Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas.

We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC⁰-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest.

We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC⁰ circuit lower bounds.