threshold-mux2

2:1 multiplexer. The fundamental building block for data selection, routing one of two inputs to the output based on a select signal.

Circuit

    d0      d1
    │       │
    │       │       s (select)
    │       │       │
    ├───────┼───────┤
    │       │       │
    ▼       │       │
┌───────┐   │       │
│ sel0  │◄──┼───────┤
│d0∧¬s  │   │       │
└───────┘   │       │
    │       ▼       │
    │   ┌───────┐   │
    │   │ sel1  │◄──┘
    │   │ d1∧s  │
    │   └───────┘
    │       │
    └───┬───┘
        ▼
    ┌───────┐
    │  OR   │
    └───────┘
        │
        ▼
      output

Truth Table

s	d0	d1	out	Description
0	0	0	0	Select d0 (0)
0	0	1	0	Select d0 (0)
0	1	0	1	Select d0 (1)
0	1	1	1	Select d0 (1)
1	0	0	0	Select d1 (0)
1	0	1	1	Select d1 (1)
1	1	0	0	Select d1 (0)
1	1	1	1	Select d1 (1)

Formula: out = (d0 AND NOT s) OR (d1 AND s)

Mechanism

Layer 1 - Selection Gates:

The circuit uses two AND-with-complement gates that fire only when their respective data input is selected:

Gate	Weights [d0, d1, s]	Bias	Function
sel0	[1, 0, -1]	-1	d0 AND NOT(s)
sel1	[0, 1, 1]	-2	d1 AND s

sel0 fires when d0=1 AND s=0:

• The negative weight on s means s=1 inhibits firing
• Bias -1 requires d0=1 to reach threshold when s=0

sel1 fires when d1=1 AND s=1:

• Both d1 and s must be 1 to overcome bias -2

Layer 2 - Output:

Simple OR gate combines the two selection paths.

Architecture

Component	Weights	Bias	Function
sel0	[1, 0, -1]	-1	d0 AND NOT(s)
sel1	[0, 1, 1]	-2	d1 AND s
or	[1, 1]	-1	OR(sel0, sel1)

Parameters

Inputs	3 (d0, d1, s)
Outputs	1
Neurons	3
Layers	2
Parameters	11
Magnitude	10

Building Larger Multiplexers

The MUX2 is the primitive for constructing larger multiplexers:

MUX4  = MUX2(MUX2(d0,d1,s0), MUX2(d2,d3,s0), s1)
MUX8  = MUX2(MUX4(d0-d3), MUX4(d4-d7), s2)
MUX16 = MUX2(MUX8(d0-d7), MUX8(d8-d15), s3)

Each doubling adds one layer of MUX2 gates.

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def mux2(d0, d1, s):
    inp = torch.tensor([float(d0), float(d1), float(s)])

    # Layer 1: Selection
    sel0 = int((inp @ w['sel0.weight'].T + w['sel0.bias'] >= 0).item())
    sel1 = int((inp @ w['sel1.weight'].T + w['sel1.bias'] >= 0).item())

    # Layer 2: Combine
    l1 = torch.tensor([float(sel0), float(sel1)])
    return int((l1 @ w['or.weight'].T + w['or.bias'] >= 0).item())

# Examples
print(mux2(1, 0, 0))  # 1 (selects d0)
print(mux2(1, 0, 1))  # 0 (selects d1)
print(mux2(0, 1, 1))  # 1 (selects d1)

Applications

• Data routing in ALUs
• Register file read ports
• Conditional assignment in hardware
• Building blocks for barrel shifters
• Crossbar switches

Files

threshold-mux2/
├── model.safetensors
├── model.py
├── create_safetensors.py
├── config.json
└── README.md

License

MIT