Verifying the determinant in parallel

In this paper, we investigate the complexity of verifying problems whose co mputation is equivalent to the determinant, both in the Boolean arithmetic circuit and in the Boolean circuit model. We observe that for a few problem s, there exists an easy (NC1) verification algorithm. To characterize the h arder ones; we define the class of problems that are reducible to the verif ication of the determinant, under two different reductions, and establish a list of complete problems in these classes. In particular, we prove that c omputing the rank is equivalent under ACO reductions to verifying the deter minant. We show in the Boolean case that none of the complete problems can be recognized in NC1 unless L = NL. On the other hand, we show that for fun ctions, there exists an NC1 checker even if they are hard to verify, and th at they can be extended into functions whose verification is easy.