The Query Complexity of Program Checking by Constant-Depth Circuits

In this paper we study program checking (in the sense of Blum and Kannan 7]) using AC 0 circuits as checkers. Our focus is on the number of queries made by the checker to the program being checked and we term this as the query complexity of the checker for the given problem. We study the query complexity of both deterministic and randomized AC 0 checkers. We show that, for each > 0, (n 1?) is a lower bound to the query complexity of deterministic AC 0 checkers for Parity and certain P-complete and NC 1-complete problems, where n is the input size. On the other hand, we show that Parity and suitably encoded complete problems for P, NL, and NC 1 have randomized AC 0 checkers of constant query complexity. The latter results are proved using techniques from the PCP(n 3 ; 1) protocol for 3-SAT in 4].