Title: A Complete Classification of the Toffoli Gate with Ancillary bits

Abstract:

The Toffoli gate is a universal gate for classical reversible computation. This means that if we are allowed to fix the values of certain inputs and outputs (called ancillary bits), we can simulate any Boolean function from with a circuit from wires consisting only of Toffoli gates (with extra ignored outputs).

*ancillary bits*. I also provide a normal form for these circuits and prove an equivalence of categories into a subcategory of