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:

1. Two independent input fields encode binary values (0 or 1) as localized activation peaks.
2. The output field should exhibit a self-stabilized peak if exactly one input is active.
3. 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:

• Fields: 4 (two inputs, one hidden, one output)

• Hidden fields: typically 1–4 (mean indicates consistent use of internal structure)

• Connections: dense multi-path coupling (including inhibitory / Mexican-hat interactions)

• 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

phenotype-xor@320x

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)

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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年02月11日 11h49m16s	0.981327
Lowest max fitness	2026年02月10日 05h03m46s	0.950413
Fastest to threshold	2026年02月10日 01h43m26s	12.0
Slowest to threshold	2026年02月10日 13h32m01s	164.0
Most hidden fields	2026年02月10日 13h32m01s	4.0
Most enabled connections	2026年02月10日 13h32m01s	14.0

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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

• Field kernels:

  • Gaussian: 130 505 (79.2%)
  • Mexican-hat: 34 180 (20.8%)

• Interaction kernels:

  • Gaussian: 89 503 (63.5%)
  • Mexican-hat: 51 403 (36.5%)

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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

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XOR.mp4

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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.