Energy-Efficient Circuit Design

We initiate the theoretical investigation of energy-efficient circuit design.\ud We assume that the circuit design specifies the circuit layout as well as the\ud supply voltages for the gates. To obtain maximum energy efficiency, the circuit\ud design must balance the conflicting demands of minimizing the energy used per gate,\ud and minimizing the number of gates in the circuit; If the energy supplied to the\ud gates is small, then functional failures are likely, necessitating a circuit layout\ud that is more fault-tolerant, and thus that has more gates. \ud \ud By leveraging previous\ud work on fault-tolerant circuit design, we show general upper and lower bounds on\ud the amount of energy required by a circuit to compute a given relation. We show\ud that some circuits would be asymptotically more energy-efficient if heterogeneous\ud supply voltages were allowed, and show that for some circuits the most energy-efficient\ud supply voltages are homogeneous over all gates.\ud \ud In the traditional approach to circuit design the\ud supply voltages for each transistor/gate are set sufficiently high so that with\ud sufficiently high probability no transistor fails.\ud We show that if there is a better (in terms of worst-case relative error with respect to energy) method than the traditional approach \ud then $P=NP$, \ud and thus there is a complexity theoretic obstacle to achieving energy savings with Near-Threshold computing. \ud \ud We show that almost all\ud Boolean functions require circuits that use exponential energy. This is not an immediate\ud consequence of Shannon's classic result that most functions require exponential\ud sized circuits of faultless gates because, as we show, the same circuit layout can\ud compute many different functions, depending on the value of the supply voltage.\ud \ud If the error bound must vanish as the number of inputs increases, we show that a natural class of functions can be computed with asymptotically less energy using heterogeneous supply voltages than is possible using homogeneous supply voltages.\ud We also prove upper bounds on the asymptotic energy savings achieved by using heterogeneous supply voltages over homogeneous supply voltages for a class of functions, and also show a relation that can bypass this bound