Two's complement negation

README.md

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+---
+license: mit
+tags:
+- pytorch
+- safetensors
+- threshold-logic
+- neuromorphic
+- arithmetic
+- signed
+---
+
+# threshold-twos-complement
+
+4-bit two's complement negation circuit. Computes -A for signed integers.
+
+## Circuit
+
+```
+A[3:0] ──► Negate ──┬──► N[3:0]    (-A)
+                    └──► overflow  (when A = -8)
+```
+
+## Algorithm
+
+```
+-A = ~A + 1
+
+1. Invert all bits
+2. Add 1
+```
+
+## Truth Table
+
+| A (unsigned) | A (signed) | N (signed) | Overflow |
+|--------------|------------|------------|----------|
+| 0000 (0) | +0 | +0 | 0 |
+| 0001 (1) | +1 | -1 | 0 |
+| 0111 (7) | +7 | -7 | 0 |
+| 1000 (8) | -8 | -8 | 1 |
+| 1111 (15) | -1 | +1 | 0 |
+
+## Overflow
+
+The value -8 (1000) cannot be negated in 4-bit two's complement:
+- Range: -8 to +7
+- -(-8) = +8, which exceeds +7
+- Result wraps to -8, overflow flag set
+
+## Architecture
+
+| Component | Count | Neurons |
+|-----------|-------|---------|
+| Bit inverters | 4 | 4 |
+| Half-adder chain | 4 | 16 |
+| Overflow detect | 1 | 2 |
+
+**Total: 22 neurons, 63 parameters, 4 layers**
+
+## Examples
+
+```
+-(+5) = ~(0101) + 1 = 1010 + 1 = 1011 = -5 ✓
+-(-3) = ~(1101) + 1 = 0010 + 1 = 0011 = +3 ✓
+-(-8) = ~(1000) + 1 = 0111 + 1 = 1000 = -8 (overflow!)
+```
+
+## Usage
+
+```python
+from safetensors.torch import load_file
+
+w = load_file('model.safetensors')
+
+# All 16 cases verified
+# Overflow only when A = -8 (1000)
+```
+
+## Files
+
+```
+threshold-twos-complement/
+├── model.safetensors
+├── create_safetensors.py
+├── config.json
+└── README.md
+```
+
+## License
+
+MIT

config.json

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+{
+  "name": "threshold-twos-complement",
+  "description": "4-bit two's complement negation",
+  "inputs": 4,
+  "outputs": 5,
+  "neurons": 22,
+  "layers": 4,
+  "parameters": 63
+}

create_safetensors.py

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+import torch
+from safetensors.torch import save_file
+
+weights = {}
+
+# 4-bit Two's Complement Negation
+# Inputs: a3,a2,a1,a0 (4 inputs)
+# Outputs: n3,n2,n1,n0, overflow (5 outputs)
+#
+# -A = ~A + 1
+# Overflow when A = -8 (1000), since -(-8) = 8 can't be represented
+
+# Invert inputs
+for i in range(4):
+    weights[f'inv{i}.weight'] = torch.tensor([[-1.0]], dtype=torch.float32)
+    weights[f'inv{i}.bias'] = torch.tensor([0.0], dtype=torch.float32)
+
+# Add 1 using half-adder chain
+def add_xor(name):
+    weights[f'{name}.or.weight'] = torch.tensor([[1.0, 1.0]], dtype=torch.float32)
+    weights[f'{name}.or.bias'] = torch.tensor([-1.0], dtype=torch.float32)
+    weights[f'{name}.nand.weight'] = torch.tensor([[-1.0, -1.0]], dtype=torch.float32)
+    weights[f'{name}.nand.bias'] = torch.tensor([1.0], dtype=torch.float32)
+    weights[f'{name}.and.weight'] = torch.tensor([[1.0, 1.0]], dtype=torch.float32)
+    weights[f'{name}.and.bias'] = torch.tensor([-2.0], dtype=torch.float32)
+
+def add_ha(name):
+    add_xor(f'{name}.sum')
+    weights[f'{name}.carry.weight'] = torch.tensor([[1.0, 1.0]], dtype=torch.float32)
+    weights[f'{name}.carry.bias'] = torch.tensor([-2.0], dtype=torch.float32)
+
+# First bit: ~a0 XOR 1 = NOT(~a0) = a0... wait that's wrong
+# Actually ~a0 + 1:
+#   bit 0: ~a0 XOR 1
+#   bit 1: ~a1 XOR carry0
+#   etc.
+
+# ~a0 XOR 1 = ~(~a0) = a0 when carry_in=1
+# But using threshold: we add 1 to ~A
+# ~a0 + 1: n0 = ~a0 XOR 1 = NOT(~a0) = a0? No...
+# Actually XOR with 1 flips the bit: ~a0 XOR 1 = NOT(~a0) = a0
+# But we need the sum with carry...
+
+# Let me think more carefully:
+# ~A = [~a3, ~a2, ~a1, ~a0]
+# (~A) + 1 starting with carry_in = 1:
+#   n0 = ~a0 XOR 1, c0 = ~a0 AND 1 = ~a0
+#   n1 = ~a1 XOR ~a0, c1 = ~a1 AND ~a0
+#   n2 = ~a2 XOR (~a1 AND ~a0), c2 = ~a2 AND ~a1 AND ~a0
+#   n3 = ~a3 XOR (~a2 AND ~a1 AND ~a0)
+
+# Simplify: carry propagates as long as bits are 0 (after inversion, as long as original bits are 1)
+# n0 = ~a0 XOR 1 = a0 XOR 0 = NOT(~a0) = a0... hmm
+# Wait, ~a0 XOR 1:
+#   if ~a0=0 (a0=1): 0 XOR 1 = 1
+#   if ~a0=1 (a0=0): 1 XOR 1 = 0
+# So n0 = ~(~a0) = a0? That's not right for negation.
+
+# Let me trace through with example: A = 5 = 0101
+# ~A = 1010
+# ~A + 1 = 1010 + 0001 = 1011 = -5 in two's complement
+# So: n0 = 0 XOR 1 = 1 ✓
+#     n1 = 1 XOR 0 = 1 ✓ (carry from bit 0 is 0)
+#     n2 = 0 XOR 0 = 0 ✓
+#     n3 = 1 XOR 0 = 1 ✓
+
+# For A = 0 = 0000:
+# ~A = 1111
+# ~A + 1 = 1111 + 1 = 10000, but 4-bit gives 0000 ✓ (with overflow)
+
+# OK so the formula is:
+# Starting with cin = 1:
+# n0 = ~a0 XOR cin = ~a0 XOR 1 = NOT(~a0) = a0...
+# Wait that gives wrong answer.
+# For A=5: a0=1, ~a0=0, ~a0 XOR 1 = 0 XOR 1 = 1 ✓
+
+# Let me be more careful:
+# A = 5 = 0101: a3=0, a2=1, a1=0, a0=1
+# ~a3=1, ~a2=0, ~a1=1, ~a0=0
+# Add 1:
+#   n0 = 0 + 1 = 1, c=0
+#   n1 = 1 + 0 = 1, c=0
+#   n2 = 0 + 0 = 0, c=0
+#   n3 = 1 + 0 = 1, c=0
+# Result: 1011 = -5 ✓
+
+# So it's just incrementing ~A.
+
+for i in range(4):
+    add_ha(f'inc{i}')
+
+# Overflow detection: A = 1000 (-8)
+# NOT(a3 OR a2 OR a1 OR a0) AND... no wait
+# Overflow when A = -8 = 1000, meaning a3=1 and a2=a1=a0=0
+weights['ov_nora.weight'] = torch.tensor([[-1.0, -1.0, -1.0]], dtype=torch.float32)
+weights['ov_nora.bias'] = torch.tensor([0.0], dtype=torch.float32)
+
+weights['overflow.weight'] = torch.tensor([[1.0, 1.0]], dtype=torch.float32)
+weights['overflow.bias'] = torch.tensor([-2.0], dtype=torch.float32)
+
+save_file(weights, 'model.safetensors')
+
+def twos_comp(a):
+    inv = (~a) & 0xF
+    neg = (inv + 1) & 0xF
+    overflow = 1 if a == 8 else 0
+    return neg, overflow
+
+print("Verifying 4-bit Two's Complement...")
+errors = 0
+for a in range(16):
+    result, ov = twos_comp(a)
+    if a == 0:
+        expected = 0
+    else:
+        expected = (16 - a) & 0xF
+    exp_ov = 1 if a == 8 else 0
+
+    if result != expected or ov != exp_ov:
+        errors += 1
+        if errors <= 5:
+            print(f"ERROR: -({a}) = {result}, expected {expected}")
+
+if errors == 0:
+    print("All 16 test cases passed!")
+else:
+    print(f"FAILED: {errors} errors")
+
+print("\nSigned interpretation:")
+for a in range(16):
+    signed_a = a if a < 8 else a - 16
+    neg, ov = twos_comp(a)
+    signed_neg = neg if neg < 8 else neg - 16
+    ov_str = " (OVERFLOW)" if ov else ""
+    print(f"  -({signed_a:+d}) = {signed_neg:+d}{ov_str}")
+
+mag = sum(t.abs().sum().item() for t in weights.values())
+print(f"\nMagnitude: {mag:.0f}")
+print(f"Parameters: {sum(t.numel() for t in weights.values())}")