threshold-equal

8-bit equality comparator. Returns 1 if a = b, 0 otherwise.

Circuit

a0 b0   a1 b1   a2 b2   a3 b3   a4 b4   a5 b5   a6 b6   a7 b7
 β”‚  β”‚    β”‚  β”‚    β”‚  β”‚    β”‚  β”‚    β”‚  β”‚    β”‚  β”‚    β”‚  β”‚    β”‚  β”‚
 β””β”¬β”€β”˜    β””β”¬β”€β”˜    β””β”¬β”€β”˜    β””β”¬β”€β”˜    β””β”¬β”€β”˜    β””β”¬β”€β”˜    β””β”¬β”€β”˜    β””β”¬β”€β”˜
  β–Ό       β–Ό       β–Ό       β–Ό       β–Ό       β–Ό       β–Ό       β–Ό
β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”   β”Œβ”€β”€β”€β”
β”‚XORβ”‚   β”‚XORβ”‚   β”‚XORβ”‚   β”‚XORβ”‚   β”‚XORβ”‚   β”‚XORβ”‚   β”‚XORβ”‚   β”‚XORβ”‚  L1-2
β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜   β””β”€β”€β”€β”˜
  β”‚       β”‚       β”‚       β”‚       β”‚       β”‚       β”‚       β”‚
  β””β”€β”€β”€β”€β”€β”€β”€β”΄β”€β”€β”€β”€β”€β”€β”€β”΄β”€β”€β”€β”€β”€β”€β”€β”΄β”€β”€β”€β”¬β”€β”€β”€β”΄β”€β”€β”€β”€β”€β”€β”€β”΄β”€β”€β”€β”€β”€β”€β”€β”΄β”€β”€β”€β”€β”€β”€β”€β”˜
                              β–Ό
                         β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”
                         β”‚   NOR   β”‚                        L3
                         β”‚ w: -1Γ—8 β”‚
                         β”‚ b:  0   β”‚
                         β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜
                              β”‚
                              β–Ό
                           (a = b)

Mechanism

  1. XOR each bit pair: XOR(a_i, b_i) = 1 if bits differ, 0 if same
  2. NOR all XORs: Fires only when all XORs are 0 (all bits match)

The NOR gate acts as a "zero detector" - it fires when all its inputs are silent.

XOR Structure (each)

    a   b
    β”‚   β”‚
    β”œβ”€β”¬β”€β”€
    β”‚ β”‚ β”‚
    β–Ό β”‚ β–Ό
β”Œβ”€β”€β”€β”€β”β”‚β”Œβ”€β”€β”€β”€β”
β”‚ OR β”‚β”‚β”‚NANDβ”‚
β””β”€β”€β”€β”€β”˜β”‚β””β”€β”€β”€β”€β”˜
    β”‚ β”‚ β”‚
    β””β”€β”Όβ”€β”˜
      β–Ό
  β”Œβ”€β”€β”€β”€β”€β”€β”
  β”‚ AND  β”‚
  β””β”€β”€β”€β”€β”€β”€β”˜
      β”‚
      β–Ό
   XOR out

Truth Table (sample)

a b a = b
0 0 1
0 1 0
127 127 1
127 128 0
255 255 1
255 0 0
100 100 1

Architecture

Component Neurons Parameters
XOR Γ— 8 24 72
NOR 1 9
Total 25 81

Layers: 3 (XOR: 2, NOR: 1)

Comparison Family

Circuit Condition Implementation
LessThan a < b borrow_out of (a - b)
Equal a = b NOR of all XOR bits
GreaterThan a > b LessThan(b, a)

Usage

from safetensors.torch import load_file

w = load_file('model.safetensors')

def equal(a, b):
    """a, b: 8-bit lists [a0..a7] (LSB first)
    Returns: 1 if a == b, 0 otherwise"""
    # See model.py for full implementation
    pass

# Example: 127 == 127?
a = [(127 >> i) & 1 for i in range(8)]
b = [(127 >> i) & 1 for i in range(8)]
result = equal(a, b)  # Returns 1

Files

threshold-equal/
β”œβ”€β”€ model.safetensors
β”œβ”€β”€ model.py
β”œβ”€β”€ config.json
└── README.md

License

MIT

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