Rename from tiny-XOR-verified

README.md

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+---
+license: mit
+tags:
+- pytorch
+- safetensors
+- threshold-logic
+- neuromorphic
+- multi-layer
+---
+
+# threshold-xor
+
+The function that broke single-layer perceptrons. XOR is not linearly separable - no single threshold neuron can compute it. This circuit uses the minimal two-layer solution.
+
+## Circuit
+
+```
+      x       y
+      │       │
+      ├───┬───┤
+      │   │   │
+      ▼   │   ▼
+  ┌──────┐│┌──────┐
+  │  OR  │││ NAND │   Layer 1
+  │w:1,1 │││w:-1,-1│
+  │b: -1 │││b: +1 │
+  └──────┘│└──────┘
+      │   │   │
+      └───┼───┘
+          ▼
+      ┌──────┐
+      │ AND  │         Layer 2
+      │w: 1,1│
+      │b: -2 │
+      └──────┘
+          │
+          ▼
+       XOR(x,y)
+```
+
+## Why Two Layers?
+
+Plot the XOR truth table on a plane:
+
+```
+    y
+    1 │ 1   0
+      │
+    0 │ 0   1
+      └────────
+        0   1  x
+```
+
+No single line can separate the 1s from the 0s. This is linear inseparability. Minsky and Papert's 1969 proof of this contributed to the first AI winter.
+
+The solution: compute OR and NAND in parallel, then AND them together.
+
+## Trace
+
+| x | y | OR | NAND | AND(OR,NAND) |
+|---|---|-----|------|--------------|
+| 0 | 0 | 0 | 1 | 0 |
+| 0 | 1 | 1 | 1 | 1 |
+| 1 | 0 | 1 | 1 | 1 |
+| 1 | 1 | 1 | 0 | 0 |
+
+OR catches "at least one." NAND catches "not both." Their intersection is "exactly one."
+
+## Parameters
+
+| Layer | Weights | Bias |
+|-------|---------|------|
+| OR | [1, 1] | -1 |
+| NAND | [-1, -1] | +1 |
+| AND | [1, 1] | -2 |
+| **Total** | | **9** |
+
+## Properties
+
+- Commutative: XOR(x,y) = XOR(y,x)
+- Associative: XOR(x,XOR(y,z)) = XOR(XOR(x,y),z)
+- Self-inverse: XOR(x,x) = 0
+- Parity: XOR is addition mod 2
+
+## Usage
+
+```python
+from safetensors.torch import load_file
+import torch
+
+w = load_file('model.safetensors')
+
+def xor_gate(x, y):
+    inp = torch.tensor([float(x), float(y)])
+
+    or_out = int((inp * w['layer1.neuron1.weight']).sum() + w['layer1.neuron1.bias'] >= 0)
+    nand_out = int((inp * w['layer1.neuron2.weight']).sum() + w['layer1.neuron2.bias'] >= 0)
+
+    l1 = torch.tensor([float(or_out), float(nand_out)])
+    return int((l1 * w['layer2.weight']).sum() + w['layer2.bias'] >= 0)
+```
+
+## Files
+
+```
+threshold-xor/
+├── model.safetensors
+├── model.py
+├── config.json
+└── README.md
+```
+
+## License
+
+MIT

config.json

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+{
+  "model_type": "threshold_network",
+  "task": "xor_gate",
+  "architecture": "2 -> 2 -> 1",
+  "input_size": 2,
+  "hidden_size": 2,
+  "output_size": 1,
+  "num_neurons": 3,
+  "num_layers": 2,
+  "num_parameters": 9,
+  "activation": "heaviside",
+  "weight_constraints": "integer",
+  "verification": {
+    "method": "coq_proof",
+    "exhaustive": true,
+    "inputs_tested": 4
+  },
+  "accuracy": {
+    "all_inputs": "4/4",
+    "percentage": 100.0
+  },
+  "properties": ["not_linearly_separable", "minimal_depth"],
+  "github": "https://github.com/CharlesCNorton/coq-circuits"
+}

model.py

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+"""
+Threshold Network for XOR Gate
+
+A formally verified two-layer threshold network computing exclusive or.
+XOR is not linearly separable, requiring at least 2 layers.
+Architecture: Layer1 computes OR and NAND, Layer2 computes AND of results.
+"""
+
+import torch
+from safetensors.torch import load_file
+
+
+class ThresholdXOR:
+    """
+    XOR gate implemented as a 2-layer threshold network.
+
+    Layer 1: OR (neuron 1) and NAND (neuron 2) in parallel
+    Layer 2: AND of layer 1 outputs
+
+    XOR(x,y) = AND(OR(x,y), NAND(x,y))
+    """
+
+    def __init__(self, weights_dict):
+        self.l1_n1_weight = weights_dict['layer1.neuron1.weight']
+        self.l1_n1_bias = weights_dict['layer1.neuron1.bias']
+        self.l1_n2_weight = weights_dict['layer1.neuron2.weight']
+        self.l1_n2_bias = weights_dict['layer1.neuron2.bias']
+        self.l2_weight = weights_dict['layer2.weight']
+        self.l2_bias = weights_dict['layer2.bias']
+
+    def __call__(self, x1, x2):
+        inputs = torch.tensor([float(x1), float(x2)])
+
+        or_out = ((inputs * self.l1_n1_weight).sum() + self.l1_n1_bias >= 0).float()
+        nand_out = ((inputs * self.l1_n2_weight).sum() + self.l1_n2_bias >= 0).float()
+
+        layer1_out = torch.tensor([or_out, nand_out])
+        output = ((layer1_out * self.l2_weight).sum() + self.l2_bias >= 0).float()
+
+        return output
+
+    @classmethod
+    def from_safetensors(cls, path="model.safetensors"):
+        return cls(load_file(path))
+
+
+def forward(x1, x2, weights):
+    """
+    Forward pass with Heaviside activation.
+
+    Args:
+        x1, x2: Input values (0 or 1)
+        weights: Dict with layer weights and biases
+
+    Returns:
+        XOR(x1, x2)
+    """
+    inputs = torch.tensor([float(x1), float(x2)])
+
+    or_out = ((inputs * weights['layer1.neuron1.weight']).sum() + weights['layer1.neuron1.bias'] >= 0).float()
+    nand_out = ((inputs * weights['layer1.neuron2.weight']).sum() + weights['layer1.neuron2.bias'] >= 0).float()
+
+    layer1_out = torch.tensor([or_out, nand_out])
+    output = ((layer1_out * weights['layer2.weight']).sum() + weights['layer2.bias'] >= 0).float()
+
+    return output
+
+
+if __name__ == "__main__":
+    weights = load_file("model.safetensors")
+    model = ThresholdXOR(weights)
+
+    print("XOR Gate Truth Table:")
+    print("-" * 25)
+    for x1 in [0, 1]:
+        for x2 in [0, 1]:
+            out = int(model(x1, x2).item())
+            expected = x1 ^ x2
+            status = "OK" if out == expected else "FAIL"
+            print(f"XOR({x1}, {x2}) = {out}  [{status}]")

model.safetensors

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+oid sha256:01a1a18f425725466c246b1268548b0d5ce3a0d4d368c4911fa670adfc084f46
+size 476