However, if we insist, for example, that the error probability is less than 2 ?n 2 , then PSPACE-complete languages do not have sub-quadratic knowledge complexity, unless PSPACE= P 3. In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions ; , the veriier runs in time polynomial in 1== and log(1==) and fails with probability at most to detect an additive error in the entropy. We believe that this protocol is of independent interest.