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Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure $A$ for some function $f$, we compose $f$ with a carefully chosen gadget $g$ and get essentially the same lower bound on a complexity measure $B$ for the lifted function $f \diamond g$. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, and so on.

One of the main question in the context of lifting is how to choose a suitable gadget $g$. Generally, to get better results, that is, to minimize the loss when transferring lower bounds, we need the gadget to be of constant size (number of inputs). Unfortunately, in many settings we only know lifting results for gadgets whose size grows with the size of $f$, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible.

In this paper, we systematically study the question “For which gadgets does the lifting result hold?” in the following four settings:

1. Lifting from decision tree depth to decision tree size.
2. Lifting from conjunction DAG width to conjunction DAG size.
3. Lifting from decision tree depth to parity decision tree depth and size.
4. Lifting from block sensitivity to deterministic and randomized communication complexity.

In all cases, we give a complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there are no intermediate cases—for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as $f \diamond \mathrm{OR} \diamond \mathrm{XOR}$ for some function $f$.