Energy and depth of threshold circuits

Size–energy tradeoffs for unate circuits computing symmetric Boolean functions

“…Our tradeoff between s and e holds for arbitrary unate circuits. It should be noted that the inequality d ≤ e does not necessarily holds for unate circuits, and that if a Boolean function f can be computed by a polynomial-size unate circuit C of energy e then the function f can be computed by a polynomial-size unate circuit C ′ of depth d ′ ≤ 2e + 1 [17].…”

“…However, if f has low communication complexity as the case of the Parity function, then the result in [18] does not yield any interesting tradeoff. On the other hand, it has been shown that the following relation holds on the depth and size of threshold circuits for an arbitrary Boolean function f : if f can be computed by a polynomial-size threshold circuit C of energy e then f can be computed by a polynomial-size threshold circuit C ′ of depth d ′ ≤ 2e + 1 [17].…”

New Bounds for Energy Complexity of Boolean Functions

“…More precisely, they defined the energy complexity of threshold circuits and gave some sufficient conditions for certain functions to be computed by small energy threshold circuits. In a sequence of works, Uchizawa et al [19,21] related energy complexity of Boolean functions under the threshold basis to the other well-studied parameters like circuit size and depth for interesting classes of Boolean functions. In a culminating result, Uchizawa and Takimoto [20] showed that constant depth thresholds circuits of unbounded weights with the energy restricted to n o (1) needs exponential size to compute the Boolean inner product function 3 .…”

Energy-Efficient Threshold Circuits for Comparison Functions

“…Recently, Uchizawa et al, introduced a new perspective to research on threshold circuits [13]: They focused on the biological fact that a neuron consumes substantially more energy to emit a ''spike'' (i.e., an electrical signal) than not to emit one [2,6], and proposed a complexity measure, named the energy complexity, for energy consumption of a threshold circuit. Based on the fact above, the energy complexity of a threshold circuit C is defined to be the maximum number of gates outputting ''1''s during computation, where the maximum is taken over all input assignments to C. In the past research, it was shown that the energy complexity has close relation to other complexity measures, such as size and depth, of threshold circuits [11,12,[14][15][16], where the size and depth correspond to the number of gates and parallel computation time of a threshold circuit, respectively. In particular, Suzuki et al, prove that one can construct an energy-efficient circuit computing a MOD function by expending its size [12]; more specifically, it is shown that a MOD function is computable by a threshold circuit of small energy e (e.g., e ¼ Oð1Þ) if it is allowable to use a large number s of gates (e.g., s ¼ ðnÞ).…”