POKA Identification and Structural Complexity:
Recovering Minimal Cognitive Architectures from Dynamical Systems
Abstract
We introduce a quantitative framework for identifying and comparing adaptive systems through structural factorization. Building upon the POKA architecture—Perception, Organization, Knowledge/Selection, and Action—we define conditions under which a closed-loop dynamical system admits a POKA decomposition.
Rather than treating intelligence as a performance metric, we study the existence, identification, and complexity of informationally mediated architectures. We introduce the concepts of canonical POKA representation, structural complexity, and minimal latent factorization, providing a mathematical basis for comparing biological, artificial, and engineered systems.
The framework yields three principal contributions:
A formal identification problem for recovering POKA architectures from observed dynamics.
A canonical representation theorem establishing equivalence classes of minimal decompositions.
A structural complexity index measuring the minimal representational resources required to implement a system's behavior.
1. Introduction
Most contemporary approaches evaluate intelligent systems through performance measures such as prediction accuracy, reward maximization, or task completion.
We pursue a different question:
Given a dynamical system, what is the minimal internal architecture required to produce its observed behavior?
To address this question, we introduce the POKA framework.
POKA describes systems through four informational stages:
[
\text{Perception}
\rightarrow
\text{Organization}
\rightarrow
\text{Knowledge/Selection}
\rightarrow
\text{Action}
]
or
[
\Omega=(P,O,K,A).
]
The central objective of this paper is not to define intelligence ontologically, but to characterize and recover minimal informational architectures underlying adaptive behavior.
2. POKA Architecture
Let
[
X
]
be a measurable state space.
Define auxiliary spaces:
[
P(X),
\quad
H,
\quad
Y.
]
The four structural operators are
Perception
[
P:X\rightarrow P(X)
]
Organization
[
O:P(X)\rightarrow H
]
Knowledge / Selection
[
K:H\rightarrow Y
]
Action
[
A:Y\rightarrow X
]
The induced system dynamics are
[
I=A\circ K\circ O\circ P.
]
We refer to
[
\Omega=(P,O,K,A)
]
as a POKA decomposition of the dynamical system.
3. The POKA Identification Problem
Suppose observations reveal only the external dynamics
[
I:X\rightarrow X.
]
The internal spaces
[
H
]
and
[
Y
]
are unknown.
The identification problem is:
Find
[
\Omega=(P,O,K,A)
]
such that
[
I=A\circ K\circ O\circ P.
]
subject to structural constraints.
Definition 3.1 (Admissible POKA Decomposition)
A decomposition is admissible if:
Non-trivial Action Space
[
|Y|>1
]
State-Dependent Selection
[
\exists h_1,h_2\in H
]
such that
[
K(h_1)\neq K(h_2)
]
Informative Representation
[
I(H;A(Y))>0.
]
4. Canonical Representation
Many distinct decompositions may reproduce the same dynamics.
We therefore seek minimal representations.
Definition 4.1 (Minimal Representation)
A POKA decomposition is minimal if no decomposition with lower-dimensional latent spaces reproduces the same dynamics.
Formally,
[
\dim(H)
+
\dim(Y)
]
is minimal among all admissible decompositions.
Theorem 1 (Canonical Representation Theorem)
Let
[
I:X\rightarrow X
]
admit at least one admissible POKA decomposition.
Then there exists a minimal decomposition
[
\Omega^\ast.
]
Furthermore, any two minimal decompositions
[
\Omega_1^\ast
]
and
[
\Omega_2^\ast
]
are equivalent up to latent-space isomorphism.
That is, there exist isomorphisms
[
\phi_H:H_1\rightarrow H_2
]
and
[
\phi_Y:Y_1\rightarrow Y_2
]
such that the induced diagrams commute.
Proof Sketch
The collection of admissible decompositions induces a partially ordered set under latent-space dimension.
Minimal elements exist by dimension minimization.
Equivalent minimal decompositions preserve identical state-transition structure.
Consequently, their latent spaces differ only by invertible coordinate transformations.
∎
5. Structural Complexity
We now define a quantitative measure of architectural sophistication.
Definition 5.1 (Structural Complexity Index)
Let
[
\mathcal F(I)
]
denote the family of admissible POKA decompositions of
[
I.
]
Define
[
C(S)
\inf_{\Omega\in\mathcal F(I)}
\left[
\dim(H)+\dim(Y)
\right].
]
The quantity
[
C(S)
]
measures the minimal representational and decision-making resources required to implement the observed dynamics.
Interpretation
Small values of
[
C(S)
]
correspond to simple control systems.
Large values correspond to systems requiring rich internal representations and extensive action repertoires.
Examples:
Thermostat: low complexity.
Bacterial chemotaxis: moderate complexity.
Mammalian cognition: high complexity.
Large language models: potentially very high complexity.
6. Complexity-Regularized Identification
Given observational data
[
D={(x_t,x_{t+1})},
]
we define
[
L(\Omega)
\text{PredictionError}(\Omega)
+
\lambda C(S).
]
The first term rewards fidelity to observations.
The second penalizes unnecessary architectural complexity.
The recovered decomposition is
[
\Omega^\ast
\arg\min_\Omega
L(\Omega).
]
This provides a practical identification framework suitable for machine-learning optimization.
7. Structural Taxonomy
The framework yields a hierarchy of systems.
Level 0
Pure dynamics
[
x_{t+1}=f(x_t).
]
No meaningful decomposition.
Level 1
Fixed POKA architecture.
[
\Omega_t=\Omega.
]
Classical controllers.
Level 2
Adaptive POKA architecture.
[
\Omega_t\neq\Omega_{t+1}.
]
Learning systems.
Level 3
Meta-POKA systems.
The architecture itself becomes an adaptive object:
[
\Psi:(\Omega_t,M_t,X_t)\rightarrow\Omega_{t+1}.
]
Self-restructuring systems.
8. Discussion
The POKA framework shifts attention from performance toward structural organization.
Instead of asking:
"How intelligent is the system?"
we ask:
"What is the minimal architecture required to generate its behavior?"
This perspective permits direct comparison between biological organisms, engineered devices, and artificial agents using common structural principles.
9. Future Directions
Future work includes:
Information-theoretic bounds on structural complexity.
Hierarchical POKA decompositions.
Efficient identification algorithms.
Neural and robotic applications.
Formal analysis of Meta-POKA adaptation dynamics.
10. Conclusion
We introduced a quantitative theory for identifying minimal informational architectures underlying adaptive systems.
By defining canonical POKA representations and a structural complexity index, the framework transforms POKA from a conceptual architecture into a measurable theory of adaptive organization.
The resulting theory provides a substrate-independent basis for comparing and analyzing dynamical systems through the minimal structures required to perceive, organize, select, and act.
0 σχόλια:
Δημοσίευση σχολίου