Upper bound for output patterns of energy-bounded boolean circuits, and its applications

Abstract

Consider a Boolean circuit over a basis \(\{ \wedge , \vee , \lnot \}\), where \(\wedge \) and \(\vee \) are the conjunction of unbounded fan-in and disjunction of unbounded fan-in, respectively, and \(\lnot \) is the negation. The energy complexity of a circuit is defined as the number of gates outputting ones in the circuit, where the maximum is taken over all the input assignments. We prove that the number of output patterns (the set of possible outputs of all the internal gates in the circuit) in a circuit of energy e is at most \(2^{e \log e + 4e}\) vastly improving the trivial bound of \(\sum _{i=0}^e {s \atopwithdelims ()i}\). where s is the size of the circuit, which contrasts with our observation that a threshold circuit achieves the trivial bound. Building on this tool, we prove the following three applications: (i) Any Boolean circuit C of energy e can be converted to a depth-3 circuit of size \(s2^{e\log e + 4e}\). (ii) The problem of counting output patterns of a given circuit is known to be \(\#\textrm{P}\)-complete in general, but is fixed-parameter tractable when parameterized by energy e (over the basis \(\mathcal {B}_2 = \{\wedge _2,\vee _2,\lnot \}\)). We also show other FPT algorithms; (iii) If a function \(f:\{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}\) is computed by a circuit of energy e, then the unbounded-error communication complexity U(f) is at most \(O(e \log e)\).