Reductions that lie

All of the reductions currently used in complexity theory (≤p, ≤γ, ≤R) have the property that they are honest. If A ≤ B then whatever machine M reduces A to B is such that: if on input x, M outputs y then x ε A ↔ y ε B. It would appear that this membership preserving property is intrinsic to the notion of reduction. We will see that it is not. We introduce reductions that lie and sometimes produce outputs y ε B when x ? A. We will use these reductions to further clarify the computational complexity of some problems raised by Gauss.