Limiting Negations in Formulas

Negation-limited circuits have been studied as a circuit model between general circuits and monotone circuits. In this paper, we consider limiting negations in formulas. The minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that log2(n + 1) NOT gates are sufficient to compute any Boolean function on n variables. We determine the inversion complexity of every Boolean function in formulas, i.e., the minimum number of NOT gates (NOT operators) in a Boolean formula computing (representing) a Boolean function, and particularly prove that n/2 NOT gates are sufficient to compute any Boolean function on n variables. Moreover we show that if there is a polynomial-size formula computing a Boolean function f, then there is a polynomial-size formula computing f with at most n/2 NOT gates. We consider also the inversion complexi...

• Type: Conference
• Year: 2009
• Where: ICALP
• Authors: Hiroki Morizumi
• Keywords: Boolean Formula, Boolean Function, Boolean Function F, ICALP 2009, Theoretical Computer Science