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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 thisdoi:10.1016/j.tcs.2011.05.050 fatcat:wy7i2fwrurapnphsvrz5bufvjq