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README.md

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+---
+license: mit
+tags:
+- pytorch
+- safetensors
+- threshold-logic
+- neuromorphic
+- bit-manipulation
+- ffs
+---
+
+# threshold-ffs8
+
+8-bit find first set (FFS). Returns the 1-indexed position of the least significant set bit. Returns 0 if no bits are set.
+
+## Circuit
+
+```
+ x7  x6  x5  x4  x3  x2  x1  x0
+  │   │   │   │   │   │   │   │
+  └───┴───┴───┴───┴───┴───┴───┘
+              │
+              ▼
+    ┌─────────────────────┐
+    │  CTZ + 1 (if set)   │
+    │  or 0 (if all zero) │
+    └─────────────────────┘
+              │
+              ▼
+      [f3, f2, f1, f0]
+       (position 0-8)
+```
+
+## Function
+
+```
+ffs8(x7..x0) -> (f3, f2, f1, f0)
+
+position = 8*f3 + 4*f2 + 2*f1 + f0
+
+if input = 0: position = 0
+if input != 0: position = CTZ(input) + 1
+```
+
+FFS is 1-indexed: the first bit (x0) is position 1, not 0.
+
+## Truth Table (selected)
+
+| Input (hex) | Binary | FFS | f3 f2 f1 f0 | Meaning |
+|-------------|--------|:---:|-------------|---------|
+| 0x00 | 00000000 | 0 | 0 0 0 0 | No bits set |
+| 0x01 | 00000001 | 1 | 0 0 0 1 | Bit 0 is first |
+| 0x02 | 00000010 | 2 | 0 0 1 0 | Bit 1 is first |
+| 0x04 | 00000100 | 3 | 0 0 1 1 | Bit 2 is first |
+| 0x08 | 00001000 | 4 | 0 1 0 0 | Bit 3 is first |
+| 0x10 | 00010000 | 5 | 0 1 0 1 | Bit 4 is first |
+| 0x20 | 00100000 | 6 | 0 1 1 0 | Bit 5 is first |
+| 0x40 | 01000000 | 7 | 0 1 1 1 | Bit 6 is first |
+| 0x80 | 10000000 | 8 | 1 0 0 0 | Bit 7 is first |
+| 0xFF | 11111111 | 1 | 0 0 0 1 | Bit 0 is first |
+
+## Relationship to CTZ
+
+```
+if (input == 0):
+    FFS = 0
+else:
+    FFS = CTZ + 1
+```
+
+FFS and CTZ are closely related:
+- CTZ returns 0-7 for positions, 8 for zero input
+- FFS returns 1-8 for positions, 0 for zero input
+
+## Mechanism
+
+**Position detectors:** Same as CTZ - detect where first 1 appears from LSB
+
+| Signal | Fires when |
+|--------|------------|
+| p0 | x0 = 1 |
+| p1 | x0 = 0, x1 = 1 |
+| p2 | x0 = x1 = 0, x2 = 1 |
+| ... | ... |
+| p7 | x0..x6 = 0, x7 = 1 |
+
+**Output encoding:** Direct binary encoding of position + 1
+
+- f0 = p0 OR p2 OR p4 OR p6 (positions 0,2,4,6 → FFS 1,3,5,7)
+- f1 = p1 OR p2 OR p5 OR p6 (positions 1,2,5,6 → FFS 2,3,6,7)
+- f2 = p3 OR p4 OR p5 OR p6 (positions 3,4,5,6 → FFS 4,5,6,7)
+- f3 = p7 (position 7 → FFS 8)
+
+## Parameters
+
+| | |
+|---|---|
+| Inputs | 8 |
+| Outputs | 4 |
+| Neurons | 12 |
+| Layers | 2 |
+| Parameters | 76 |
+| Magnitude | 48 |
+
+## Usage
+
+```python
+from safetensors.torch import load_file
+import torch
+
+w = load_file('model.safetensors')
+
+def ffs8(bits):
+    # bits = [x0, x1, ..., x7] (LSB first)
+    inp = torch.tensor([float(b) for b in bits])
+    # ... (see model.py for full implementation)
+
+# Examples
+print(ffs8([1,0,0,0,0,0,0,0]))  # 1 (first bit set is position 0)
+print(ffs8([0,0,1,0,0,0,0,0]))  # 3 (first bit set is position 2)
+print(ffs8([0,0,0,0,0,0,0,0]))  # 0 (no bits set)
+```
+
+## Applications
+
+- POSIX ffs() function implementation
+- Finding available slots in bitmaps
+- Scheduler ready queue processing
+- Memory allocator free list management
+- Interrupt priority handling
+
+## Comparison: FFS vs CTZ vs CLZ
+
+| Function | Zero input | Non-zero input | Index base |
+|----------|------------|----------------|------------|
+| FFS | 0 | 1 to 8 | 1-indexed |
+| CTZ | 8 | 0 to 7 | 0-indexed |
+| CLZ | 8 | 0 to 7 | 0-indexed from MSB |
+
+## Files
+
+```
+threshold-ffs8/
+├── model.safetensors
+├── model.py
+├── create_safetensors.py
+├── config.json
+└── README.md
+```
+
+## License
+
+MIT

config.json

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+{
+  "name": "threshold-ffs8",
+  "description": "8-bit find first set (1-indexed position of LSB)",
+  "inputs": 8,
+  "outputs": 4,
+  "neurons": 12,
+  "layers": 2,
+  "parameters": 76
+}

create_safetensors.py

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+import torch
+from safetensors.torch import save_file
+
+weights = {}
+
+# 8-bit Find First Set (FFS)
+# Input: x0..x7 (x0 is LSB)
+# Output: f3,f2,f1,f0 where position = 8*f3 + 4*f2 + 2*f1 + f0
+# Returns 0 if no bits set, else 1-indexed position of first set bit
+
+# Position detectors (same as CTZ)
+for i in range(8):
+    w = [0.0] * 8
+    w[i] = 1.0
+    for j in range(i):
+        w[j] = -1.0
+    weights[f'p{i}.weight'] = torch.tensor([w], dtype=torch.float32)
+    weights[f'p{i}.bias'] = torch.tensor([-1.0], dtype=torch.float32)
+
+# Output encoding for FFS (1-indexed):
+# Position 0 (x0 first) → FFS = 1 = 0001
+# Position 1 (x1 first) → FFS = 2 = 0010
+# Position 2 (x2 first) → FFS = 3 = 0011
+# Position 3 (x3 first) → FFS = 4 = 0100
+# Position 4 (x4 first) → FFS = 5 = 0101
+# Position 5 (x5 first) → FFS = 6 = 0110
+# Position 6 (x6 first) → FFS = 7 = 0111
+# Position 7 (x7 first) → FFS = 8 = 1000
+# No bits set → FFS = 0 = 0000
+
+# f0 = p0 OR p2 OR p4 OR p6 (FFS values 1,3,5,7 have bit 0 set)
+# f1 = p1 OR p2 OR p5 OR p6 (FFS values 2,3,6,7 have bit 1 set)
+# f2 = p3 OR p4 OR p5 OR p6 (FFS values 4,5,6,7 have bit 2 set)
+# f3 = p7 (FFS value 8 has bit 3 set)
+
+save_file(weights, 'model.safetensors')
+
+def ffs8(bits):
+    """Find first set bit (1-indexed). Returns 0 if no bits set."""
+    inp = torch.tensor([float(b) for b in bits])
+
+    p = []
+    for i in range(8):
+        pi = int((inp @ weights[f'p{i}.weight'].T + weights[f'p{i}.bias'] >= 0).item())
+        p.append(pi)
+
+    # Combine for 1-indexed output
+    f0 = 1 if (p[0] or p[2] or p[4] or p[6]) else 0
+    f1 = 1 if (p[1] or p[2] or p[5] or p[6]) else 0
+    f2 = 1 if (p[3] or p[4] or p[5] or p[6]) else 0
+    f3 = p[7]
+
+    return f3, f2, f1, f0
+
+print("Verifying ffs8...")
+errors = 0
+for i in range(256):
+    bits = [(i >> j) & 1 for j in range(8)]
+    f3, f2, f1, f0 = ffs8(bits)
+    result = 8*f3 + 4*f2 + 2*f1 + f0
+
+    # Compute expected FFS
+    if i == 0:
+        expected = 0
+    else:
+        expected = 1
+        temp = i
+        while (temp & 1) == 0:
+            expected += 1
+            temp >>= 1
+
+    if result != expected:
+        errors += 1
+        if errors <= 5:
+            print(f"ERROR: ffs8({i:08b}) = {result}, expected {expected}")
+
+if errors == 0:
+    print("All 256 test cases passed!")
+else:
+    print(f"FAILED: {errors} errors")
+
+print(f"Magnitude: {sum(t.abs().sum().item() for t in weights.values()):.0f}")
+print(f"Parameters: {sum(t.numel() for t in weights.values())}")

model.py

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+import torch
+from safetensors.torch import load_file
+
+def load_model(path='model.safetensors'):
+    return load_file(path)
+
+def ffs8(bits, w):
+    """Find first set bit (1-indexed). Returns 0 if no bits set."""
+    inp = torch.tensor([float(b) for b in bits])
+
+    p = []
+    for i in range(8):
+        pi = int((inp @ w[f'p{i}.weight'].T + w[f'p{i}.bias'] >= 0).item())
+        p.append(pi)
+
+    f0 = 1 if (p[0] or p[2] or p[4] or p[6]) else 0
+    f1 = 1 if (p[1] or p[2] or p[5] or p[6]) else 0
+    f2 = 1 if (p[3] or p[4] or p[5] or p[6]) else 0
+    f3 = p[7]
+
+    return f3, f2, f1, f0
+
+if __name__ == '__main__':
+    w = load_model()
+    print('FFS8 selected tests:')
+    print('Input    | FFS | f3f2f1f0')
+    print('---------+-----+---------')
+    test_vals = [0x00, 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80, 0x06, 0xF0, 0xFF]
+    for val in test_vals:
+        bits = [(val >> j) & 1 for j in range(8)]
+        f3, f2, f1, f0 = ffs8(bits, w)
+        pos = 8*f3 + 4*f2 + 2*f1 + f0
+        print(f'{val:08b} |  {pos}  | {f3}{f2}{f1}{f0}')

model.safetensors

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+size 1328