tiny-NAND-verified

Formally verified NAND gate. Single threshold neuron computing negated conjunction with 100% accuracy.

Architecture

Component	Value
Inputs	2
Outputs	1
Neurons	1
Parameters	3
Weights	[-1, -1]
Bias	1
Activation	Heaviside step

Key Properties

• 100% accuracy (4/4 inputs correct)
• Coq-proven correctness
• Single threshold neuron
• Integer weights
• Commutative: NAND(x,y) = NAND(y,x)
• Functionally complete (can build any Boolean function)
• Self-dual: NAND(x,x) = NOT(x)

Usage

import torch
from safetensors.torch import load_file

weights = load_file('nand.safetensors')

def nand_gate(x, y):
    # Heaviside: weighted_sum + bias >= 0
    inputs = torch.tensor([float(x), float(y)])
    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
    return int(weighted_sum >= 0)

# Test
print(nand_gate(0, 0))  # 1
print(nand_gate(0, 1))  # 1
print(nand_gate(1, 0))  # 1
print(nand_gate(1, 1))  # 0

Verification

Coq Theorem:

Theorem nand_correct : forall x y, nand_circuit x y = negb (andb x y).

Proven axiom-free with properties:

• Commutativity
• Self-duality (NAND(x,x) = NOT(x))
• Functional completeness
• Identity with true gives NOT
• Absorption with false gives true

Full proof: coq-circuits/Boolean/NAND.v

Circuit Operation

Input combination produces weighted sum:

• (0,0): 0*(-1) + 0*(-1) + 1 = 1 >= 0 → 1
• (0,1): 0*(-1) + 1*(-1) + 1 = 0 >= 0 → 1
• (1,0): 1*(-1) + 0*(-1) + 1 = 0 >= 0 → 1
• (1,1): 1*(-1) + 1*(-1) + 1 = -1 < 0 → 0

Fails to fire only when both inputs are true.

Functional Completeness

NAND is functionally complete - any Boolean function can be built from NAND gates alone. This makes it particularly important for circuit composition.

Citation

@software{tiny_nand_prover_2025,
  title={tiny-NAND-verified: Formally Verified NAND Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-NAND-verified},
  year={2025}
}