Structured Dynamics in the Algorithmic Agent

Abstract

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether's theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent's constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain.

Keywords: AI; Kolmogorov theory; Lie groups and pseudogroups; algorithmic information theory (AIT); computational neuroscience; conservation laws; control theory; groups; manifold hypothesis; neural networks; neurophenomenology; symmetry.

Figure 7

Agent tracking a computer-generated moving hand. (a) An agent is observing and tracking a moving human hand generated by some simple physics-based computational model (such as Blender [99]). (b) World tracking neural network, where the input is compared to the read out—see Equation (11). We don’t detail here how the Comparator is implemented in the network or how its output is used to regulate the network—see Section 4.4 for a more in-depth discussion. (The image in (a) is AI-generated).

Figure 1

KT’s algorithmic agent, symmetry, and dynamics. The algorithmic agent [6,7,8] interacts dynamically with the World (structure, symmetry, compositional data). The Modeling Engine (compression) runs the current Model (which encodes found structure/symmetry) and makes predictions of future (compositional, coarse-grained) data and then evaluates the prediction error in the Comparator (world-tracking constraint monitoring) to update the Model. The Planning Engine runs counterfactual simulations and selects plans for the next (compositional) actions (agent outputs). The Updater receives prediction errors from the Comparator as inputs to improve the Model. The Comparator is a key agent element monitoring the success of the modeling engine in matching input data. We reflect this process mathematically as a world-tracking constraint on the dynamics (Equation (11); see also Section 4.4 and Appendix A).

Figure 2

An illustration of the relationship between underlying mathematical model Structure, Algorithm features (such as algorithmic complexity $\mathcal{K}$, i.e., program length, compositionally, recursion, cyclomatic complexity [26], etc.), Dynamics, first-person Experience (neurophenomenology), and in more detail some of the tools used in studying Dynamics, i.e., stability theory, bifurcation theory, chaos, geometry, topology, complexity (entropy, algorithmic), and symmetry, and Experience (including psychophysics, phenomenology and AI methods to characterize the structure of experience).

Figure 3

Variations from canonical cat (left) or diatom (right). An illustration of cat and diatom images (AI-generated in Ernst Haeckel’s style) derived from a central archetype (center) via the transformative action of a Lie group in latent space. In such a generative model, any image can be used as an archetype due to the transitivity of the acting group.

Figure 4

Cat robot and generated cat image. A representation of the generative model of a cat derived from a robotic construction using the product of exponentials formula (Equation (3)) in robot kinematics. The robot consists of a set of joints controlled by a Lie group. Left: cat robot using joints to control pose and expression. Right, projection into an image using a camera. (These images are themselves AI-generated).

Figure 5

Image classification task. Classification of cat images can be seen as learning the invariances of a generative model. Such a network can also be implemented using an autoencoder of cat images with a skip connection to detect anomalies.

Figure 6

Group acting on solutions space and the (generalized) Noether’s theorem. The action of a group on ODE trajectories can be used to provide a labeling system for them. A reference solution trajectory $s_e(t)$ (e is the identity element in G) is moved to $s_{\gamma }$ under the action of element $\gamma \in G$. This gives rise to conserved quantity $C(s)$, which is labeled by group elements (functions from phase space to group elements after the choice of some reference solutions—convention). [Figure inspired by the illustration of the action of the Galois group on polynomial equation solutions in Tom Leinster’s Galois Theory (Figure 2.1) [94]].

Figure 8

World-tracking neurodynamics. Tracking the world, represented here by a set of frames of a moving hand, is essentially the task of a compressive autoencoder, which can be described by Equation (11). The top panel displays a feedforward autoencoder, while the bottom provides a more general recurrent (RNN) autoencoder (connections going backward are added to highlight the potential use of predictive coding). Both realizations display an algorithmic information bottleneck (latent space), where input data are mapped to generative model parameter space.

Figure 9

Hierarchical modeling: step-by-step drawing of a cat demonstrating hierarchical constraints: starting with the basic form of a cat (“it’s a cat”), refining it with more specific features, and finally adding details like “has white fur and blue eyes”, progressively narrowing the state space to match more specific characteristics. (Image is partially AI-generated).