Magnitude Index
Results of exhaustive magnitude enumeration on threshold logic circuits.
Summary
All circuits listed below have been tested via exhaustive enumeration. "Min Mag" is the minimum magnitude at which valid configurations were found.
Single-Layer Gates (Linearly Separable)
| Circuit | Inputs | Params | Min Mag | Solutions | Configs Tested |
|---|---|---|---|---|---|
| threshold-not | 1 | 2 | 1 | 1 | 5 |
| threshold-nor | 2 | 3 | 2 | 1 | 7 |
| threshold-implies | 2 | 3 | 2 | 1 | 25 |
| threshold-or | 2 | 3 | 3 | 1 | 25 |
| threshold-nand | 2 | 3 | 3 | 1 | 25 |
| threshold-and | 2 | 3 | 4 | 1 | 129 |
| threshold-nor3 | 3 | 4 | 3 | 1 | 129 |
| threshold-or3 | 3 | 4 | 4 | 1 | 321 |
| threshold-nand3 | 3 | 4 | 5 | 1 | 681 |
| threshold-and3 | 3 | 4 | 6 | 1 | 1,289 |
| threshold-demux | 2 | 6 | 7 | 1 | 19,825 |
| threshold-majority | 8 | 9 | 13 | 1 | 27,298,155 |
Multi-Layer Gates (Not Linearly Separable)
| Circuit | Inputs | Params | Original Mag | Min Mag | Solutions | Reduction |
|---|---|---|---|---|---|---|
| threshold-xor | 2 | 9 | 10 | 7 | 6 | 30% |
| threshold-xnor | 2 | 9 | 9 | 7 | 2 | 22% |
| threshold-mux | 3 | 11 | 10 | 7 | 4 | 30% |
| threshold-xor3 | 3 | 16 | 14 | 10 | 18 | 29% |
Magnitude-Minimized Variants
These repos contain the minimum-magnitude weights found:
threshold-xor-mag7- 6 solutions at magnitude 7threshold-xnor-mag7- 2 solutions at magnitude 7threshold-mux-mag7- 4 solutions at magnitude 7threshold-xor3-mag10- 18 solutions at magnitude 10 (flat architecture)
Pending / In Progress
| Circuit | Params | Status |
|---|---|---|
| threshold-halfadder | 12 | Running (expected min: 11) |
| threshold-mod4 | 9 | Running |
| threshold-biimplies | 9 | Not yet tested (same as XNOR) |
| threshold-halfsubtractor | 12 | Not yet tested |
Methodology
Exhaustive search enumerates all integer weight configurations by magnitude level (0, 1, 2, ...) until valid solutions are found. This finds the minimum magnitude within the search space.
For circuits with >12 parameters, exhaustive search becomes impractical. Use evolutionary or simulated annealing instead.
Key Findings
Single-layer threshold gates appear to have unique minimum-magnitude representations. All tested single-layer gates have exactly 1 solution at their minimum magnitude.
Multi-layer gates can have solution families. XOR has 6 solutions at magnitude 7, XOR3 has 18 solutions at magnitude 10.
Non-linearly-separable functions benefit most from optimization. XOR/XNOR/MUX achieved 22-30% magnitude reduction.
Architecture matters. XOR3 flat architecture (mag 10) beats cascade architecture (mag 14) by 29%.
Last Updated
2026-01-23