System Intelligence Index (SII)¶

Status: task-relative reporting instrument for selected prediction, regulation, adaptation, and component-activity measures. It is not a general definition or validated index of intelligence.

1. Motivation¶

When we talk about intelligence, we often think of agents: humans, animals, machines. But many of the phenomena that interest us in complex systems look "intelligent" without any explicit agent:

ecosystems that regulate themselves
markets that react to shocks
infrastructures that re-route around failures
technical systems that maintain service under changing load

Instead of asking "Is this system intelligent, yes or no?", it can be more useful to ask:

"To what degree does this system behave intelligently?"

The System Intelligence Index (SII) is a tentative way to structure that question.

It is not meant as a precise metric, but as a conceptual scaffold: a way to decompose "intelligence" into dimensions that can be observed, simulated, and eventually quantified.

2. Intelligence as a System Property¶

In this context, a system is any configuration of interacting components:

cells in a cellular automaton
services in a software architecture
species in an ecosystem
agents in an economy

For a declared benchmark, the instrument records:

Predictive structure – it encodes regularities of its environment or of its own dynamics.
Regulatory capacity – it can maintain some internal variables within viable bounds.
Adaptive flexibility – it can change its own behaviour or structure in response to perturbations.

The System Intelligence Index combines these three selected aspects. Systems can have other capacities, and no theorem makes this list necessary or sufficient for intelligence.

3. Three Core Dimensions¶

We can think of the SII as a product (or composition) of three factors:

\[ \text{SII} \approx P \times R \times A \]

where:

$ P $ = Predictive Power
$ R $ = Regulation Ability
$ A $ = Adaptive Capacity

Each factor can be explored separately in simulations and models.

3.1 Predictive Power (P)¶

Question:
How well does the system anticipate what will happen next?

In many models, "prediction" appears as:

internal transition matrices that match the world's dynamics
low prediction error over time
internal representations that compress regularities in the environment

Possible operationalisations:

inverse of long-term prediction error
mutual information between internal state and future environment state
convergence of internal models to external dynamics

In your nested-learning demos, $ P $ is high when the observer's internal transition matrix closely matches the true Markov process.

3.2 Regulation Ability (R)¶

Question:
How well does the system keep key variables within viable ranges?

Examples:

homeostatic cellular automata maintaining target density
control systems keeping temperature, load, or latency stable
organisms maintaining internal variables (pH, glucose, temperature)

Possible operationalisations:

inverse of variance around a target value
time spent within a defined "viability corridor"
robustness under perturbations (how quickly it returns to viable ranges)

In your ecosystem-regulation models, $ R $ increases when the system can compensate for shocks (e.g. random deaths) without collapsing or exploding.

3.3 Adaptive Capacity (A)¶

Question:
How capable is the system of changing its own behaviour or structure when conditions change?

This includes:

updating internal models when the environment shifts
changing parameters (learning rates, thresholds)
reconfiguring topology (who interacts with whom)

Possible operationalisations:

speed of re-convergence after a regime change
capacity to maintain performance across multiple environments
diversity of internal models or strategies

In nested-learning setups, $ A $ shows up when a meta-learner adjusts the learning rate or strategy of a lower-level learner in response to non-stationary dynamics.

4. A Simple Conceptual Formula¶

For many exploratory simulations, a simple multiplicative structure is enough:

\[ \text{SII} = f(P, R, A) \approx P \times R \times A \]

Why consider a multiplicative toy score?

if any factor is near zero, the chosen product becomes low by construction
a system that predicts well but cannot regulate anything
a system that regulates well but never learns
if all three are moderately high, the system feels "intelligent" in a systems sense, even without consciousness or explicit goals

In practice, each of $ P, R, A $ could be normalised to $[0, 1]$ or to some bounded interval, and the resulting SII would also live in a bounded range.

This is not a definition of intelligence, but a working reporting choice to connect:

predictive models (nested learning)
regulatory structures (homeostasis)
adaptation mechanisms (meta-learning, plasticity)

5. Examples Across Models¶

To make SII less abstract, here are some informal sketches:

5.1 Simple Homeostatic CA¶

Predictive Power (P):
None in the internal sense – it just applies a rule, no explicit model.
→ $ P $ ≈ low
Regulation Ability (R):
High, if density hovers around a target with low variance.
→ $ R $ ≈ high
Adaptive Capacity (A):
Low, if the rule set is fully fixed.
→ $ A $ ≈ low

Result:
SII is non-zero (the system is "smart" in regulating density),
but not very high (no learning, no internal modelling).

5.2 Nested Learning Two-State Model¶

Predictive Power (P):
High, if the learned transition matrix converges to the true dynamics.
→ $ P $ ≈ high
Regulation Ability (R):
Minimal – the model doesn't regulate a target variable, it only predicts.
→ $ R $ ≈ low
Adaptive Capacity (A):
Medium – it can adapt its model if the environment changes, but only via a simple learning rule.
→ $ A $ ≈ medium

Result:
SII is driven by prediction and some adaptation, but lacks explicit regulation.

5.3 Meta-Learning System (Future Work)¶

A model where:

a lower-level learner predicts the world
a meta-learner tunes the learner (e.g. learning rate, prior assumptions)
a regulatory loop keeps performance or error within bounds

could exhibit:

high $ P $ (accurate internal model)
medium to high $ R $ (stable performance under noise)
high $ A $ (robust under shifting regimes)

Such systems are good candidates for high SII in simulations.

6. Limitations and Open Questions¶

The System Intelligence Index is intentionally incomplete.
Open issues include:

Metric choice
How exactly should $ P, R, A $ be measured in different models?
Non-multiplicative combinations
In some systems, a weighted sum or more complex function might reflect "intelligence" better than a product.
Scale and scope
Intelligence at the component level vs. subsystem vs. whole system – SII might need to be defined at multiple scales.
Goal ambiguity
Regulation requires choosing "what matters". Who or what defines the target variables in a given model?
Relation to human notions of intelligence
SII says nothing about consciousness, experience, or meaning – it is a systems-level view, not a psychological one.

7. Why Bother?¶

Even if SII never becomes a "standard metric", it is useful as a lens:

it encourages thinking about prediction, regulation, and adaptation as separable dimensions
it helps compare very different systems (ecosystems, automata, networks) along a common conceptual axis
it creates a bridge between your code experiments and your philosophical questions about intelligence and systems

Future work in this repository can:

implement concrete estimators for $ P, R, A $ in specific models
explore how changes in architecture or feedback affect SII
connect SII to ideas from control theory, information theory, and active inference

For now, the System Intelligence Index is a thinking tool – a way to structure intuitions and guide the design of new experiments.

8. Optional Fourth Report: Component Activity (IP)¶

For agent tasks with a preregistered set of governing components, the repository also reports an instrument historically called Identity Persistence. Let $\mathcal{O}(t)$ contain the declared components whose causal intervention changes the action distribution at step $t$:

\[ \operatorname{IP}(t)=\frac{|\mathcal{O}(t)|}{n}. \]

The result depends on the component list, intervention, threshold, time resolution, and task. It reports detected component activity. It does not establish a self, consciousness, or metaphysical identity. Exp5–7 also show that consultation without joint satisfaction can score highly, so IP alone does not identify the Chord regime it was intended to describe.

For dashboards that intentionally require all four selected dimensions, one may report

\[ \operatorname{SII}_{PRAIP}=P\,R\,A\,\operatorname{IP}. \]

A zero collapses this product because multiplication was chosen; the result does not classify a system as merely a tool or as integrated intelligence. Other aggregation rules encode different trade-offs and should be compared.

Cross-system examples such as Boids, an Ising model, an ecosystem, and an agent should not receive shared scores until each axis has commensurable units and a common task. Magnetization at the Ising critical point is not, by itself, identity persistence.

The Lerchner Boundary note states the appropriate test: compare sequential and jointly constrained architectures under matched resources and perturbations. Whether a sharp boundary exists is open.

See the Lerchner Boundary for the proposed test protocol and the glossary for the maintained instrument definition.