Abstract
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 complexity in formulas of negation normal form and prove that the inversion complexity is at most polynomials of n for every Boolean function on n variables.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
