Counting classes and the fine structure between [formula omitted] and [formula omitted

Abstract

The class N C 1 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 N C 1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes P NC 1 and C = NC 1 , 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 P NC 1 and C = NC 1 . 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 P NC 1 collapses to its first level P NC 1 , and the constant-depth oracle hierarchy over C = NC 1 collapses to its second level.