Verifying the determinant in parallel

Abstract.

In this paper, we investigate the complexity of verifying problems whose computation is equivalent to the determinant, both in the Boolean arithmetic circuit and in the Boolean circuit model. We observe that for a few problems, there exists an easy (NC ¹) verification algorithm. To characterize the harder ones, we define the class of problems that are reducible to the verification of the determinant, under two different reductions, and establish a list of complete problems in these classes. In particular, we prove that computing the rank is equivalent under AC ⁰ reductions to verifying the determinant. We show in the Boolean case that none of the complete problems can be recognized in NC ¹ unless L = NL. On the other hand, we show that for functions, there exists an NC ¹ checker even if they are hard to verify, and that they can be extended into functions whose verification is easy.