Size–energy tradeoffs for unate circuits computing symmetric Boolean functions

Abstract

a b s t r a c tA unate gate is a logical gate computing a unate Boolean function, which is monotone in each variable. Examples of unate gates are AND gates, OR gates, NOT gates, threshold gates, etc. A unate circuit C is a combinatorial logic circuit consisting of unate gates. Let f be a symmetric Boolean function of n variables, such as the Parity function, MOD function, and Majority function. Let m 0 and m 1 be the maximum numbers of consecutive 0's and consecutive 1's in the value vector of f , respectively, and let l = min{m 0 , m 1 } and m = max{m 0 , m 1 }. Let C be a unate circuit computing f . Let s be the size of the circuit C , that is, C consists of s unate gates. Let e be the energy of C , that is, e is the maximum number of gates outputting ''1'' over all inputs to C . In this paper, we show that there is a tradeoff between the size s and the energy e of C . More precisely, we show that (n + 1 − l)/m ≤ s e .We also present lower bounds on the size s of C represented in terms of n, l and m. Our tradeoff immediately implies that log n ≤ e log s for every unate circuit C computing the Parity function of n variables.

Lower Bounds for Threshold Circuits of Bounded Energy

“…We can also obtain a more generalized version of the second lower bound by almost the same argument: a similar lower bound holds not only for PARITY but for all the symmetric functions [29]. The generalized lower bound is given as follows.…”

“…The lower bounds are presented in a line of papers [26,28,29], but we simplify the original statements so that the proofs highlight ideas and are self-contained. We mainly consider two Boolean functions, PARITY and INNER-PRODUCT.…”

New Bounds for Energy Complexity of Boolean Functions

“…We describe an application of this converse in obtaining size-energy trade-offs in Boolean circuits. Recalling from Table 1, Uchizawa et al [21] showed that for any symmetric function f computed by a unate circuit C, the number of firing patterns is lower bounded by (n + 1 − a f )/b f . On the other hand, for any circuit C (not necessarily unate) of size s and energy e, the number of firing patterns of C is at most s e + 1 [21].…”

Neural computing for scientific computing applications

“…The former corresponds to hardware resources while the latter corresponds to computation time. Energy is another critical resource, and Uchizawa, Douglas, and Maass have initiated the examination of tradeoffs between energy expenditure and other resources in threshold circuits [37][38][39]. Other more biologically motivated resource measures have also been proposed [19].…”