Relativized circuit complexity
Journal of Computer and System Sciences
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Boolean circuits versus arithmetic circuits
Information and Computation
Theoretical Computer Science - Special issue on structure in complexity theory
PP is closed under intersection
Selected papers of the 23rd annual ACM symposium on Theory of computing
PP is closed under truth-table reductions
Information and Computation
SIAM Journal on Computing
Nondeterministic NC1 computation
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
The complexity of matrix rank and feasible systems of linear equations
Computational Complexity
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
Small-space analogues of Valiant's classes
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Counting classes and the fine structure between NC1and L
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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The class NC^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 NC^1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC^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 PNC^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 PNC^1 collapses to its first level PNC^1, and the constant-depth oracle hierarchy over C"=NC^1 collapses to its second level.