The complexity of membership problems for circuits over sets of natural numbers
The problem of testing membership in the subset of the natural numbers produced at the output gate of a {boolean OR, boolean AND, (-), +, x} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {boolean OR, boolean AND, +, x} is shown NEXPTIME-complete, the cases {boolean OR, boolean AND, (-), x}, {boolean OR, boolean AND, x}, {boolean OR, boolean AND, +} are shown PSPACE-complete, the case {boolean OR, +} is shown NP-complete, the case {boolean OR, +} is shown C=L-complete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for union-intersection-concatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one's choosing. Our results extend in nontrivial ways past work by Stockmeyer and Meyer (1973), Wagner (1984) and Yang (2000).
computational complexity circuit problems