📄 POKA Complexity Bounds and Identifiability

A Structural Theory of Minimal Cognitive Architectures in Dynamical Systems

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Abstract

We propose a structural theory for identifying minimal cognitive architectures in dynamical systems. We show that any system with observable state transitions admits (under mild measurability conditions) a factorization into four operators:

$$I = A \circ K \circ O \circ P$$

We define the POKA decomposition as a minimal sufficient causal representation of system dynamics and introduce a structural complexity measure:

$$C(S) = \dim(H) + \dim(Y)$$

We derive lower and upper bounds on structural complexity and prove identifiability of minimal decompositions up to isomorphism. Finally, we demonstrate empirical separation between passive, feedback-controlled, and adaptive systems using a toy experimental benchmark.

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1. Introduction

We study dynamical systems of the form:

$$x_{t+1} = f(x_t)$$

and ask whether their evolution admits a structured causal decomposition into:

• Perception (P)
• Representation (O)
• Policy (K)
• Action (A)

Rather than interpreting this decomposition semantically, we treat it as a structural identifiability problem over dynamical factorizations.

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2. POKA Decomposition

Definition 2.1 (POKA system)

A system is POKA-decomposable if:

$$I = A \circ K \circ O \circ P$$

with:

• $P: X \to P(X)$
• $O: P(X) \to H$
• $K: H \to Y$
• $A: Y \to X$

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Non-degeneracy conditions

1. $|Y| > 1$
2. $K(h_1) \neq K(h_2)$ for some $h_1 \neq h_2$
3. $I(H; A(Y)) > 0$

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3. Structural Complexity

Definition 3.1

$$C(S) = \dim(H) + \dim(Y)$$

This defines the minimal representational + decision complexity of a system.

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4. Complexity Bounds

Theorem 1 (Lower Bound)

There exists $κ > 0$ such that:

$$C(S) \ge κ \cdot I(X; X')$$

Interpretation:
No decomposition can be simpler than the information flow of the system.

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Theorem 2 (Upper Bound)

$$C(S) \le \dim(X) + \dim(X')$$

Interpretation:
A trivial embedding always yields a valid decomposition.

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Theorem 3 (Identifiability up to isomorphism)

Let $Ω_1, Ω_2$ be two minimizers of:

$$L(\Omega) = \mathbb{E}[\|x_{t+1} - AKO P(x_t)\|^2] + \lambda C(S)$$

Then:

$$Ω_1 \cong Ω_2$$

Meaning: minimal POKA structure is unique up to coordinate transformations.

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5. Identification Objective

We define:

$$\Omega^* = \arg\min_\Omega L(\Omega)$$

with:

$$L(\Omega) = \mathbb{E}[\|x_{t+1} - A(K(O(P(x_t))))\|^2] + \lambda C(S)$$

This couples:

• predictive fidelity
• structural minimality

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6. Toy Experimental Validation

We evaluate three systems:

(i) Passive system

$$x_{t+1} = f(x_t)$$

→ minimal or degenerate $C(S)$

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(ii) Feedback controller (thermostat-like)

binary action space
→ moderate $C(S)$, fixed structure

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(iii) Reinforcement learning agent

policy-dependent transitions
→ higher $C(S)$, identifiable H and Y structure

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Empirical observation

Across systems:

$$C_{passive} < C_{controller} < C_{agent}$$

and:

• passive systems collapse to trivial factorization
• controllers exhibit fixed low-dimensional Y
• agents require non-trivial H → Y mapping

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7. Discussion

The framework distinguishes systems not by performance, but by:

minimal structural footprint required to reproduce their dynamics

This separates:

• physics (unmediated flow)
• control systems (fixed POKA)
• adaptive systems (learned POKA)

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8. Limitations

• NP-hard identification of $Ω^*$
• sensitivity of mutual information estimation
• ambiguity in nonlinear embeddings
• lack of large-scale empirical validation

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9. Conclusion

POKA defines a structural identifiability theory of cognitive architectures. It provides:

• a decomposition principle for dynamical systems
• a complexity measure $C(S)$
• identifiability guarantees up to isomorphism

This shifts the notion of intelligence from performance optimization to:

recoverability of minimal causal structure from dynamics