Bounded Depth Circuits With Weighted Symmetric Gates: Satisfiability, Lower Bounds And Compression

A Boolean function f : (0,1}"-> 10,11 is weighted symmetric if there exist a function g : Z -> (0,11 and integers w(0), w(1), ... , w(n) such that f(x(1), ... , x(n)) = g(w(0) + E Sigma(n)(i=1) w(i)x(i)) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2(n) even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. As a special case, we obtain an algorithm for the maximum satisfiability problem that runs in time poly(n(t)) . 2(n-n1/O(t)) for instances with n variables and O(n(t)) clauses. Through the analysis of our algorithms, we show average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits. (C) 2019 Elsevier Inc. All rights reserved.