3-input AND gate. Fires when all three inputs are active. The 3-of-3 threshold gate.
Circuit
a b c
│ │ │
└───┼───┘
│
▼
┌─────────┐
│ w: 1,1,1│
│ b: -3 │
└─────────┘
│
▼
AND(a,b,c)
The Unanimity Requirement
3-input AND is the smallest non-trivial unanimity gate:
Inputs
Sum
Output
000
-3
0
001
-2
0
010
-2
0
011
-1
0
100
-2
0
101
-1
0
110
-1
0
111
0
1
Only when all three contribute does the sum reach the threshold.
Generalization from 2-input AND
Gate
Weights
Bias
Threshold
AND(a,b)
[1, 1]
-2
2 votes needed
AND(a,b,c)
[1, 1, 1]
-3
3 votes needed
AND(a,b,c,d)
[1, 1, 1, 1]
-4
4 votes needed
The pattern: n inputs, all weight +1, bias -n.
Single Point of Veto
Any single 0 input blocks the gate:
a=0: max sum = 0 + 1 + 1 - 3 = -1 < 0
b=0: max sum = 1 + 0 + 1 - 3 = -1 < 0
c=0: max sum = 1 + 1 + 0 - 3 = -1 < 0
Each input has veto power. This is unanimous consent.
Parameters
Component
Value
Weights
[1, 1, 1]
Bias
-3
Total
4 parameters
Optimality
Exhaustive enumeration of all 1,289 weight configurations at magnitudes 0-6 confirms this circuit is already at minimum magnitude (6). There is exactly one valid configuration at magnitude 6, and no valid configurations exist below it.
