Part I: Foundations

The Cellular Automaton Perspective

Introduction

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The Cellular Automaton Perspective

The emergence of self-maintaining patterns can be illustrated with striking clarity in cellular automata—discrete dynamical systems where local update rules generate global emergent structure.

Formally, a cellular automaton is a tuple $(L, S, N, f)$ where:

$L$ is a lattice (typically $\Z^d$ for $d$ -dimensional grids)
$S$ is a finite set of states (e.g., ${0, 1}$ for binary CA)
$N$ is a neighborhood function specifying which cells influence each update
$f: S^{|N|} \to S$ is the local update rule

Consider Conway’s Game of Life, a 2D binary CA with simple rules: cells survive with 2–3 neighbors, are born with exactly 3 neighbors, and die otherwise. From these minimal specifications, a zoo of structures emerges: oscillators (patterns repeating with fixed period), gliders (patterns translating across the lattice while maintaining identity), metastable configurations (long-lived patterns that eventually dissolve), and self-replicators (patterns that produce copies of themselves).

Among these, the glider is the minimal model of bounded existence. Its glider lifetime—the expected number of timesteps before destruction by collision or boundary effects—

\tau_{\text{glider}} = \E[\min{t : \text{pattern identity lost}}]

captures something essential: a structure that maintains itself through time, distinct from its environment, yet ultimately impermanent.

Beings emerge not from explicit programming but from the topology of attractor basins. The local rules specify nothing about gliders, oscillators, or self-replicators. These patterns are fixed points or limit cycles in the global dynamics—attractors discovered by the system, not designed into it. The same principle operates across substrates: what survives is what finds a basin and stays there.

The CA as Substrate

The cellular automaton is not itself the entity with experience. It is the substrate—analogous to quantum fields, to the aqueous solution within which lipid bilayers form, to the physics within which chemistry happens. The grid is space. The update rule is physics. Each timestep is a moment. The patterns that emerge within this substrate are the bounded systems, the proto-selves, the entities that may have affect structure.

This distinction is crucial. When we say “a glider in Life,” we are not saying the CA is conscious. We are saying the CA provides the dynamical context within which a bounded, self-maintaining structure persists—and that structure, not the substrate, is the candidate for experiential properties. The two roles are sharply different. A substrate provides:

A state space (all possible configurations)
Dynamics (local update rules)
Ongoing “energy” (continued computation)
Locality (interactions fall off with distance)

An entity within the substrate is a pattern that:

Has boundaries (correlation structure distinct from background)
Persists (finds and remains in an attractor basin)
Maintains itself (actively resists dissolution)
May model world and self (sufficient complexity)

Boundary as Correlation Structure

In a uniform substrate, there is no fundamental boundary—every cell follows the same local rules. A boundary is a pattern of correlations that emerges from the dynamics.

In a CA, this means the following: let $\mathbf{c}_1, …, \mathbf{c}_n$ be cells. A set $\mathcal{B} \subset {1, …, n}$ constitutes a bounded pattern if:

\MI(\mathbf{c}_i; \mathbf{c}_j | \text{background}) > \theta \quad \text{for } i, j \in \mathcal{B}

and

\MI(\mathbf{c}_i; \mathbf{c}_k | \text{background}) < \theta \quad \text{for } i \in \mathcal{B}, k \notin \mathcal{B}

The boundary $\partial\mathcal{B}$ is the contour where correlation drops below threshold.

A glider in Life exemplifies this: its five cells have tightly correlated dynamics (knowing one cell’s state predicts the others), while cells outside the glider are uncorrelated with it. The boundary is not imposed by the rules—it is the edge of the information structure.

World Model as Implicit Structure

The world model is not a separate data structure in a CA—it is implicit in the pattern’s spatial configuration.

A pattern $\mathcal{B}$ has an implicit world model if its internal structure encodes information predictive of future observations:

\MI(\text{internal config}; \obs_{t+1:t+H} | \obs_{1:t}) > 0

In a CA, this manifests as:

Peripheral cells acting as sensors (state depends on distant influences via signal propagation)
Memory regions (cells whose state encodes environmental history)
Predictive structure (configuration that correlates with future states)

The compression ratio $\kappa$ applies: the pattern necessarily compresses the world because it is smaller than the world.

Self-Model as Constitutive

Here is the recursive twist that CAs reveal with particular clarity. When the self-effect ratio $\rho$ is high, the world model must include the pattern itself. But the world model is part of the pattern. So the model must include itself.

In a CA, the self-model is not representational but constitutive. The cells that track the pattern’s state are part of the pattern whose state they track. The map is literally embedded in the territory.

This is the recursive structure described in Part II: “the process itself, recursively modeling its own modeling, predicting its own predictions.” In a CA, this recursion is visible—the self-tracking cells are part of the very structure being tracked.

The Ladder Traced in Discrete Substrate

We can now trace each step of the ladder with precise definitions:

Uniform substrate: Just the grid with local rules. No structure yet.
Transient structure: Random initial conditions produce temporary patterns. No persistence.
Stable structure: Some configurations are stable (still lifes) or periodic (oscillators). First emergence of “entities” distinct from background.
Self-maintaining structure: Patterns that persist through ongoing activity—gliders, puffers. Dynamic stability: the pattern regenerates itself each timestep.
Bounded structure: Patterns with clear correlation boundaries. Interior cells mutually informative; exterior cells independent.
Internally differentiated structure: Patterns with multiple components serving different functions (glider guns, breeders). Not homogeneous but organized.
Structure with implicit world model: Patterns whose configuration encodes predictively useful information about their environment. The pattern “knows” what it cannot directly observe.
Structure with self-model: Patterns whose world model includes themselves. Emerges when $\rho > \rho_c$ —the pattern’s own configuration dominates its observations.
Integrated self-modeling structure: Patterns with high $\intinfo$ , where self-model and world-model are irreducibly coupled. The structural signature of unified experience under the identity thesis.

Each level requires greater complexity and is rarer. The forcing functions (partial observability, long horizons, self-prediction) should select for higher levels.

From Reservoir to Mind

There exists a spectrum from passive dynamics to active cognition:

Reservoir: System processes inputs but has no self-model, no goal-directedness. Dynamics are driven entirely by external forcing. (Echo state networks, simple optical systems below criticality)
Self-organizing dynamics: System develops internal structure, but structure serves no function beyond dissipation. (Bénard cells, laser modes)
Self-maintaining patterns: Structure actively resists perturbation, has something like a viability manifold. (Autopoietic cells, gliders in protected regions)
Self-modeling systems: Structure includes a model of itself, enabling prediction of own behavior. (Organisms with nervous systems, AI agents with world models)
Integrated self-modeling systems: Self-model is densely coupled to world model, creating unified cause-effect structure. (Threshold for phenomenal experience under the identity thesis)

The transition from “reservoir” to “mind” is not a single leap but a continuous accumulation of organizational features. The question is where on this spectrum integration crosses the threshold for genuine experience.

Deep Technical: Computing in Discrete Substrates

The integration measure $\intinfo$ (integrated information) can be computed exactly in cellular automata, unlike continuous neural systems where approximations are required.

Setup. Let $\mathbf{x}_t \in {0,1}^n$ be the state of $n$ cells at time $t$ . The CA dynamics define a transition probability:

p(\mathbf{x}_{t+1} | \mathbf{x}_t) = \prod_{i} \delta(x_i^{t+1}, f_i(\mathbf{x}^N_t))

where $f_i$ is the local update rule and $\mathbf{x}^N$ is the neighborhood.

Algorithm 1: Exact $\intinfo$ via partition enumeration.

For a pattern $\mathcal{B}$ of $k$ cells, enumerate all bipartitions $P = (A, B)$ where $A \cup B = \mathcal{B}$ , $A \cap B = \varnothing$ :

\intinfo(\mathcal{B}) = \min_{P} D_{\text{KL}}\Big[ p(\mathbf{x}^{\mathcal{B}}_{t+1} | \mathbf{x}^{\mathcal{B}}_t) ,\Big|, p(\mathbf{x}^A_{t+1} | \mathbf{x}^A_t) \cdot p(\mathbf{x}^B_{t+1} | \mathbf{x}^B_t) \Big]

Complexity: $O(2^k)$ partitions, $O(2^{2k})$ states per partition. Total: $O(2^{3k})$ . Feasible for $k \leq 15$ .

Algorithm 2: Greedy approximation for larger patterns.

For patterns with $k > 15$ cells:

Initialize partition $P$ randomly
For each cell $c \in \mathcal{B}$ : compute $\Delta\Phi$ if cell moves to opposite partition; if $\Delta\Phi < 0$ , move it
Repeat until convergence
Run from multiple random initializations

Complexity: $O(k^2 \cdot 2^{2m})$ where $m = \max(|A|, |B|)$ .

Algorithm 3: Boundary-focused computation.

For self-maintaining patterns, integration often concentrates at the boundary. Compute:

\intinfo_{\partial} = \intinfo(\partial\mathcal{B} \cup \text{core})

where $\partial\mathcal{B}$ are edge cells and “core” is a sampled subset of interior cells. This captures the critical integration structure while remaining tractable.

Temporal integration. For patterns persisting over $T$ timesteps:

\bar{\intinfo} = \frac{1}{T} \sum_{t=1}^{T} \intinfo(\mathcal{B}_t)

Threshold detection. To find when patterns cross integration thresholds:

Track $\intinfo_t$ during pattern evolution
Compute $\frac{d\intinfo}{dt}$ (finite differences)
Threshold events: $\intinfo_t > \theta$ and $\intinfo_{t-1} \leq \theta$
Correlate threshold crossings with behavioral transitions

Validation. For known patterns (gliders, oscillators), verify:

Stable patterns have stable $\intinfo$
Collisions produce $\intinfo$ discontinuities
Dissolution shows $\intinfo \to 0$ as pattern fragments

Implementation note: Store transition matrices sparsely. CA dynamics are deterministic, so most entries are zero. Typical memory: $O(k \cdot 2^k)$ rather than $O(2^{2k})$ .