Template:Families of Boolean functions/sharpness

possible bijections

The quantities suggest two kinds of bijection for arities greater 1:

From the union of sharp and haploid families to blunt families. Haploid families are themselves blunt. So they are assigned to themselves.
This leaves a bijection from sharp families to diploid blunt families.

From the union of sharp and haploid families to super-families. Haploid families are themselves complete super-families. So they are assigned to themselves.
This leaves a bijection from sharp families to diploid super-families – which include sharp super-families.

E.g. the light sharp families could be assigned to the super-families containing them, and the heavy sharp families to the diploid blunt super-families.
This seems silly. There is probably no meaningful bijection of this kind.

sharp to diploid blunt

These images show the 32 diploid families for arity 3 within their super-families.
On the left are the 16 (diploid) sharp families. On the right are the 16 diploid blunt families

truth tables
	0 255 3, 12, 48, 192 63, 207, 243, 252 5, 10, 80, 160 95, 175, 245, 250 6, 9, 96, 144 111, 159, 246, 249 17, 34, 68, 136 119, 187, 221, 238 18, 33, 72, 132 123, 183, 222, 237 20, 40, 65, 130 125, 190, 215, 235 24, 36, 66, 129 126, 189, 219, 231

Zhegalkin indices
The families on the left belong to two villages, which correspond to the rows of the 2×8 grid. Those on the right belong to four villages, which correspond to the columns.
128 130 132 134 144 146 148 150	0 1 64, 68, 80, 85 65, 69, 81, 84 32, 34, 48, 51 33, 35, 49, 50 96, 102, 112, 119 97, 103, 113, 118 8, 10, 12, 15 9, 11, 13, 14 72, 76, 90, 95 73, 77, 91, 94 40, 42, 60, 63 41, 43, 61, 62 106, 108, 120, 127 107, 109, 121, 126

haploid

These images show the 14 haploid (blunt) families for arity 3.
On the left are the linear functions except contradiction and tautology. Their Walsh indices 1...7 go from top left to bottom right.
On the right are the families with size 8. Their consuls correspond to the Walsh indices on the left.

truth tables
85, 170 51, 204 102, 153 15, 240 90, 165 60, 195 105, 150

Zhegalkin indices
2, 3 4, 5 6, 7 16, 17 18, 19 20, 21 22, 23

twin prefects of blunt families
blunt Boolean functions The twin prefect (prefect of the twin) is the essentially the consul, but refined by a sign. The eight images below are in the same 2×8 grid as the matrices shown above. (Here truth tables and Zhegalkin indices are shown within the same image.) The blunt families shown above (extroverted and introverted) are within these patterns. 0 ¬1 ¬2 3 ¬4 5 6 ¬7

Families of Boolean functions#blunt and sharp