As an algorithmic problem, the possible questions are to find if a given integer is an element of the output node or if two circuits compute the same set.
The decidability is still an open question, but there are results on restriction of those circuits.
It is a natural extension of the circuits over sets of natural numbers when the considered set contains also negative integers, the definitions, which does not change, will not be repeated on this page.
The computational complexity of this problem depends on the type of gates allowed in the circuit C.[1] The table below summarizes the computational complexity of the membership problem for various classes of integer circuits.
(O) denotes the classes defined by O-formulae, which are O-circuits with maximal fan-out 1.