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anonymous-author-submissions edited this page Feb 12, 2026 · 5 revisions

AND

Task description

The AND task evaluates whether an evolved Dynamic Neural Field (DNF) architecture can implement a logical conjunction over two binary inputs.

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 only when both inputs are active.
  3. If either input is inactive, the output must remain at baseline.

Highest-fitness minimal phenotype

The AND task converges reliably to a minimal three-field architecture:

  • Fields: 3 (two inputs, one output)

  • Hidden fields: 0

  • Connections: 2 feedforward couplings (one from each input to the output)

  • Dynamics:

    • Input fields: self-stabilized peaks representing binary values
    • Output field: self-stabilized peak gated by convergent excitation
and-phenotype@320x

The figure above illustrates:

  • Individual input activations (top and bottom)
  • Convergent coupling into the output field
  • Output activation occurring only when both inputs are active

This implements logical AND without explicit thresholding or symbolic rules.

Experiment-level statistics across runs

Analysed 100 runs; 100 reached the fitness threshold (100.0% success, threshold = 0.950)

Generations to threshold (successful runs)

  • Mean: 3.15
  • Median: 3.00
  • Std: 0.78

Convergence speed

  • Mean convergence rate (fitness gain/gen): 0.2407
  • Mean fitness improvement/gen: 0.2407

Architecture (successful solutions)

  • Hidden fields (mean ± std): 0.00 ± 0.00
  • Enabled connections (mean ± std): 2.00 ± 0.00

Notable runs

label run value
Highest max fitness 2026-02-07 20h47m11s 0.988201
Lowest max fitness 2026-02-07 20h05m46s 0.950229
Fastest to threshold 2026-02-07 20h03m03s 2.0
Slowest to threshold 2026-02-07 20h17m54s 5.0
Most hidden fields 2026-02-07 20h01m58s 0.0
Most enabled connections 2026-02-07 20h01m58s 2.0

Perfect — here is the Best-run evolutionary dynamics section for the AND page, fully aligned with the exact structure and tone used in Detection, Memory, Selection, DMTS, and IOR.

You can paste this directly under the experiment-level statistics on the AND page.

Best-run evolutionary dynamics

This section reports detailed statistics from the evolutionary run that produced the highest-fitness solution for the AND task.

Fitness statistics

Final generation: g = 4

  • Best fitness: 0.9882
  • Target fitness: 0.950
  • Average fitness: 0.4391

Overall run statistics

  • Max best fitness: 0.9882 (reached at generation 4)
  • Mean best fitness over run (AUC): 0.6375
  • Mean average fitness over run (AUC): 0.3112
  • Longest stagnation period: 0 generations
  • Target fitness first reached: generation 4 (best fitness ≈ 0.9882)

Species statistics

Final generation (g = 4)

  • Species: 1
  • Active species: 1

Across run

  • Total distinct species created: 1
  • Species extinct by final generation: 0
  • Average species per generation: 1.00
  • Average active species per generation: 1.00
  • Max active species in a generation: 1 (at g = 0)

Species lifetime & size

  • Average species lifespan: 4.00 generations
  • Longest-lived species: 0 (lifespan 4)
  • Average max members per species: 1000.00
  • Average offspring per species: 3000.00

Topology statistics

Final generation (g = 4)

  • Avg genome size: 4.23
  • Avg field genes: 3.00
  • Avg connection genes: 1.23

Growth over run

  • Genome size change: +1.23 (≈ +0.308 / gen)
  • Field genes change: +0.00 (≈ +0.000 / gen)
  • Connection genes change: +1.23 (≈ +0.308 / gen)

Ratios

  • Avg connections per field (final gen): 0.41

Population-level kernel usage

  • Field kernels:

    • Gaussian: 9 227 (76.9%)
    • Mexican-hat: 2 773 (23.1%)

  • Interaction kernels:

    • Gaussian: 1 190 (84.0%)
    • Mexican-hat: 227 (16.0%)

Genome of the highest-performing solution

Generation: 4 Fitness: 0.9882

Field genes

Field Role Kernel type Field parameters Kernel parameters
nf 1 Input Mexican-hat h = -12.28, τ = 14.13 A_exc = 21.25, σ_exc = 5.01, A_inh = 5.21, σ_inh = 14.98, A_glob = -0.10
nf 2 Input Gaussian h = -12.15, τ = 32.63 A = 22.31, σ = 9.73, A_glob = -0.09
nf 3 Output Gaussian h = -9.36, τ = 37.63 A = 7.19, σ = 1.95, A_glob = -0.16

Interaction genes

Interaction gene From → To Kernel parameters
gk cg 2-3-0 nf 2 → nf 3 A = 5.36, σ = 7.60, A_glob = 0.00
mhk cg 1-3-7 nf 1 → nf 3 A_exc = 21.69, σ_exc = 9.76, A_inh = 17.94, σ_inh = 14.32, A_glob = 0.00

Interpretation

The AND task shows a distinct evolutionary signature compared to the instability tasks:

  • No speciation pressure is required — a single lineage dominates the run.
  • Evolution focuses on parameter refinement rather than structural exploration.
  • The output field integrates convergent excitation from both inputs, naturally implementing logical conjunction.

This highlights that neat-dnfs can express discrete logical structure while remaining fully grounded in continuous neural field dynamics.

AND.mp4

Takeaway

The AND task demonstrates that neat-dnfs can:

  • Solve symbolic logical operations using continuous neural dynamics
  • Converge rapidly and reliably
  • Produce fully interpretable, minimal architectures

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