Compositional Symmetry as Compression: Lie Pseudogroup Structure in Algorithmic Agents

TL;DR

This work frames algorithmic agents as compressive systems whose structure is governed by compositional symmetry implemented through finite-parameter Lie pseudogroups acting on low-dimensional latent manifolds. By embedding generative models in a Lie-pseudogroup framework and leveraging the Spencer formalism, it derives symmetry-based constraints: equivariant world-tracking dynamics, conserved readouts, and reduced invariant manifolds, all organized into a hierarchical tower of nested quotients. The predictive processing view is recast as residual transformations along unresolved symmetry directions, enabling a bottom-up/upward information flow that preserves structure while progressively refining representations. The approach links deep-model efficiency to symmetry-aware architectures, guides design of equivariant networks, and offers a pathway to symmetry discovery and practical predictive coding implementations, e.g., via a Blender-rig analogy and potential empirical tests on symmetry-aware systems.

Abstract

In the algorithmic (Kolmogorov) view, agents are programs that track and compress sensory streams using generative programs. We propose a framework where the relevant structural prior is simplicity (Solomonoff) understood as \emph{compositional symmetry}: natural streams are well described by (local) actions of finite-parameter Lie pseudogroups on geometrically and topologically complex low-dimensional configuration manifolds (latent spaces). Modeling the agent as a generic neural dynamical system coupled to such streams, we show that accurate world-tracking imposes (i) \emph{structural constraints} -- equivariance of the agent's constitutive equations and readouts -- and (ii) \emph{dynamical constraints}: under static inputs, symmetry induces conserved quantities (Noether-style labels) in the agent dynamics and confines trajectories to reduced invariant manifolds; under slow drift, these manifolds move but remain low-dimensional. This yields a hierarchy of reduced manifolds aligned with the compositional factorization of the pseudogroup, providing a geometric account of the ``blessing of compositionality'' in deep models. We connect these ideas to the Spencer formalism for Lie pseudogroups and formulate a symmetry-based, self-contained version of predictive coding in which higher layers receive only \emph{coarse-grained residual transformations} (prediction-error coordinates) along symmetry directions unresolved at lower layers.

Figures (3)

• Figure 1: The algorithmic agent interacts with the World (structure, symmetry, compositional data) ruffiniAlgorithmicInformationTheory2017ruffiniAITFoundationsStructured2022ruffiniAlgorithmicAgentPerspective2024. The Modeling Engine (compression) runs the current Model (which encodes found structure/symmetry), makes hierarchical/compositional predictions of future coarse-grained (pooled) data, and then evaluates the prediction error in the Comparator (world-tracking constraint monitoring) to hierarchically update the Model. The comparison process is carried hierarchically, and the output is coarse-grained to feed the next level. We reflect this process mathematically as a world-tracking constraint on the dynamics (Equation (\ref{['eq:track']}) Hierarchical modeling engine (Comparator). Level $k$ predicts $\hat{I}_k=\hat{\gamma}_k\!\cdot I_0$, compares to its incoming datum (raw image at $k=0$; canonicalized, coarse‑grained residual $m_{k-1\to k}$ for $k\ge1$), updates from error $E_k$, and forwards only the residual after canonicalization and coarse‑graining, $m_{k\to k+1}=\mathcal{C}_{k\to k+1}(\hat{\gamma}_k^{-1}\!\cdot\space\text{input}-I_0)$. Generators shrink along $H_0\supset H_1\supset H_2 \cdots$; residuals live in quotient directions and carry “what’s left to explain” to coarser scales. The Planning Engine runs counterfactual simulations and selects plans for the next (compositional) actions (agent outputs to world and self). The Updater receives hierarchical prediction errors from the Comparator as inputs to improve the Model. All modules can be implemented hierarchically.
• Figure 2: From generative symmetry to compositional reduced dynamics. Top: a Lie pseudogroup $G$ acts locally on configuration $\mathcal{C}$ and observation space, with Spencer providing a hierarchy of compatibility constraints. Middle: the agent enforces world tracking (Eq. \ref{['eq:track']}); equivariance (Eq. \ref{['eq:equiv']}) imposes structural constraints and, for static inputs, yields conserved labels (Noether‑style), confining trajectories to reduced leaves. Bottom: a flag $G=H_0\supset\cdots\supset H_L$defines the nested invariant manifolds $\{\mathcal{M}_k\}$ (solid bent arrow); residual messages $m_{k\to k+1}$ parameterize quotient directions $\varepsilon_k\in H_{k-1}/H_k$ and induce drift/updates on these leaves (dashed arrow), yielding a hierarchy of compositional reduced dynamical manifolds.
• Figure 3: Blender cat example hierarchy as a Lie‑pseudogroup ladder. Solid arrows (left) show the generative order (coarse $\to$ fine). A right‑hand fine $\to$ coarse error rail aggregates bottom‑up prediction residuals (dashed arrows) with short connectors from each level. Each level $A_k$ is a local group/pseudogroup factor; the flag $H_k:=A_{k+1}\cdots A_7$ induces nested quotients $\mathcal{M}_k$.

Theorems & Definitions (2)

• definition 1: Generative model
• definition 2: Lie generative model