Counting classes and the fine structure between NC1 and L

The class NC1 of problems solvable by bounded fan-in circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC1 and C=NC1, defined by a test for positivity and a test for zero, respectively, of arithmetic circuit families of logarithmic depth, sit in this complexity interval. We study the landscape of Boolean hierarchies, constant-depth oracle hierarchies, and logarithmic-depth oracle hierarchies over PNC1 and C=NC1. We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses. In particular, the constant-depth oracle hierarchy over PNC1 collapses to its first level PNC1, and the constant-depth oracle hierarchy over C=NC1 collapses to its second level. (C) 2011 Elsevier B.V. All rights reserved.