Abstract
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.
N. Barcelo—This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1247842.
K. Pruhs—Supported in part by NSF grants CCF-1115575, CNS-1253218, CCF-1421508, and an IBM Faculty Award.
M. Scquizzato—Work done while at the University of Pittsburgh.
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- 1.
In previous work faulty and noisy are often used as synonyms, however, in order to provide additional clarity in regards to which model is currently being referred to, we use noisy when referring to gates in the fault-tolerant model, and faulty when referring to gates in the near-threshold model.
- 2.
If we fix which gates fail, then the output of C on I is fixed to either 1 or 0. A fixed set of q gates fail with probability \(\epsilon ^q(1-\epsilon )^{s-q}\), a polynomial of degree s in \(\epsilon \). \(C_I(\epsilon )\) can be viewed as the sum over the sets of gates that, when failing, cause C to output 1 on I, of the probability of that set failing.
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Barcelo, N., Nugent, M., Pruhs, K., Scquizzato, M. (2015). Almost All Functions Require Exponential Energy. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_8
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