Enumeration of Boolean Functions of Sensitivity Three and Inheritance of Nondegeneracy

The sensitivity of a Boolean function is the maximum, over all inputs, of the number of input bits which when flipped change the output of the function. We enumerate all Boolean functions of sensitivity at most three and investigate their properties with the aid of computers. The number of NPN equivalence classes of nondegenerate n-variable Boolean functions of sensitivity three is 7, 80, 4215, 190221, 65694, 8873, 848, 64 and 8 for n = 3, 4,..., 11 and zero for n >= 12. We verify that, over all these functions, the maximum of block sensitivity, certificate complexity, decision tree complexity and degree is 6, 6, 9 and 9, respectively. A key to making this enumeration possible is the fact that, for every nondegenerate Boolean function f, a subfunction f vertical bar x(i)=0 or f vertical bar x(i)=1 is nondegenerate for some variable x(i), which was recently shown by Lee, Lokam, Tsai and Yang [in Proc. ISIT '15, pages 501-505]. We extend this result by showing that, the minimum number of nondegenerate subfunctions in {f vertical bar x(i)=0, f vertical bar x(i)=1} 1 <= i <= n is, in fact, four.