The Query Complexity of Program Checking by Constant-Depth Circuits

Abstract

We study program result checking using AC⁰ circuits as checkers. We focus 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 deterministic and randomized AC⁰ checkers for certain P-complete and NC1-complete problems. We show that for each ε > 0, Ω(n ^{1 ε}) is a lower bound to the query complexity of deterministic AC⁰ checkers for the considered problems, for inputs of length n. On the other hand, we show that suitably encoded complete problems for P and NC¹ have randomi- zed AC⁰ checkers of constant query complexity. The latter results are proved using techniques from the PCP(n ³, 1) protocol for 3-SAT in [4].

Part of the work done while at IMSc.