Complexity of Counting Output Patterns of Logic Circuits

Let C be a logic circuit consisting of s gates g1, g2, . . . , gs, then the output pattern of C for an input x ∈ {0, 1} is defined to be a vector (g1(x), g2(x), . . . , gs(x)) ∈ {0, 1} of the outputs of g1, g2, . . . , gs for x. For each f : {0, 1} → {0, 1}, we define an f -circuit as a logic circuit where every gate computes f , and investigate computational complexity of the following counting problem: Given an f -circuit C, how many output patterns arise in C? We then provide a dichotomy result on the counting problem: We prove that the problem is solvable in polynomial time if f is PARITY or any degenerate function, while the problem is #P-complete even for constant-depth f -circuits if f is one of the other functions, such as AND, OR, NAND and NOR.