Almost All Functions Require Exponential Energy

One potential method to attain more energy-efficient circuits with the current technology is Near-Threshold Computing, which means using less energy per gate by designing the supply voltages to be closer to the threshold voltage of transistors. However, this energy savings comes at a cost of a greater probability of gate failure, which necessitates that the circuits must be more fault-tolerant, and thus contain more gates. Thus achieving energy savings with Near-Threshold Computing involves properly balancing the energy used per gate with the number of gates used. The main result of this paper is that almost all Boolean functions require circuits that use exponential energy, even if allowed circuits using heterogeneous supply voltages. This is not an immediate consequence of Shannon's classic result that almost all functions require exponential sized circuits of faultless gates because, as we show, the same circuit layout can compute many different functions, depending on the value of the supply voltages. The key step in the proof is to upper bound the number of different functions that one circuit layout can compute. We also show that the Boolean functions that require exponential energy are exactly the Boolean functions that require exponentially many faulty gates.