XOR
The XOR task evaluates whether an evolved Dynamic Neural Field (DNF) architecture can implement exclusive-or, a classic benchmark requiring non-linear separability.
The desired behavior is:
- Two independent input fields encode binary values (0 or 1) as localized activation peaks.
- The output field should exhibit a self-stabilized peak if exactly one input is active.
- If both inputs are inactive (0,0) or both active (1,1), the output must remain at baseline.
This task tests whether neat-dnfs can evolve non-linear computational structure using continuous-time neural field dynamics.
Unlike AND, XOR reliably requires additional structure to implement a non-linear mapping:
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Fields: 4 (two inputs, one hidden, one output)
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Hidden fields: typically 1–4 (mean indicates consistent use of internal structure)
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Connections: dense multi-path coupling (including inhibitory / Mexican-hat interactions)
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Dynamics:
- Input fields encode the two bits as peaks
- A hidden field constructs an internal intermediate representation
- The output field combines direct and indirect pathways to realize exclusive-or
The figure above highlights the core mechanism:
- Multiple coupling routes from inputs to output
- A hidden-field “pre-shape” that modulates the output
- Inhibitory/excitatory interactions that suppress the (1,1) case while preserving (1,0) and (0,1)
Analysed 100 runs; 100 reached the fitness threshold (100.0% success, threshold = 0.950)
- Mean: 41.33
- Median: 35.00
- Std: 23.86
- Mean convergence rate (fitness gain/gen): 0.0234
- Mean fitness improvement/gen: 0.0250
- Hidden fields (mean ± std): 1.75 ± 0.65
- Enabled connections (mean ± std): 7.04 ± 2.20
- Average run duration: 0.40 h
- Avg. time per generation: 29.39 s/gen
| label | run | value |
|---|---|---|
| Highest max fitness | 2026-02-11 11h49m16s | 0.981327 |
| Lowest max fitness | 2026-02-10 05h03m46s | 0.950413 |
| Fastest to threshold | 2026-02-10 01h43m26s | 12.0 |
| Slowest to threshold | 2026-02-10 13h32m01s | 164.0 |
| Most hidden fields | 2026-02-10 13h32m01s | 4.0 |
| Most enabled connections | 2026-02-10 13h32m01s | 14.0 |
This section reports detailed statistics from the evolutionary run that produced the highest-fitness solution for the XOR task.
Final generation: g = 42
- Best fitness: 0.9731
- Target fitness: 0.950
- Average fitness: 0.3949
Overall run statistics
- Max best fitness: 0.9731 (reached at generation 42)
- Mean best fitness over run (AUC): 0.7430
- Mean average fitness over run (AUC): 0.4228
- Longest stagnation period: 4 generations
- Target fitness first reached: generation 42 (best fitness ≈ 0.9731)
Final generation (g = 42)
- Species: 308
- Active species: 12
Across run
- Total distinct species created: 304
- Species extinct by final generation: 296
- Average species per generation: 85.86
- Average active species per generation: 41.49
- Max active species in a generation: 195 (at g = 39)
Species lifetime & size
- Average species lifespan: 11.13 generations
- Longest-lived species: 0 (lifespan 42)
- Average max members per species: 23.39
- Average offspring per species: 134.87
Final generation (g = 42)
- Avg genome size: 9.00
- Avg field genes: 4.00
- Avg connection genes: 5.00
Growth over run
- Genome size change: +6.00 (≈ +0.143 / gen)
- Field genes change: +1.00 (≈ +0.024 / gen)
- Connection genes change: +5.00 (≈ +0.119 / gen)
Ratios
- Avg connections per field (final gen): 1.25
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Field kernels:
- Gaussian: 130 505 (79.2%)
- Mexican-hat: 34 180 (20.8%)
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Interaction kernels:
- Gaussian: 89 503 (63.5%)
- Mexican-hat: 51 403 (36.5%)
Generation: 42 Fitness: 0.9731
| Field | Role | Kernel type | Field parameters | Kernel parameters |
|---|---|---|---|---|
| nf 1 | Input | Gaussian | h = -11.08, τ = 24.33 | A = 8.67, σ = 7.69, A_glob = -0.20 |
| nf 2 | Input | Mexican-hat | h = -14.00, τ = 5.00 | A_exc = 22.02, σ_exc = 10.01, A_inh = 26.26, σ_inh = 12.33, A_glob = -0.01 |
| nf 3 | Output | Gaussian | h = -7.99, τ = 40.00 | A = 9.55, σ = 6.69, A_glob = -0.10 |
| nf 4 | Hidden | Gaussian | h = -5.48, τ = 31.38 | A = 13.76, σ = 5.52, A_glob = -0.20 |
| Interaction gene | From → To | Kernel parameters |
|---|---|---|
| mhk cg 1-3-3 | nf 1 → nf 3 | A_exc = -15.00, σ_exc = 30.00, A_inh = 7.16, σ_inh = 21.80, A_glob = 0.00 |
| gk cg 1-4-4 | nf 1 → nf 4 | A = 7.91, σ = 4.39, A_glob = 0.00 |
| mhk cg 4-3-5 | nf 4 → nf 3 | A_exc = 15.00, σ_exc = 5.00, A_inh = 1.69, σ_inh = 21.86, A_glob = -0.05 |
| gk cg 2-3-11 | nf 2 → nf 3 | A = 16.98, σ = 5.85, A_glob = -0.05 |
| mhk cg 2-4-37 | nf 2 → nf 4 | A_exc = -17.16, σ_exc = 14.30, A_inh = 28.46, σ_inh = 20.25, A_glob = 0.00 |
XOR.mp4
XOR is a strong contrast case to AND:
- It requires non-minimal structure (a hidden field and multiple interaction pathways).
- Evolution explores a very large architectural space (high species counts and extinctions).
- Despite this, solutions remain interpretable: a hidden intermediate representation plus structured excitation/inhibition implements the exclusive-or constraint.
In the overall suite, XOR sits closer to DMTS/IOR than to AND in terms of evolutionary effort and architectural complexity, making it a useful bridge between symbolic and cognitive benchmarks.