---
license: mit
tags:
- formal-verification
- coq
- threshold-logic
- neuromorphic
- arithmetic
- adder
---

# tiny-RippleCarry2Bit-verified

Formally verified 2-bit ripple carry adder. Chains two full adders to add two 2-bit numbers with 100% accuracy.

## Architecture

| Component | Value |
|-----------|-------|
| Inputs | 4 (a1, a0, b1, b0) |
| Outputs | 3 (cout, s1, s0) |
| Neurons | 8 (2 full adders × 4 neurons each) |
| Parameters | 24 (2 full adders × 12 params each) |
| Layers | 2 (chained full adders) |
| Activation | Heaviside step |

## Key Properties

- 100% accuracy (16/16 input combinations correct)
- Coq-proven correctness
- Compositional construction from verified full adders
- Produces 3-bit output (sum can be 0-6, requiring 3 bits)
- Compatible with neuromorphic hardware

## Circuit Structure

```
a1 a0    b1 b0
 |  |     |  |
 |  +--+--+  |
 |     |     |
 |   FA0 (cin=0)
 |     |     |
 |    s0    c0
 |     |     |
 +--+--+-----+
    |
   FA1
    |
   s1  c1
```

First full adder adds least significant bits (a0 + b0 + 0), producing sum bit s0 and carry c0. Second full adder adds most significant bits with the carry (a1 + b1 + c0), producing s1 and final carry cout.

## Usage

```python
import torch
from safetensors.torch import load_file

weights = load_file('ripplecarry2bit.safetensors')

def full_adder_sim(a, b, cin):
    sum_out = a ^ b ^ cin
    carry_out = (a & b) | (cin & (a ^ b))
    return sum_out, carry_out

def ripple_carry_2bit(a1, a0, b1, b0):
    s0, c0 = full_adder_sim(a0, b0, 0)
    s1, cout = full_adder_sim(a1, b1, c0)
    return cout, s1, s0

# Test
print(ripple_carry_2bit(1, 1, 1, 0))  # 3 + 2 = 5 -> (1, 0, 1)
print(ripple_carry_2bit(1, 0, 1, 0))  # 2 + 2 = 4 -> (1, 0, 0)
print(ripple_carry_2bit(0, 1, 0, 1))  # 1 + 1 = 2 -> (0, 1, 0)
```

## Verification

**Coq Theorem**:
```coq
Theorem ripple_carry_2bit_correct : forall a1 a0 b1 b0,
  ripple_carry_2bit a1 a0 b1 b0 = ripple_carry_2bit_spec a1 a0 b1 b0.
```

Proven axiom-free via exhaustive case analysis on all 16 input combinations.

Full proof: [coq-circuits/Arithmetic/RippleCarry2Bit.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Arithmetic/RippleCarry2Bit.v)

## Properties

- **Commutative**: Adding A + B equals B + A
- **Identity**: Adding 0 preserves the value
- **Compositional**: Built from two verified FullAdder circuits

## Citation

```bibtex
@software{tiny_ripplecarry2bit_verified_2025,
  title={tiny-RippleCarry2Bit-verified: Formally Verified 2-Bit Ripple Carry Adder},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-RippleCarry2Bit-verified},
  year={2025}
}
```