Intelligence Entropy

Can a system make goal-directed environmental progress efficiently, and keep doing so across diverse situations? $I_{cd}$ is a behavioral diagnostic of that demonstrated pragmatic competence.

The measure

Producing order is not enough: crystals, thermostats, and other fixed processes can reduce disorder within one narrow set of conditions. The proposed diagnostic therefore combines efficient action with breadth across contexts.

$$I_{cd} = \underbrace{\frac{-\Delta S}{W}}_{\text{efficiency }\eta} \times \underbrace{H(C)}_{\text{context diversity}}$$

Term	Meaning
$-\Delta S$	Reduction in domain-defined, goal-relative environmental disorder
$W$	Environmental action cost, not compute, energy, or runtime
$H(C)$	Shannon diversity of evaluator-defined contexts in which behavior is observed

The definitions of $S$, $W$, and the context partition belong to an evaluation. Scores are comparable within a fixed operationalization, not directly across unrelated domains.

Why breadth changes the result

In the sliding-puzzle experiment, every restricted agent uses the same greedy Manhattan-distance heuristic on its allowed grid sizes and random actions elsewhere. The Easy specialist is slightly more efficient, but the Full agent demonstrates useful behavior across more contexts. Accounting for that context diversity flips their ranking.

Four experiments, one progression

Experiment	Role in the argument	Implementation
Coupled regulation	Isolates efficiency while every agent sees the same three contexts	run_all.py
Sliding puzzle	Varies efficiency and competence range using restricted versions of the same heuristic	run_sliding_puzzle_heuristic.py
IPC 2023 planning	Tests efficiency and domain coverage using archival competition data	run_ipc.py
ARC	Uses compressed goal residuals to distinguish structured near-misses hidden by exact-match accuracy	run_arc_compression.py

Note

$I_{cd}$ is a post-hoc behavioral diagnostic, not a training objective. It measures demonstrated environmental competence, not learning speed, computation, understanding, creativity, consciousness, or latent capability.

Try the measure

Install the project and development dependencies:

uv sync --extra dev

from ie.measure import i_cd

score = i_cd(
    delta_s=-2.0,
    work=1.0,
    contexts=[0, 1, 2, 3],
)
print(score)  # 2.772588722239781

Run the test suite:

uv run pytest tests/ -v --tb=short

Reproducing the experiments

The experiments have different runtime and data requirements. ARC rule-based results can be recomputed from the included tasks; the published LLM caches are not tracked. IPC requires the external benchmark repository and competition properties. The full synthetic pipeline trains several agents over 50 seeds.

See Reproducing the paper for exact commands, inputs, outputs, and caveats.

Paper

Efficient Entropy Reduction as a Substrate-Independent Measure of Pragmatic Intelligence
Dario Cazzani, Artificial General Intelligence: 19th International Conference, AGI 2026, Springer LNCS 16854, pp. 154–171.

Read the repository copy · View the source

Citation

@inproceedings{cazzani2026entropy,
  author    = {Cazzani, Dario},
  editor    = {Iklé, Matthew and Franz, Arthur and Kemendo, Andrew and Lowy, Haley},
  title     = {Efficient Entropy Reduction as a Substrate-Independent Measure of Pragmatic Intelligence},
  booktitle = {Artificial General Intelligence},
  series    = {Lecture Notes in Computer Science},
  volume    = {16854},
  pages     = {154--171},
  publisher = {Springer Nature Switzerland},
  address   = {Cham},
  year      = {2026},
  doi       = {10.1007/978-3-032-33010-9_10},
  isbn      = {978-3-032-33010-9},
  url       = {https://doi.org/10.1007/978-3-032-33010-9_10}
}

License

MIT