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Abstract: In this note, we consider the minimum number of NOT operators in a Booleanformula representing a Boolean function. In circuit complexity theory, theminimum number of NOT gates in a Boolean circuit computing a Boolean function$f$ is called the inversion complexity of $f$. In 1958, Markov determined theinversion complexity of every Boolean function and particularly proved that$\lceil \log 2n+1 ceil$ NOT gates are sufficient to compute any Booleanfunction on $n$ variables. As far as we know, no result is known for inversioncomplexity in Boolean formulas, i.e., the minimum number of NOT operators in aBoolean formula representing a Boolean function. The aim of this note isshowing that we can determine the inversion complexity of every Booleanfunction in Boolean formulas by arguments based on the study of circuitcomplexity.



Author: Hiroki Morizumi

Source: https://arxiv.org/



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