An inverter is a circuit which outputs ¬ x1, ¬ x2, ..., ¬ xn for any Boolean inputs x1, x2, ..., xn. We consider constructing an inverter with AND gates and OR gates and a few NOT gates. Beals, Nishino and Tanaka have given a construction of an inverter which has size O(nlog n) and depth O(log n) and uses ⌈ log (n+1) ⌉ NOT gates. In this paper we give a construction of an inverter which has size O(n) and depth log 1+o(1)n and uses log 1+o(1)n NOT gates. This is the first negation-limited inverter of linear size using only o(n) NOT gates. We also discuss implications of our construction for negation-limited circuit complexity.

The circuit satisfiability problem has been intensively studied since Ryan Williams showed a connection between the problem and lower bounds for circuit complexity. In this letter, we present a #SAT algorithm for synchronous Boolean circuits of n inputs and s gates in time $2^{nleft(1 - rac{1}{2^{O(s/n)}} ight)}$ if s=o(n log n).