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.

1. Introduction

Understanding the fundamental principles of agency—spanning cognition, planning, valence, and behavior—and the shaping of subjective experience (perceived structure and emotion) remains a central challenge in neuroscience and artificial intelligence (AI), with broad socioethical ramifications beyond homo sapiens [1,2].

In a series of papers [3,4,5,6,7,8,9], we proposed a framework anchored in Algorithmic Information Theory (AIT) to study the phenomenon of structured experience. This term refers to the organized spatial, temporal, and conceptual structure of our subjective experience [10], encompassing both our perception of the world and our self-awareness as agents engaged with it, and serves as a critical bridge between first-person subjective experience and third-person scientific perspectives. Named after Kolmogorov complexity ($\mathcal{K}$), Kolmogorov Theory (KT) posits that a key function of the brain is information compression [3,4]. This ability to compress information is equivalent to having access to a model of the world—unveiling its underlying structure—a critical foundation for cognition and the perception of structured reality [5].

KT is related to and complements existing theories such as Active Inference and Predictive Coding and intersects with themes like the Information Bottleneck, grounded in Shannon Information theory [7,11,12,13]. AIT is the generalization of Shannon information theory pioneered by Kolmogorov, Chaitin, and Solomonoff using computation theory (Turing) [9,14,15]. While Shannon’s entropy provides a probabilistic approximation to algorithmic complexity under noisy conditions [9,15,16], Kolmogorov complexity provides a universal, computation-theoretic lens whose role in defining and implementing model building has been highlighted in related domains, including AI and compression in Nature [16,17,18]. These principles resonate with the longstanding pursuit of simplicity (low Kolmogorov Complexity), emphasized throughout history by thinkers such as Pythagoras, Plato, Aristotle, Epicurus, Occam, Leibniz (with special clarity in his Discours de métaphysique (1696), as pointed out by Chaitin [19], Newton, Hume, Kant, and Einstein [20].

Algorithmic agents are the central players in KT. They are defined as computational constructs using compressive world models, goals, and planning to interact effectively with their environment (Figure 1). Our definition of agent is inspired but not limited to natural agents. It characterizes them as information-processing systems equipped with features essential for evolutionary success: homeostasis (preservation of self) and, ultimately, telehomeostasis (preservation of kin). Examples of agents include animals, plants, and life in general, as well as properly designed AI systems combining a world modeling engine, an objective function, and a planning module [6,7].

Definition 1. 

An algorithmic agent is an information processing system with an objective function (goals) that interacts bidirectionally with the external world, inferring and running compressive models, planning, and acting to maximize its objective function.

Agent elements of relevance to our discussion are the Modeling Engine and the Comparator, which evaluates model prediction errors by comparing model-generated data with sensor data (see Figure 1). The existence of a Modeling Engine in the agent follows from the Regulator Theorem, a key result in the field of cybernetics that states that “Every good regulator of a system must be a model of that system” [21,22]. For a regulator to effectively steer a system toward its goals, it must rely—implicitly or explicitly—on a model that captures the system’s relevant properties. In AIT terms, this requirement manifests as high mutual algorithmic information (MAI) between the agent and its environment, indicating the degree to which they share the underlying algorithmic structure.

KT is both a theory of cognition (behavior, third-person observations) and experience (phenomenology, first-person science), where the qualitative aspects of structured experience ($\mathcal{S}$) are associated with the algorithmic features of agent world models and their associated valence ($\mathcal{V}$) [8]. It hypothesizes that the presence of qualia of structured experience in algorithmic agents emerges from the successful comparison of (compressive) model-generated data with world data [4,6,7] and that the structure of all experience reflects the structure of models. Since successful models reflect structure in world data, KT predicts that the structure of experience ultimately stems from the structure inherent in the underlying generative model of world data. This raises the important challenge of characterizing the structure of world data and its corresponding agent models, with a focus on compositionally and recursion—key ingredients for compressive and efficient representations. But how is this expressed in dynamical terms?

The structure of models, central to KT, can be explored by instantiating the agent as a dynamical system. This approach is well-suited because computation can be understood as a dynamical process [23,24]—Turing machines can be interpreted as discrete dynamical systems characterized by transitions between states evolving over time. From this perspective, models and other components of the agent are programs that are “run”, and model and agent structure map directly onto dynamical structure [7,8], where dynamics refers to the time evolution of variables in the agent program. The connection between algorithmic and dynamical perspectives opens avenues for analytical methodologies. Tools such as bifurcation theory, differential geometry, and topology provide insights into algorithmic agents and the data they produce. These approaches are particularly relevant for exploring the brain state dynamics and attractor landscapes that can deepen our understanding of the mechanisms underlying neuropsychiatric phenomena and neurophenomenology [25], as depicted in Figure 2.

Viewing the brain through the lenses of computation and compression provides valuable insights for both theoretical and empirical research. The computational viewpoint of brain function directly suggests that the brain operates as a dynamical system with unique computational and informational characteristics emerging at the critical points in complex systems [24,27,28,29,30]. Criticality is increasingly recognized as a guiding principle for understanding the brain’s complex behavior [7,31,32,33,34,35,36,37,38,39,40,41,42,43,44] (although some of its features, e.g., power-law neuronal avalanches, may stem from other dynamical mechanisms [45,46,47]). Neural systems are hypothesized to reach such states through evolutionary and developmental mechanisms driven by homeostatic plasticity [25,48,49,50,51].

Regarding the role of compression, we have argued elsewhere that, from the perspective of natural agents, the world—defined as whatever is generating data on the other side of the agent—is, to a significant extent, simple and compressible [6,7]. This assumption, which mirrors the “unreasonable effectiveness of mathematics” [52] and Occam’s razor [15,16], underlies our rationale for the definition of agents as systems seeking to derive models of the world for survival, guided by a prior bias toward simplicity (compressibility, in the algorithmic information-theoretic sense).

AIT provides the foundational definition of compression: Kolmogorov complexity is defined as the length of the shortest program capable of generating a dataset (e.g., a stack of images of a hand [15]). As we discuss here, compression fundamentally relies on the presence of symmetry or invariance [3]. The ability to compress a dataset is equivalent to having a model of the data, which implies the existence of patterns, invariances, or regularities that can be exploited [5]. Intuitively, the length and structure of such a minimal program or model will be closely related to the symmetries within the dataset. A model that successfully compresses data essentially builds on and reflects the underlying symmetries or invariances. In this sense, the model is the invariant entity.

A variety of methodologies exist to characterize the structure of algorithms beyond traditional Kolmogorov-style measures. Classical software engineering metrics (e.g., cyclomatic complexity, Halstead metrics, and Henry–Kafura’s information flow) capture factors like control-flow branching, operator usage, and module coupling [26,53]. In addition to these, one can examine recursion depth and nested compositionality [54], parallel or circuit depth [55], and category-theoretic notions of compositional hierarchy [56]. Furthermore, functional and inductive approaches highlight structures such as iterated function composition [57] and structural recursion [58], offering granular insights into how algorithms decompose data and transform it through layered abstractions.

While structure in algorithms manifests as compositionality and recursion, we expect that the dynamical systems arising from agent computations using these models—as observed in their structure, behavior, or neural activity—inherit and reflect the structure derived from the models. Consequently, by studying the relationship between the structure and symmetries of world data, the computational features of the models being run by the agent (model structure), the structure of neurophenomenological (first-person reports), behavioral, and physiological data (third-person records), can shed light on the hypothesis that an agent has (or at least reports) structured experience in a way that mirrors the structure of the compressive models it uses to track (structured) world data.

As part of this ambitious research program, we focus in this paper on the relationship between the structure of input (World) data and the computational (constitutive, structural) and dynamical features of the successful world-tracking agent. Through this analysis, we aim to provide a new perspective on phenomena such as changes in the complexity of neural data [25,39,40,41,59] or the manifold hypothesis, which holds that natural data (including neuroimaging data) lies on lower-dimensional manifolds in its embedding space [60,61,62]. We include insights from the compositional (hierarchical) nature of world data and neural processing and the associated necessity of coarse-graining.

Since invariance, structure, and symmetry are central to the problem at hand, we are naturally led to group theory and, more specifically, to the theory of continuous groups—Lie groups and their generalization. Next, we provide an informal definition of Lie groups and pseudogroups. Formal versions are provided in the Appendix B.

Definition 2. 

A Lie group can be thought of as a set of symmetries that forms both a group (where we can combine symmetries and invert them) and a smooth manifold (so these operations behave nicely in a “smooth” sense). Concretely, its group multiplication $(g,h)\mapsto gh$ and inverse $g\mapsto g^{-1}$ are smooth maps.

In a Lie pseudogroup, we only require this structure to hold locally on a manifold. In other words, each transformation is only defined on a neighborhood within the manifold, but the usual group properties (identity, composition, inverse) still apply wherever these neighborhoods overlap.

The fundamental principles of symmetry and Lie groups, which have provided the foundation for modern physics, including relativity theory and quantum mechanics [63,64], are now being adopted in machine learning (see, e.g., [65,66,67,68,69]) after pioneering work in the context of convolutional networks [70,71]. This paper is a parallel development in the context of KT and, as we shall see, with applications in both computational neuroscience and machine learning.

In this paper, we provide a unifying framework to define and study structure across three domains: computation (via AIT), dynamics, and subjective experience. Specifically, we present a general model in which agents identify and exploit patterns in world data to build compressive representations, drawing on AIT as a foundation for compositionality and recursive structures [15,16]. We link these AIT principles to invariance and symmetry, introducing a Lie Generative Model that connects these ideas with Lie group theory and pseudogroups [72,73]. By showing how symmetries in the environment constrain the agent’s internal dynamics through a Comparator mechanism, we demonstrate that world-tracking conditions naturally induce reduced manifold structures in the agent’s state space.

The main novel contributions of this work are: (i) establishing a formal correspondence between compression-based theories of representation and Lie-theoretic approaches to invariance; (ii) presenting a unified Lie Generative Model framework that captures how environmental symmetries shape both the agent’s constitutive and dynamical structures; and (iii) illustrating how these links inform compositional and recursive perspectives on cognition and learning.

In the next sections, we start from the mathematical framework of Lie groups to formalize the concept of structure and provide a definition of a generative model. This foundation allows us to explore how the algorithmic simplicity of a model is linked to representations of compositional or hierarchical Lie groups. We then demonstrate that neural networks, such as feedforward networks, inherently acquire structural constraints from the symmetry properties of the data they are trained on. Translating this to the context of recurrent neural networks (RNNs) leads us to discuss the central role of symmetry and conservation laws, which provide the basis for the concept of compatible world-tracking constraints. Building on this, we show that when a dynamical system aims to track the world, it is compelled to mirror the symmetries present in the data. This mirroring results in structural constraints and the emergence of reduced manifolds, which are lower-dimensional spaces that encapsulate the system’s essential features while respecting the imposed constraints. The compositional nature of world data gives rise to the notion of compositional or hierarchical constraints and manifolds. Our discussion is high-level and does not distinguish between natural and artificial neural networks, with implications for neuroscience, AI design, and brain modeling.

2. Generative Models as Lie Groups

Prior work in computer vision has highlighted the role of symmetries in image data [74,75,76], and in particular those associated with the group of rotations, translations, and dilations in n-dimensional Euclidean space. These can be considered as an extension of the Special Euclidean group $SE(n)$ to include scaling transformations and are often referred to as the Similarity group, denoted as $\mathrm{Sim}(n)$ [77]. We recall that groups act on manifolds to effect transformations (see the Appendix A for various definitions).

Artificial neural networks used for classification can be understood as representing invariance with respect to a group of transformations [5]. This group defines a structure under which instances that are similarly classified form equivalence classes, capturing the shared features or invariances among those instances. This perspective offers a principled way to understand the representations learned by the neural network and the invariance properties it might acquire during training, potentially facilitating the analysis and visualization of the learned representations in terms of the group actions or guiding the design of network architectures. We aim to extend these ideas to more general generative models—the starting point in KT.

2.1. Classifying Model-Generated Cat Images

To make the discussion tangible but without loss of generality, we continue with the example of a classifier or autoencoder of cat images. We will work with several mathematical objects: the universe of all images ${\mathbb{R}}^X$ (with X the number of pixels), the subset of cat images (a very large or infinite set), a generative model of this set, and a symmetry group G transforming cat images into cat images.

2.1.1. Generative Models

Our first assumption is that the space of cat images can be produced using a simple, smooth, generative model of cat images mapping points from a (relatively) low-dimensional manifold $\mathcal{C}$ (the compressed, “cat image latent space”) to a larger image space—see Figure 3. This aligns with the view that the notion of “cat” is intrinsically a compressive model, an invariance encoded as a generative model using a few parameters. We formalize this in the following definition.

Definition 3. 

By a generative model of images, we mean a smooth function mapping points in the M-dimensional configuration space manifold to X-dimensional image space, $f:\mathcal{C}\to {\mathbb{R}}^X$ so that $I_c=f(c)\in {\mathbb{R}}^X$ for $c\in \mathcal{C}$, with $M<<X$.

We can locally represent the generative model using coordinates on the manifold, which we refer to as ‘patch’ coordinates, denoted by $\theta \in {\mathbb{R}}^M$, typically with $M<<X$. In this context, $I_{\theta }=f(\theta )\in {\mathbb{R}}^X$ represents the image generated by the model, where M is the number of parameters defining the model, and  X is the number of pixels in the generated image $I_{\theta }$. Parameters are coordinates of cat space manifold $\mathcal{C}$ (locally ${\mathbb{R}}^M$)—cat image configuration space.

A subset of the space of all possible images corresponds to the category of “cats”. Suppose there is a group G that acts on the set of images of cats, or, equivalently, their corresponding parameters in configuration (latent) space and transforms them into other images of cats, formally defined as the maps $G\times \mathcal{C}\to \mathcal{C}$, and $G\times {\mathbb{R}}^X\to {\mathbb{R}}^X$, corresponding to $\gamma ·\theta ={\theta }^{\prime }$ (configuration space) and $\gamma ·I\to I^{\prime }$, with $\gamma \in G$ (image space).

The largest group of transformations that can act on cat images while preserving their structure is the group of automorphisms, which in this case refers to all possible permutations of cat images [5]. Within this large group, there exists a specific subgroup known as the diffeomorphism group, denoted as Diff($\mathcal{C}$). This subgroup is an infinite-dimensional Lie group [78] that consists of smooth (i.e., continuously differentiable) transformations or deformations of the configuration space manifold, mapping cat images to other cat images through the generative model.

To further formalize the notion of a generative model, we start from a reference image of a “cat” and define continuous transformations to other images of cats. A desirable property of a generative model defined this way is that it should not depend on the choice of a reference image—the concept of “cat” should not hinge on a specific canonical cat. To achieve this, we impose a transitivity requirement, meaning that the group of transformations must be able to link any cat image to any other cat image.

Since we can appeal to the infinite-dimensional group Diff($\mathcal{C}$), we know we can always associate a Lie group with the generative model. However, to keep the generative model simple, particularly by ensuring it has a finite number of parameters, we make the stronger assumption that the generative world model of cat images can, at least locally, be associated with a finite Lie group G.

Finite Lie groups capture the essence of continuous symmetries in a compact, finite-dimensional form, making them highly amenable for theoretical study and practical computation. In contrast to the general diffeomorphism group $\mathrm{Diff}(\mathcal{C})$, which is often infinite-dimensional, finite Lie groups provide a structured approach to understanding global behavior from local information near the group identity element. This makes them particularly useful as candidates for simple generative models of world data, as they encapsulate key symmetries and invariances in a tractable manner through a few group generators. They also capture naturally the algorithmic notions of recursion and compositionality.

2.1.2. Recursion and Compositionally in Lie Group Action

To illustrate this, recall that a small transformation generated by a Lie group element $\gamma$ can be described as a perturbation around the identity element as $\gamma =1_G+ϵT,$ where $\gamma$ is a group element representing a small transformation, $1_G$ is the identity element representing no transformation, $ϵ$ is a small parameter representing the “size” of the transformation, and T is an element of the Lie algebra, inducing infinitesimal transformations in the considered space (in our context, the “cat space”). This formulation offers a linear approximation of $\gamma$ in the vicinity of the identity element, utilizing the Lie algebra structure derived from the Lie group.

Any element within the connected component of the identity can be represented as a product of infinitesimally perturbed identity elements, denoted by $\gamma =1+{\sum }_k{ϵ}_k{T}^k$,

$$g=\underset{n\to \infty }{lim}{\left(1+{\displaystyle \frac{1}{n}}\sum _k{\theta }_k{T}^k\right)}^n=exp[\sum _k{\theta }_k{T}^k].$$

(1)

where ${\theta }_k$ are parameters and $T^k$ are the generators of the Lie algebra [72]. If G is a connected, compact matrix Lie group, the exponential map for G is surjective (covers the entire group) [72]. This recursive expression encapsulates the fundamental property of Lie groups that any group element can be decomposed into an (infinite) series of infinitesimal transformations. Compositionality extends this by including the sequential action of different group elements. In algorithmic terms, recursion is a for loop, and composition is a set of nested for loops.

2.1.3. Lie Pseudogroups and Generative Models

However, in a topologically complex latent space, finite Lie groups (globally defined transformations) may be too restrictive as they require a single structure acting globally. A Lie pseudogroup, by contrast, focuses on partial transformations with restricted domains that can be iteratively composed when their domains overlap, providing both local flexibility and global reach. This is particularly suited for “walking” the latent space of a generative model in iterative steps—chaining local diffeomorphisms across patches in a manner that respects the underlying manifold structure. Consequently, a pseudogroup formalism often emerges as the most robust solution to handling high-dimensional, topologically intricate configuration spaces.

This motivates the following definition, closely related to the notion of a group of transformations acting on a manifold (see the Appendix A):

Definition 4. 

An r-parameter generative model $I=f(c)$, $c\in \mathcal{C}$, is a (Lie) generative model if it can be written in the form

$$I=f(c)=f(\gamma ·c_0)=\gamma ·I_0,\gamma \in G$$

(2)

where $c_0$ is an arbitrary reference point, $I_0\in {\mathbb{R}}^X$ is an arbitrary reference image, f is a smooth function, and G is an r-dimensional Lie pseudogroup.

For example, in the case of cat images in a particular manifold patch, we may write $f(\theta )=h({\gamma }_{\theta }·P_0)={\gamma }_{\theta }·I_0$, where h is a projection function (describing the operation of a camera, which, for simplicity, we may assume to be fixed, with a uniform background, etc.), and $P_0$ is an arbitrary reference point-cloud capturing the 3D form of a particular reference cat. In the second equality, $I_0$ is the corresponding reference cat image, and ${\gamma }_{\theta }$ is a representation of the group G on that object. The action of the group element $\gamma$ varies depending on the type of object it is applied to—whether a point cloud, an image, or a parameter—resulting in different representations of the group for the same transformation. For instance, when dealing with point cloud data and rotations, the group’s action can be described by matrices, giving us a linear representation in a vector space. However, linearity of representations may not generally hold—the representations found in natural settings are often non-linear, outside the domain of classical representation theory [73]. Thus, although we refer to the map from the latent (or object) space to sensor measurements (e.g., images) as an “induced representation”, it may generally be a non-linear group action—and, due to the non-injectivity of the sensor projection, it may also be multi-valued.

The function f must be smooth to leverage the recursive and compositional nature of the Lie action, ensuring that the resulting generative model is compressive (e.g., it cannot be a giant look-up table). This connection ties the structure of the Lie generative model to the notion of simplicity as defined by Kolmogorov complexity in algorithmic information theory.

A Lie pseudogroup structure enables the navigation of the configuration space manifold in a succinct manner. By using Lie groups, models can capture the essential characteristics of objects in a mathematically rigorous and computationally efficient manner. This approach facilitates both the understanding of the learned representations in neural networks and the practical implementation of generative models.

The assumption of finite dimensionality is not restrictive when the generative model depends on only one continuous parameter. This applies to processes associated with rotations, translations, and dilations since these correspond to finite Lie groups. It also holds for transformations of shape space, e.g., if the cat image generator source is a robot with any number of rigid joints, which can be described using the special Euclidean group SE$(n)$ [79]. More broadly, while it is not always clear whether images generated by models like generative neural networks or 3D modeling programs (e.g., Blender [80]) possess the group structure of a finite Lie group, generative models exhibiting transitivity, smoothness, and a finite number of parameters may often be described by a finite Lie pseudogroup [76,81,82]. We discuss the limitations of the finite-dimensional assumption in Appendix C.

In summary, data produced by a generative model can always be locally generated by a finite-dimensional Lie group. Some generative models admit global finite-dimensional Lie pseudogroup actions and are, therefore, Lie generative models in our definition. This will hinge on the geometry of the generative model configuration space $\mathcal{C}$. It is nevertheless always possible to locally cast generative models as Lie generative models.

2.2. Implications of Invariance Under Group Action

To illustrate a Lie recursive and compositional generative model and study the implications of invariance, imagine generating cat images by taking pictures of a jointed cat robot (see Figure 4) for training a deep neural network classifier. In this context, invariance means that when we apply transformations, such as rotations or pose changes that do not alter the semantic identity of the object, the output of the model should remain unchanged—i.e., the classifier should still recognize the object as “cat”. The robot’s varying state is expressed through generative compositionality and is described by the Product of Exponentials formula from robot kinematics [83],

$$T=\prod _{n\in \mathrm{parents}}{e}^{\left[{\mathbf{S}}_n\right]{\theta }_n}M$$

(3)

where $T \in$ $SE$(3) is the final position and orientation of the end effector (e.g., a particular cat claw) in the special Euclidean group, ${\theta }_n$ is the vector of joint variables for the n-th joint, $[{\mathbf{S}}_n]$ is the skew-symmetric matrix representing the screw axis of the n-th joint, $e^{\left[{\mathbf{S}}_n\right]{\theta }_n}$ is the matrix exponential representing the action of the n-th joint’s transformation, and M is the initial (home) configuration of the end effector.

Here, SE(3) is realized as a matrix Lie group, so the Lie algebra $\mathfrak{se}$(3) becomes a finite-dimensional vector space of $4\times 4$ matrices (with a specific block structure). Accordingly, each joint’s screw axis $[{\mathbf{S}}_n]$ lies in $\mathfrak{se}(3)$, and the map $e^{[{\mathbf{S}}_n]{\theta }_n}$ is the usual matrix exponential from the Lie algebra to the group. In this sense, we move from the manifold description of SE(3) to its vector-space representation (the Lie algebra) for the sake of computational convenience.

In this sequential composition, the order of joint transformations is crucial and reflects the non-commutative nature of the Lie group $SE(3)$. Each joint’s transformation is applied in a fixed sequence, ensuring that the resulting configuration accurately represents the compounded effect of all joint movements. This ordered product does not assume that the individual transformations commute; instead, it leverages the inherent structure of the Lie group to maintain the correct spatial relationships between joints.

To represent the full state of the robot, a similar expression is applied to each joint and, by extension, to the entire robot. Each joint corresponds to a representation of a Lie group, and the calculations—including those involving the generators—are compositional. Leveraging this Lie-group compositional structure, we can generate an arbitrarily large yet highly compressible dataset of robot pose images [5].

This principle may extend to other generative models of the world, such as more realistic images of cats. Fundamental physical laws often exhibit symmetry principles, such as covariance in general relativity or gauge invariance in quantum field theory. While it is not immediately clear how these symmetries propagate after the coarse-graining of phase space by an agent’s sensory system, it is plausible that they persist in some form. Alternatively, natural selection may favor agents that coarse-grain the world in a way that allows them to exploit compositional representations [9].

Our network classifier aims to output “1” if an image contains a cat and “0” otherwise. While the universal approximation theorem [84] guarantees that a sufficiently large network can approximate this classification function, it does not address how efficiently such a network can learn. Neural networks achieve this task efficiently, as shown in [85,86], if (1) the data are compositional and (2) the architecture leverages that compositionality through a sufficiently deep structure. The former hinges on the presence of a succinct generative model—akin to having a “short program” (recursive, compositional) in algorithmic information theory. The latter leads to the design of deep or truly recursive architectures. The same reasoning applies to compressive autoencoders, where exploiting compositionality enables learning compact latent representations.

The equations for a general feedforward neural network are

$$x^{(l+1)}=h\left(W^{(l)}{x}^{(l)}+b^{(l)}\right),l=0,1,\dots ,L-1$$

(4)

where $x^{(l)}$ is the vector of neuron activations at layer l, $W^{(l)}$ and $b^{(l)}$ is the weight matrix and bias vector at layer l, respectively, $h()$ is the activation function applied element-wise, and the input data are fed as $x^{(0)}$. The equations can be compactly expressed as $x=h(Wx+b),$ where x is a vector encompassing the activations of all nodes in the network, structured in a way that respects the layered organization of the network, and W is a block matrix describing the connections between the nodes, including the weights associated with each connection.

We require the classifier to be invariant under the action of the group G, whether it is finite or an infinite Lie group. Specifically, if the network classifies a point $\theta \in \mathcal{C}$ (via the image $I_{\theta }$ generated using parameters $\theta$) as a “cat”, it should also classify the point $\gamma ·\theta$ as a “cat” for any $\gamma \in G$. If the group is a finite Lie group, it is sufficient to study the finite number of generators of the group—linear operators in the group’s Lie algebra. An important property of Lie groups is that any element can be expressed in terms of actions near the identity element, although this applies only to the part of the group connected with the identity. The composition of group actions from elements near the identity covers the connected part. The recursive nature of Lie groups ensures that to establish invariance or equivariance of a function under group action, it is sufficient to verify this property for the generators $T_k$. The behavior under infinitesimal transformations near the identity informs the behavior under the full group action. However, for disconnected Lie groups, invariance must be confirmed for each component separately.

A change in the input image from a class-preserving infinitesimal transformation will propagate through the network and affect the state x. The transformation can be described by $ϵ$ where $x\to \gamma ·x=x+\delta x$, with $\delta x=ϵTx$, where $\delta x$ is a small change in the state vector x, T is a linear transformation generator associated with the group action, and $ϵ$ is a small parameter. This induces the transformation

$$x^{\prime }=h(W(x+\delta x)+b)=h(W(x+ϵTx)+b),$$

(5)

and we seek conditions under which the output of the network is invariant to first order in $ϵ$. (See Figure 5).

Using the chain rule, we can express the transformation matrix at a given layer l in terms of the transformation matrix at the input layer and the Jacobian matrices of the transformations at all preceding layers,

$$T_l=J_{l-1}{W}_{l-1}·J_{l-2}{W}_{l-2}·\dots ·J_1{W}_1·T_0$$

(6)

where $J_i$ denotes the Jacobian matrix of the transformations at layer i, encompassing the activation functions, and $T_0$ is the transformation matrix at the input layer, the linear approximation to $I_{\theta }={\gamma }_{\theta }·I_0\approx h((1+ϵT_0){\theta }_0)$.

A necessary (but not sufficient) condition for the invariance of the neural network output with respect to the group action is that there exists at least one layer l for which the condition $J_l·W_l·T_l{x}_l=0$ holds for all cat image inputs (see [76] for similar reasoning on the conditions imposed by symmetry). This ensures that the transformed state vector $x_l+T_l{x}_l$ remains in the null space of the weight matrix $W_l$ at layer l, guaranteeing the invariance of the state of this layer with respect to the transformations generated by the group action. In other words, the space of cat images must be in the kernel of the operator $T_l$ for some layer l. This property propagates to higher layers, ensuring the network’s output remains invariant to these transformations.

For the feedforward autoencoder, the analogous equivariance requirement is that the transformation of the output matches that of the input, namely,

$$T_N=J_{L-1}{W}_{L-1}·J_{L-2}{W}_{L-2}·\dots ·J_1{W}_1·T_0=T_0$$

(7)

when acting on cat images. This sets an M-dimensional family of constraints on W requiring that the matrix $T_N=J_{L-1}{W}_{L-1}·J_{L-2}{W}_{L-2}·\dots ·J_1{W}_1$ has the space of cat images as an eigenspace corresponding to the unit eigenvalue. These are the necessary requirements for invariance or equivariance under transformations.

Systems specifically designed for invariant behavior, such as convolutional networks with pooling layers, explicitly implement some of the required symmetries. Generalizing this idea involves designing networks that “filter” data across group orbits (e.g., such as translations or rotations), then analyze the results (e.g., computing the maximum or the average) [66], or using data augmentation techniques, or finding fundamental architectural principles for group invariance [87]. The success of hierarchical, recursive, deep architectures reflects this principle [60,65].

In summary, imposing invariance of the classification of data produced by a generative model leads to equations that constrain the weight parameter matrix under transformations associated with the generative model. This approach is valuable for designing neural networks—where the weight equations can be enforced in the loss function during training—provided the generators are known. Conversely, these principles can be useful for inferring symmetries underlying world data from the structure of networks trained on it.

3. The Agent as a Dynamical System

Although feedforward networks can be viewed as dynamical systems, their computational capacity is constrained by a fixed number of sequential evaluation steps. A broader perspective, bridging both algorithmic and biological paradigms, arises from the use of ordinary differential equations (ODEs), which have the potential to instantiate universal Turing machines [88,89,90]. Differential equations can be interpreted as recursive neural networks (RNNs), modeling continuous dynamics similarly to recursive processes in NNs. This analogy becomes particularly clear when the ODE equations are discretized.

While the algorithmic perspective defines the agent as a program or Turing machine exchanging information with the environment, the dynamical view describes it less abstractly (but equivalently) and in a manner that makes more clear its connections to physics and biology—e.g., as a human brain or a neural network running on hardware. As a dynamical system, the agent is characterized by state variables and rules governing state transitions as a function of an external parameter denominated as physical time.

We define some terminology used in what follows. The phase space in dynamical systems theory is a multidimensional manifold (see Appendix A for mathematical details) where each point specifies the state of the system (the necessary and sufficient information to project the state in time). Finally, by the geometry of the phase space, we refer loosely to the structure and properties of the subspace of dynamical trajectories in space, including the presence of any special points or areas, such as attractors or repellors.

3.1. General Model

A general model of the state of neurons or other elements of a dynamical system receiving external inputs can be written in the form of a multidimensional ordinary differential equation (ODE),

$$\dot{x}=f\left(x;w,\eta (t)\right)$$

(8)

with $x\in {\mathbb{R}}^X$ is a highly dimensional state variable representing the activity of neurons or neuron populations (e.g., membrane potential or firing rate) and where w stands for connectivity parameters (and maybe others). For example, in a model of the human brain, each coordinate may represent the average firing rate of populations of neurons (a neural mass model) or a more detailed ternary code of silences, spikes, and bursts in each cell [91]. This finite-dimensional equation can be generalized to a partial differential equation (a neural field equation), but these details are not essential in what follows.

The input term $\eta (t)$ reflects inputs from the world represented as an external forcing independent of the agent’s state that makes the equations non-autonomous (i.e., explicitly time-dependent). We can imagine it representing random kicks to the trajectory or a more steady, purposeful forcing received by agent sensors (we provide a generalization of this model to include inputs and outputs reflecting interaction with World dynamics in Appendix A). External inputs—which, from the perspective of a Universal Turing machine, can be viewed as reprogramming the agent—can force changes in the state trajectory or even drive the system’s landscape through dramatic topological transitions [25,92]. For instance, such changes in topology can be induced by time-varying parameters associated with bifurcations in the system.

We call Equation (8) the “fast time” dynamics equation [25], as it governs dynamics at short time scales (seconds), where connectivity parameters w are assumed to be constant (in reality, w may be dynamic at slower time scales, reflecting plasticity phenomena). This equation fixes a dynamical landscape, the geometry of trajectories in phase space defined by the set of ODEs. More precisely, the landscape is the manifold defined by the motion of trajectories in the phase space manifold (with local coordinates $x\in {\mathbb{R}}^n$). The dynamical landscape is determined by the form of the function $f()$ and parameters w.

While we focus on deterministic ODEs for clarity in this work, future research will explore the implications of explicitly modeling $\eta (t)$ as a stochastic process. The external input $\eta (t)$ in our model can be extended to stochastic processes, leading to a stochastic differential equation (SDE) framework. For instance, $\eta (t)$ could follow an Ornstein–Uhlenbeck process, introducing biologically realistic noise into the dynamics. This extension allows the exploration of noise-induced phenomena, such as stochastic bifurcations, attractor transitions, and resonance, adding richness to the system’s behavior.

3.2. Conserved Quantities and Symmetries

Let us focus now on the case where $\eta (t)$ is constant or very slowly varying compared to the other dynamics. In the context of dynamical systems, symmetry is a transformation that maps a solution of the ODEs into another or, equivalently, maps the set of solutions into itself. More precisely, by the symmetry of an ODE, we mean that there is a group G such that for all elements $\gamma \in G$, if x is a solution to the equations, then $\gamma ·x$, a transformation of x specified by the group element, is also a solution to the ODE.

Since the only indeterminacies in the solutions to the ODEs are the initial conditions, we can also view symmetries as the action of a group on the initial conditions manifold (integration constants). Symmetries partition the solution space (or initial condition manifold) into orbits under the group action. If the group acts transitively on that space, then, indeed, one can map any solution to any other. Otherwise, solutions lie in distinct orbits, and only solutions within the same orbit can be mapped into each other. Identifying symmetries in a system simplifies its analysis by organizing solutions into equivalence classes under the symmetry group. Because solutions within each class can be mapped into one another by the group action, it suffices to study one representative solution per class, with the group element labeling how solutions transform within that class.

Symmetries can be categorized as discrete or continuous, and in the latter case they are described by local Lie groups. Paralleling the implications of Noether’s theorem [63,64,93], the existence of symmetries can help in analyzing and solving ODEs. For example, the existence of a continuous symmetry means that we can locally define a coordinate system in which some of the coordinates do not appear in the equations other than in the derivatives (they are cyclic) [93]. Furthermore, there is a deep connection between symmetries and conservation quantities, as we now describe. (See Figure 6).

A conservation law is a map from solutions to a manifold. For example, energy conservation is a map from solutions to Newton’s equations to real numbers. Momentum conservation is a map from solutions to ${\mathbb{R}}^3$.

The relationship between symmetry and conservation laws can be understood as follows. If a system is invariant under the action of a group G, then for any solution $x(t)$, applying a group element $\gamma \in G$ yields another solution, $\gamma ·x(t)$. Consequently, one can label every solution by the group element needed to map a reference solution (corresponding to the identity element $e\in G$) to it.

Under standard uniqueness conditions (guaranteed by the Picard–Lindelöf/Cauchy– Lipschitz Theorem [95]), trajectories in phase space do not intersect—apart from the trivial case of time-translation symmetry in autonomous ODEs. Hence, each point on a trajectory can be uniquely associated with a specific group element (disregarding temporal translations in autonomous systems). This induces a mapping from phase space to the group that remains constant along each trajectory. That constant constitutes the conserved quantity associated with the system’s symmetry. This construction parallels Noether’s statement: continuous symmetries imply conserved quantities.

Thus, we can think of this process as the act of creating a structured labeling system for the solution space and then the phase space, using the group. The group labels, as the output of functions from phase space to group elements, are the conserved quantities.

To illustrate this, consider the equation $\dot{x}=0$ with $x\in \mathbb{R}$. It displays translational symmetry: if $x(t)$ is a solution, then $x(t)\to \tilde{x}(t)=x(t)+\delta x$ is also a solution. The group’s action is to translate solutions along the x-axis (phase space is 1D in this example). The conserved quantity is $C(x)=x=x_0$, which is shifted by the action of the group, $C\to C+\delta x$. We provide more simple examples of symmetry transformations that shift associated conserved quantities in Appendix F.

Finally, a well-known example in classical mechanics is the Kepler problem (two-body problem in celestial mechanics), an autonomous ODE in six-dimensional phase space, where a (bound) solution may be classified using conserved quantities corresponding to symmetries: the shape of an ellipse (eccentricity), its size (the energy or total angular momentum, one parameter), and the orientation of the plane containing it (three parameters). The latter results from the rotational symmetry of the problem and allows for the classification of the solutions with the orientation of the plane (the angular momentum). We can think of the specification of a solution in a constructive way: specify first the energy (one parameter) and eccentricity (one parameter) of an ellipse with its major axis along the x-axis and a minor axis in the y-axis. Then, rotate this plane in 3D (three parameters from the SO(3) or, equivalently, the SU(2) group). The existence of these five conserved parameters in the Kepler problem or in other similar central potential problems stems from the symmetries of the equations (the O(4) group) [96].

More generally, N-dimensional ODEs have $N-1$ conserved quantities (essentially functions of the constants of the motion; see Appendix F). An ODE with constraints (an algebraic differential equation) has solutions if the constraints are functions of the constants of the motion (and hence constant). The conserved quantities can be used to create, at least locally, new coordinates, which are then constant and act as (group) labels of solutions. The symmetry of the equation is associated with transformations that shift these constants.

4. The World-Tracking Condition

Because of the time-dependent input term, it appears difficult to say very much about the behavior of the fast-time Equation (8). However, there are two important elements of the model that we can leverage. Namely, we assume that:

• Agent inputs are generated by simple rules (the world is inherently simple), i.e., by a hierarchical generative model as discussed above.
• The agent is able to “track” its inputs, i.e., use an internal generative model to approximately match them.

We now analyze the mathematical consequences of these statements.

First, the assumption that world data are the product of a simple generative model—as discussed in the previous sections—implies that the constrained dynamics of the agent can be studied from the point of view of symmetry. We may think of the external world as providing stochastic but structured input to the dynamic system.

Second, one of the objectives of the agent is to track world data through the implementation of the world model in its modeling engine, which is to be compared with the data (this task is carried out by the Comparator in the KT agent model; see Figure 1).

To make this comparison meaningful, there must be a readout from the agent’s internal state that matches the tracked world data. This necessity imposes a dynamical constraint on the system: the agent’s internal states must evolve so that their readouts correspond to the external inputs. This constraint ensures that the agent’s internal dynamics are aligned with the environment, enabling accurate tracking and adaptation.

As we will see, the implications of world-tracking are, in fact, twofold. On the one hand, the dynamics of the system are constrained to lie in reduced manifolds. On the other, the architecture of the equations implementing world tracking needs to satisfy some requirements, as we already saw above in the simpler case of a feedforward classifier.

4.1. Constrained Dynamics I

We begin with a simple toy model to study the role of symmetry and conservation laws. Consider the constrained system of differential equations (or differential algebraic equation [97]),

$$\begin{array}{ccc}\hfill \dot{x}& =& 0\hfill \\ \hfill \dot{y}& =& 1\hfill \\ \hfill x& =& C\hfill \end{array}$$

(9)

The first two equations are regular ODEs, and the last is a constraint (C is an arbitrary parameter), a toy world-tracking condition. These equations happen to be solvable for any value of C, because the general solution to the ODE subsystem is

$$\begin{array}{ccc}\hfill x& =& A\hfill \\ \hfill y& =& t+B\hfill \end{array}$$

(10)

Thus, it is possible to select the integration constants A and B that satisfy the constraint. This occurs because the constraint is not independent of the conserved quantities of the system. The same conclusion follows if the third equation in (9) is replaced by $f(x)=C$, for any function f such that C is in its range.

Another way to view the compatibility of the equations is that the ODE subsystem displays translational symmetry in x, with the symmetry $x\to x+ϵ$. This means that if x is a solution to the differential equations, then so is $x+c$, for any constant c. Hence, we can shift the trajectory x by any value we wish and still obtain a valid solution to the ODE.

Conserved quantities in an ODE, such as $x=C$ above, can be used to create new coordinates with zero derivatives, which manifest the symmetry, just like x in our example. The symmetry group shifts solutions across the conservation surface, i.e., it shifts the constant associated with the conserved law. Further examples are provided in Appendix F.

4.2. Constrained Dynamics II

Consider next an agent watching a moving human hand generated by some simple physics-based computational model, as we described in [7]—see Figure 7.

We represent the retina image by $I=I_{\theta }\in {\mathbb{R}}^Y$, with Y the number of “pixels” in the retina (e.g., $Y={10}^6$, the approximate number of fibers in the optic nerve [98]) and $\theta \in {\mathbb{R}}^M$ a small set of parameters reflecting the state of about $M={10}^2$ ligaments and tendons (as may be generated by a realistic animation [99], for example). Although the space of all possible images has a large dimensionality Y, the space of all hand-generated images has a much smaller dimensionality $M<<Y$. More precisely, the points in image space corresponding to hand images lie in an M-dimensional manifold embedded into a Y-dimensional box.

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 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).

The agent is equipped with a brain consisting of a large network of neurons, and the brain state specified by $x\in {\mathbb{R}}^X$ (X is very large, there are about $1.6\times {10}^{10}$ neurons and about 1.5 $\times {10}^{14}$ synapses in the human cerebral cortex [100,101]). The dimensions of the brain, pixel, and model space are drastically different. There are many more neurons (or synapses) than data points (pixels) and very few model parameters (hand shape and orientation parameters), $M<<Y<<X$.

The generative nature of images can be understood as the action of a continuous group $\mathcal{G}$ on the parameter set of the hand model and, therefore, of the images, $\gamma ·I_{\theta }=I_{\gamma ·\theta }$. This equation states that the action of the group on image space can be understood as its action on hand parameter space. The set of all possible hand images (the hand manifold) is invariant under the action of $\mathcal{G}$. This set is also equivalent to the semantic notion or the algorithmic pattern of “hand”.

We return to Equation (8), which becomes $\dot{x}=f\left(x;w,I_{\theta (t)}\right)$, where we have expressed the model inputs as $\eta (t)=I_{\theta (t)}$ (the image may be preprocessed in some way before its input to the network—e.g., the agent may be tracking only some features of the image, not trying to match every pixel in the image —but this is not important here).

Unless $\theta$ is fixed, our equation is non-autonomous. Even so, the range of possible inputs is now greatly constrained by their simple generative nature. This facilitates the agent’s task, which is modeling and (approximately) matching these inputs. We represent this by the world-tracking constraint: there is a projection from neural states to inputs—the neural state variable x carries a representation of the inputs via some projection operator (a variable selector or projector in practice) $p(x)$. Putting this together with the neural dynamics equation, we have the world-tracking neurodynamics equations (WTNE)

$$\begin{array}{ccc}\hfill \dot{x}& =& f\left(x;w,I_{\theta (t)}\right)\hfill \\ \hfill p(x)& \approx & I_{\theta (t)}\hfill \end{array}$$

(11)

with the inputs $I_{\theta (t)}$ produced by the generative model and some parameter dynamics (see Figure 8).

This a differential algebraic equation (it consists both of differential equations and algebraic constraints). It will not have a solution for an arbitrary set of fixed weights w: once the input is fixed, the system dynamics and, therefore, $p(x)$ are determined, which will not, in general, match the inputs.

In fact, even in the case of a static input, the only way to satisfy the constraint is if the constraint is a conserved quantity in disguise,

$${\displaystyle \frac{d}{dt}}(p(x)-I_{\theta })=0$$

(12)

We may relax the problem by asking for approximate tracking, as indicated by the “≈” symbol in the second equation. This may be implemented by replacing the constraint with an equation of the form $||I_{\theta (t)}-p(x){||}^2<ϵ$. We may also relax the problem by asking for approximate tracking after transient behavior (the constraint is then seen as a Lyapunov function of the ODE; see Section 4.4).

Another way to view Equations (11) is as an autoencoder of the input function, i.e., with the input $I_{\theta }(t)$ approximately equal to the output $p(x)$, as outlined in Figure 8. The network implementation is recurrent and reminiscent of architectures in Reservoir Computing [102,103,104], but it can also represent a feedforward system such as a variational autoencoder [105]—or something in between, as in Figure 8. In this machine-learning context, we rewrite the WTNE autoencoder equation as

$$\begin{array}{ccc}\hfill \dot{x}& =& f\left(x;w,I_{\theta (t)}\right)\hfill \\ \hfill w& =& \underset{w}{\arg \min }\int dt||p(x(t;w))-I_{\theta (t)}{||}^2\hfill \end{array}$$

(13)

where the dataset used for training is the generated image space discussed above.

In what follows, we assume the problem has been solved, with the w parameters adjusted to provide such a solution (through some training process). We will also work with Equation (11) rather than Equation (13) for the sake of simplicity, although we should keep in mind that, in a real scenario, only approximate solutions will be possible.

To represent explicitly the role of Comparator, we can provide an alternative formulation in dynamical systems’ terms with more details on the world-tracking equations, where the Comparator computes the error between the model and the data (see Figure 1) and feeds it back to adjust the dynamics so that the constraint is met after a transient period. We describe the dynamics and the approximate constraint in the general WTNE (Equation (11)) using the notion of the Lyapunov function. Lyapunov stability is a method for analyzing the behavior of dynamical systems near an equilibrium. It uses a Lyapunov function, which acts like an “energy” measure. If this function is non-increasing over time, the system is stable: small disturbances won’t cause it to diverge far from equilibrium. When the Lyapunov function strictly decreases, the system converges to a stable point or a minimum. However, if the function remains constant along certain trajectories, the system may not settle at a point but instead exhibit periodic or orbiting behavior. With this at hand, we define the dynamic world-tracking problem. World tracking problem formalization: Agent dynamics (i.e., the equation structure, weights, etc.) need to satisfy the equivariant equations

$$\begin{array}{ccc}\hfill \dot{x}& =& f(x;w,I_{\theta (t)})\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \dot{V}& \le & 0\hfill \end{array}$$

(15)

where $V(t)$ is a Lyapunov function capturing the role of the Comparator,

$$\begin{array}{ccc}\hfill V& =& {\displaystyle \frac{1}{2}}{E}^{T}E\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill E(t)& =& p(x)-I_{\theta (t)}\hfill \end{array}$$

(17)

Here x represents the state variables of the agent system, w denotes the system parameters or weights, $I_{\theta (t)}$ is the Lie-generated time-varying input from the environment, parameterized by $\theta (t)$, $p(x)$ is the model’s output (readout) based on the current state x, and $E(t)$ is the error computed by the Comparator. The goal is to achieve world-tracking after transient behavior. As usual, these equations are to be equivariant under the action of the Lie group. The Lie group control structure of time-varying input plays a key role in making the problem feasible by setting constraints on the equation parameters, including the weights w.

4.3. The Fixed Input Problem

The projection operator g in Equation (11) restricts the dynamics of the state x through dim(I) = Y time-dependent constraint equations. The conditions set by the equations must be met with fixed weights w no matter what the world is doing as long as it is under the control of the generative process. In particular, if the inputs are constant (the hand is not moving or moving very slowly), we have, for any $\theta$,

$$p(x)=I_{\theta }$$

(18)

This implies that the dynamics are characterized by a large set of conserved quantities $p_n(x)=C_n$, $n=1,\dots ,Y$ (which may not all be independent, of course). Since p is a variable selector, this means that some of the x elements will be constant (the network readout layer). The trajectories in phase space will, therefore, lie in a reduced manifold defined by the coordinates $p(x)=C$.

The existence of conserved quantities (constraints) implies the ODE has symmetry associated with them (see Appendix F). This implies the parameters w will satisfy some relations. Another way of seeing this is that the only possible way of having a solution to Equation (11) is if the world-tracking constraint is compatible with the conserved quantities in the ODE—not an independent new constraint, which would force the dynamics to be a single point. Expressed in (local) canonical coordinates, the world-tracking constraint must be a function of the cyclic (conserved) canonical variables. The world tracking condition requires that the system encode some conservation laws, which translate into local canonical variables where the variables corresponding to the conserved quantities are constant, i.e., ${\dot{Q}}_n=0$. This subspace constitutes a center manifold with zero eigenvalues. This means that the generator of world inputs must be part of the symmetry group of the ODE system. Finally, as we saw, approximate symmetries can be achieved in steady state.

Finally, Equation (18) has to hold for any value of $\theta$. Since this constraint is associated with the dynamics, $\dot{x}=f(x;w,I_{\theta })$, this imposes conditions on the w structure of the equations, as we discuss next.

Covariance of the Equations

To analyze the general case where the network is capable of tracking any fixed image generated by the group, we now split the variables x into two blocks, $x=(z,u)$, where u are the variables involved in the world-tracking constraint (see Figure 7). With some abuse of notation, since $u=I_{\theta }$ is constant ($\dot{u}=0$), we can write the generic fixed image world-tracking equations without explicit reference to u as

$$\begin{array}{ccc}\hfill \dot{z}& =& f_z\left(z;w,I_{\theta }\right)\hfill \\ \hfill I_{\theta }& \approx & u=\sigma (w_{u}z)\hfill \end{array}$$

(19)

where for simplicity, we assume a feedforward read-out layer with an element nonlinearity, with $p(x)=\sigma (w_{u}z)=u$ (recall that p is a variable selector operator selecting u, a small subset of ~${10}^6$ x variables). We use the approximate symbol in the second equation to allow for transients and approximate matching.

We assume the system is symmetric, i.e., we assume that, given the input $I_{\theta }$, z is a solution, then so is $\gamma ·z$ with input $\gamma ·I_{\theta }$, that is,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}(\gamma ·z)& =& f_z\left(\gamma ·z;w,\gamma ·I_{\theta }\right)\hfill \\ \hfill \gamma ·I_0& \approx & \sigma (w_u\gamma ·z)\hfill \end{array}$$

(20)

This is simply a consequence of the input tracking constraints and the Lie-generative nature of the world model. The agent is able, by definition, to track any version of the hand images (recognizing them as “hands”).

The action of the Lie group on z may be nonlinear, but the condition applies to an arbitrary infinitesimal transformation $\gamma =1+{ϵ}^k{T}_k(z)$, i.e., in the linear regime (with T a linear operator; see [69] for a more formal mathematical discussion). These operators commute with time differentiation ($T_k(z)$ act on configuration space), hence

$$\begin{array}{ccc}\hfill \gamma ·f_z\left(z;w,I_{\theta }\right)& =& f_z\left(\gamma ·z;w,\gamma ·I_{\theta }\right)\hfill \\ \hfill \gamma ·\sigma (w_u\gamma ·z)& \approx & \sigma (w_u\gamma ·z)\hfill \end{array}$$

(21)

for an infinitesimal transformation. Expanding $\gamma =1+{ϵ}^k{T}_k(z)$, we can write a pair of such equations for each generator $T^k$ (linear operator) since $ϵ$ is arbitrary and the generators linearly independent.

Covariance under the Lie group transformations, Equation (21), imposes specific conditions on the weights w. These conditions ensure that the dynamics $\dot{z}=f_z(z;w,I_{\theta })$ are consistent with the symmetry properties of the system and the desired conservation laws (world tracking constraints).

As a consequence of the world tracking constraints associated with inputs from a generative model, we see that the number of conservation laws or constraints is proportional to the number of group generators. This is because each generator of the Lie group introduces constraints through its action on the output variables u. A group with a large number of generators will impose tighter constraints (and more conservation laws) on the possible w space.

4.4. Tracking Time-Varying Inputs

Here, we extend the analysis to the situation where the inputs change in time but at a long time scale compared to neural dynamics to allow for transient phenomena to complete. As the input image moves slowly through changes in the M-dimensional parameters $\theta$ in the generative model, the reduced manifold of trajectories will also shift. Through this process, the manifold will increase its dimensionality, but it can only acquire a maximum of M extra dimensions. Because the world model is assumed to be simple, i.e., with M small, even in the case where the parameters $\theta$ are dynamic, the invariant manifold of the tracking system will remain dimensionally constrained.

The fact that images are generated by a simple model using a few parameters compared to the dimensionality of image space is crucial. In the example above, we analyzed the case where the image does not vary in time, i.e., a static input. If, on the contrary, the image is fully dynamic (each pixel varying in time independently of the others), there would be no reduction of the dimension of the trajectory space. There would be as many constraints as pixels, but they would be time-dependent, and possibly, no satisfactory weights w would be found in such a case. It is the intrinsic low-dimensionality of data generation that allows almost all the constraint equations to independently contribute conserved quantities: the image $I_{\theta }$ lies in a small dimensional manifold, and the operator g must project x into it.

The world-tracking equations are expected to hold in an approximate way and only after a transient time. As in the simple example in Appendix F.3, it may be the case that the world-tracking constraints are realized as attractors in the dynamics. In the case of a dynamics input generated by dynamical parameters in the generative model, we can expect to see an approximate match of the output to the input only after a transient period of time.

So far, we have not discussed how the agent system can track changing inputs. We now provide a potential solution to the problem in the form of a specific structure for Equation (14), assuming that only the error is used by the network using proportional feedback [106],

$$\begin{array}{cc}\hfill \dot{x}& =h\left(x;w\right)+\widehat{K}E(t)\hfill \end{array}$$

(22)

where $\widehat{K}$ is a linear feedback gain operator that determines the influence of the error on the system’s dynamics.

The time derivative of V is $\dot{V}=E^T\dot{E}$. Substituting the dynamics of $\dot{E}$ from Equation (22):

$$\dot{E}={\displaystyle \frac{\partial g}{\partial x}}\left[h\left(x;w\right)+\widehat{K}E(t)\right]-{\displaystyle \frac{d{I}_{\theta (t)}}{dt}}$$

(23)

To ensure $\dot{V}\le 0$, the feedback gain $\widehat{K}$ must be designed such that

$$E^T\left({\displaystyle \frac{\partial g}{\partial x}}\left[h\left(x;w,I_{\theta (t)}\right)+\widehat{K}E(t)\right]-{\displaystyle \frac{d{I}_{\theta (t)}}{dt}}\right)\le 0$$

(24)

Proper selection of $\widehat{K}$ will ensure that $V(E)$ is non-increasing, thereby driving $E(t)$ towards zero asymptotically. The choice of this operator will reflect the Lie group structure of the system since these equations need to be equivariant. We leave this analysis for further work.

Finally, these equations correspond to the fast-time equations in Ruffini et al., 2024 [25]. The Comparator error data stream $E(t)$ will feed the slow-time connectodynamics equations to govern the slow change of the structural parameters w, $\dot{w}=g(w,x,E(t))$, with $g(w,x,0)=0$.

In summary, this section extends the previous analysis of static inputs to the case of slowly varying signals mediated by the Lie group action, showing that even when the environment undergoes incremental changes, the low-dimensional structure of the generative model and equivariance can provide a tractable path for world-tracking with an appropriate selection of weights and gain operator. Thus, the same principles that apply to static generative scenarios—namely, leveraging low-dimensional structure and compositional transformations—remain central for handling time-varying inputs. This completes the foundations laid in earlier sections, formalizing how compositional and equivariant mechanisms enable agents to track and compress real-world complexity in both static and slowly evolving contexts.

4.5. Coarse-Graining, Hierarchical Constraints, and Manifolds

An important idea in the algorithmic agent framework is that all cognitive processes—modeling, planning, and objective function evaluation—are inherently hierarchical (compositional) [7]. This means decision-making and goal-setting occur in layers, ranging from simple actions to complex behaviors. This hierarchical organization naturally finds its origin in the multi-scale structure of real-world data, enabling effective modeling and decision-making. To ensure survival with limited resources, the agent must employ coarse-graining to transform complex, uncompressible data into a simpler, compressed form while retaining a non-trivial structure [9]. Techniques like spatiotemporal averaging, spectral decomposition, compressive sensing, and dimensionality reduction achieve this by reorganizing data at various levels of abstraction. This perspective aligns with the concept of the brain processing information through hierarchical coarse-graining, as supported by research in the visual [107,108] and auditory systems [109]. A similar approach is utilized by deep neural network architectures, including convolutional neural networks [65,70].

At each processing level, details are aggregated and abstracted to form higher-level representations. World-tracking at different levels corresponds to different scales of coarse-graining, progressively reducing the complexity of the system while preserving essential features. Since lower-level constraints (e.g., “it’s furry”) must be compatible with higher-level constraints (“it’s a cat”), there is a natural nested structure of constraints and corresponding manifolds (which also induces an inclusion relation among their tangent spaces). World tracking can be achieved through a hierarchy of constraints ${\left\{{\mathcal{C}}_i\right\}}_{i=1}^k$, each corresponding to a specific coarse-graining level. The constraints are nested and must be compatible, meaning each lower-level constraint operates within the subset of the state space defined by the higher-level constraints.

We can formalize this concept in a simplified manner (reality is undoubtedly more complex, involving parallel threads of constraints, for example) through a sequence of coarse-graining operators ${\left\{{\mathcal{G}}_i\right\}}_{i=1}^k$, where each operator ${\mathcal{G}}_i:{\mathbb{R}}^n\to {\mathbb{R}}^{n_i}$ maps the fine-grained state vector $\mathbf{x}$ to a coarse-grained state vector ${\mathbf{y}}_i$, with ${\mathbf{y}}_i={\mathcal{G}}_i(\mathbf{x})$ ($n_i<n$). Each ${\mathbf{y}}_i$ represents the system at a specific level of abstraction, capturing variables relevant at that scale. For each level i, there is a corresponding hierarchical constraint ${\mathcal{C}}_i$ on ${\mathbf{y}}_i$, expressed as ${\mathcal{C}}_i({\mathbf{y}}_i)=0.$ Constraints reflect the agent’s world model at different levels, ensuring that the dynamics of the coarse-grained variables align with expected behaviors.

Lower-level constraints must be compatible with higher-level constraints. This means that the solution set of a lower-level constraint ${\mathcal{C}}_{i+1}$ is a subset of the solution set of the higher-level constraint ${\mathcal{C}}_i$. Formally, $\{\mathbf{x}\in {\mathbb{R}}^n\mid {\mathcal{C}}_{i+1}({\mathcal{G}}_{i+1}(\mathbf{x}))=0\}\subseteq \{\mathbf{x}\in {\mathbb{R}}^n\mid {\mathcal{C}}_i({\mathcal{G}}_i(\mathbf{x}))=0\}$. Nesting ensures that as we move to lower levels (higher indices), the constraints become more specific, refining the state space within the context defined by higher-level constraints.

As an example, consider a visual perception task with a high-level constraint ${\mathcal{C}}_1$: the agent recognizes that the object is a cat—see Figure 9. This constraint reduces the state space to the manifold ${\mathcal{M}}_1$ of all possible cats, which is still high-dimensional.

To this, we can add a lower-level constraint ${\mathcal{C}}_2$: the agent further discerns that the cat is white with blue eyes. This constraint reduces the state space to a submanifold ${\mathcal{M}}_2$ within ${\mathcal{M}}_1$, containing only white cats with blue eyes. Here, the lower-level constraint ${\mathcal{C}}_2$ is compatible with the higher-level constraint ${\mathcal{C}}_1$, as all white cats with blue eyes are indeed cats, ${\mathcal{M}}_2\subseteq {\mathcal{M}}_1.$ Thus, higher-level constraints define broad categories or contexts (e.g., recognizing an object as a cat), while lower-level constraints provide additional specificity (e.g., identifying the cat’s color, hair type, etc). The nested structure ensures that all lower-level processing is coherent with the higher-level understanding, leading to efficient and accurate modeling of the world. The hierarchical framework of constraints is related to compositional Lie groups characterizing the invariant properties of the symmetries of the equations. In particular, the hierarchical constraints (${\mathcal{C}}_i$) on agent dynamics correspond to conserved quantities associated with the symmetries of the ODEs represented by a hierarchical Lie group structure.

By imposing these hierarchical, compatible constraints, the original high-dimensional phase space ${\mathbb{R}}^n$ is reduced in a sequence of nested manifolds, forming what we may call a hierarchical or compositional manifold,

$${\mathbb{R}}^n={\mathcal{M}}_0\stackrel{{\mathcal{C}}_1}{\to }{\mathcal{M}}_1\stackrel{{\mathcal{C}}_2}{\to }{\mathcal{M}}_2\stackrel{\dots }{\to }{\mathcal{M}}_k,$$

(25)

where each manifold is defined recursively as

$${\mathcal{M}}_i=\{\mathbf{x}\in {\mathcal{M}}_{i-1}\mid {\mathcal{C}}_i({\mathcal{G}}_i(\mathbf{x}))=0\},\mathrm{for}i=1,2,\dots ,k,$$

(26)

with ${\mathcal{M}}_0={\mathbb{R}}^n$. Each manifold ${\mathcal{M}}_i$ satisfies all constraints up to level i, ensuring that the agent’s internal states remain consistent with both high-level goals and fine-grained observations. This recursive structure ensures that ${\mathcal{M}}_i\subseteq {\mathcal{M}}_{i-1}$, reflecting the increased specificity of constraints at lower levels (this also induces an inclusion relation among their tangent spaces).

The sequence of compositionally compatible constraints leads, in turn, to a sequence of nested manifolds, each representing the state space under the accumulated constraints up to that level. The agent’s dynamics are thus confined to a reduced hierarchical manifold, ensuring that its internal states remain consistent with both high-level goals and fine-grained observations. This mirrors the hierarchical organization of the brain or deep artificial networks, where a nested structure of coarse-grained models synthesizes a multilevel emergent view of the world.

5. Discussion

In this paper, we first assumed that agent inputs from the world are generated by simple, hierarchical rules (i.e., that the world is inherently simple in algorithmic terms), i.e., by a hierarchical (compositional) Lie generative model, and that the agent can track its inputs, i.e., encodes an internal generative model to approximately match data, perhaps after transients. From this, we showed that

• As a dynamical system, the successful agent displays conserved quantities—with dynamics in a reduced hierarchical manifold—corresponding to the world-tracking constraints generated by a Lie group. For dynamical inputs, this stability is achieved after initial transients, captured by a Lyapunov function (see Section 4.4).
• The constraints force structural symmetries in the dynamical system constitutive equations that meet them: the agent carries the symmetries of the world generative model. This is encoded in the structural elements of the dynamical system (the ws in our formulation).

World-tracking conditions are satisfied by specialized, symmetric networks, as the governing equations must maintain (approximately) conserved quantities that are compatible with the required outputs. Conversely, constraint (symmetry) breaking, as detected by the agent in the Comparator (Figure 1), allows the agent to identify anomalies or novel patterns, prompting updates to its internal models and engaging in learning and adaptation.

If the inputs are static, dynamics will lie in a reduced manifold, as determined by the constraints of world tracking. If the inputs vary slowly—following the rules of a generative world model—the dimension of the reduced manifold will increase, but only slightly compared to the embedding neural dimension. If inputs violate the world model, on the other hand, tracking will fail, and the dynamics may no longer be constrained.

Our analysis thus links the manifold hypothesis with world tracking and symmetry principles. Empirical observations of dynamics in natural systems indicate that data trajectories typically lie in a manifold of dimensionality much lower than ${\mathbb{R}}^X$ (the invariant manifold), which we also loosely identify with the latent space when using autoencoders to compress neural data. The manifold hypothesis posits that high-dimensional data, such as world data collected from sensory systems (e.g., images) or neuroimaging, can be compressed into a reduced number of parameters (latent space [110]) due to the presence of a low-dimensional manifold within the high-dimensional phase space [60,62,111,112]. Such manifolds can take various forms, such as attractors and unstable manifolds.

Because the data-generating processes we consider are inherently low-dimensional, our analysis provides a link with the manifold hypothesis in agent-generated data: large-scale data sets (be they high-dimensional images or neural recordings) often occupy manifolds of significantly lower dimension than the raw data space would suggest [60,62,111,112]. Here, we see that if data truly arise from a compositional world model, then an agent tracking those data automatically inherits such a manifold structure in its own state space, effectively compressing high-dimensional inputs into a smaller number of effective degrees of freedom.

Finally, this compositional view also suggests a hierarchical structure of manifolds when world data are generated at multiple, nested scales of abstraction, yielding the notion of a hierarchical manifold. This hierarchical perspective emphasizes how multiple levels of coarse-grained, low-dimensional constraints can collectively shape both world data and agent dynamics, providing different spatiotemporal scales of shared algorithmic information (see, e.g., [113] for related concepts in the Shannon information theory context). From this vantage, compositional world data and the act of tracking them together naturally realize the manifold hypothesis in both observational data and in an agent’s neural (or computational) trajectories.

Reduced hierarchical data manifold embody the inherent simplicity of the underlying generative models that produce the observed data, as well as the effectiveness of a world-tracking system in learning these models. This observation supports the manifold hypothesis as it manifests in world-tracking systems.

5.1. Symmetry, Groups, and Algorithmic Complexity

Does group action encompass the general notion of compression—finding short programs (i.e., with low Kolmogorov Complexity)? Under what conditions does the existence of a highly compressive generative model imply the presence of underlying symmetries described by a group (or generalization of)? Although rigorously answering this will require bridging abstract algebra and algorithmic information theory, the intuition is straightforward: a compressive (short) program capable of generating a large dataset will involve repeating an operation (recursion) or a sequence of operations (compositionality) multiple times.

This suggests that in a compressive model, the recursive action of a Lie group in configuration/latent space is represented economically in the construction of the generative function. In particular, if the image-generating function is a huge lookup table, there will be no compression despite the reduction of dimensionality from image to latent space. The recursive property of Lie groups, where complex elements are constructed through repeated applications of simple algebraic operations, mirrors foundational concepts in algorithmic information theory and recursive function theory. Compositionality in this context refers to the combined action of different generators.

The presence of a Lie group structure in a generative model significantly influences its algorithmic complexity. If one can navigate a configuration space with a Lie group, any function on that space—including a generative model—can be manipulated using the Lie group’s generators. This approach can be highly efficient. By generating a single instance within our model, we can apply the Lie group’s transformations in small, discrete steps to produce a diverse set of outputs.

As we saw, this process is inherently recursive and compositional, hence enabling a compressive representation. The program only needs to encode the initial instance of the object and the transformation rules derived from the Lie group’s generators, thus enabling the generation of a vast array of images through a sequence of incremental, systematic transformations.

Finally, we recall that the compositional sparsity of the underlying target function (i.e., the world)—the task to be learned—is the key principle behind the success of machine learning, particularly deep learning architectures such as CNNs or transformers [85,114]. Compositional sparsity, the idea that a function can be computed through the composition of functions, each with a constituent function of low dimensionality [115], is equivalent in algorithmic terms to efficient computability, that is, computability in polynomial time [114]. Thus, algorithmic and computational complexity notions may converge on the role of Lie action in generative (compressive) models.

In summary, we highlighted the relationship between algorithmic complexity and Lie groups, which stems from compressive models aiming to capture and recreate complex data structures through recursion and compositionality and the natural connection of these and Lie group action.

5.2. Connections with Empirical Observations

Symmetry is an example of an algorithmic feature that systems that compute the same function must share, regardless of their specific implementation. This implies that natural agents such as human brains implement symmetries inherent in world data within their architecture and, in turn, generate data with conserved quantities. This translates into reduced dimensionality dynamics in invariant manifolds and may explain complexity features observed in neuroimaging data across brain states and neurological or psychiatric disorders. Techniques like topological data analysis may help uncover further correlates of structured experience with applications in consciousness research in natural and artificial systems.

The fundamental predictions from studying the role of symmetry in agent dynamics pertain to the structural constraints of its constitutive equations and to the emergence of reduced manifolds, reflecting the structure of compositional data the agent is tracking.

The simplicity associated with a generative model may manifest in different ways. For instance, an efficient autoencoder of neural data from a brain exposed to time-varying world inputs could regenerate the dynamics using a latent parameter space whose dimension approximately matches the dimension of the world-generative model configuration space.

Furthermore, the Comparator mechanism enables the emergence of Mutual Algorithmic Information (MAI) between the agent and the world, which KT postulates [6,7]. By aligning internal models with external data, the Comparator highlights how constraints in world-tracking impact the structure of reduced, invariant manifolds in neural dynamics. These manifold structures, in turn, are expected to vary with brain states and environmental conditions, influencing neuroimaging data across different states of consciousness (awake, REM, NREM) [110], under anesthesia, minimally conscious (MCS) or unresponsive wakeful state (UWS), Locked-In Syndrome (LIS), epilepsy, or Alzheimer’s disease (AD) [116], which should display different latent space features, including dimensionality (compressibility, complexity), or exhibit task dependence (e.g., eyes open or closed). This can be attributed, at least in part, to the world-tracking features associated with brain state or task and motivates the hypothesis that structured experience is associated with the successful match of data and model at the Comparator [6,7,8].

Constrained dynamics may also provide the context for phenomena such as increased EEG slow-frequency power in the eyes-closed state (and reduced complexity [117]), anesthesia, or completely locked-in syndrome (CLIS, where patients shift toward slower frequencies and lower signal complexity) [118,119], all of which are characterized by clamping down (setting to a null fixed value) world inputs. This may either reflect suppression of Comparator outputs or the active tracking of a predicted static input. On the other extreme, the perspective of constrained dynamics may explain how psychedelics, which interfere with sensory processing, disrupt the Comparator and modeling process (Equations (14)–(17)), with network dynamics departing from the normative invariant manifold. Such a departure manifests as an increase in complexity and reduction in alpha power [39,40,51,120,121,122,123,124].

For data analysis, approaches such as deep learning autoencoders are well-suited for uncovering latent spaces associated with brain dynamics. Complementing this, topological data analysis can reveal structural features of reduced manifolds. These techniques could provide new insights into how different states compress or expand neural dynamics, providing insights into underlying brain symmetries and their breakdown in pathology. Complexity measures, like the Lempel-Ziv algorithm, can help quantify the intrinsic dimensionality of these manifolds, revealing symmetry-related information processing characteristics [15,125].

Finally, while we have focused here on how compositional data-tracking gives rise to constrained dynamics, the same logic applies to the generation of compositional outputs by the agent, such as motor actions. Recent research has uncovered low-dimensional neural manifolds that capture significant neural variability across various brain regions and experimental conditions [126,127,128,129,130,131,132,133,134,135]. These manifolds, defined by correlated neural activity patterns of neural activity—referred to as “neural modes”—are proposed to generate motor behavior through their time-dependent activation [126,130].

This research will benefit from a similar analysis of artificial systems, as discussed next.

5.3. Connections with Neurophenomenology

The emergence of reduced manifolds governed by Lie pseudogroups, as described in the context of agent-world interactions, offers a link to neurophenomenology. When an agent tracks compositional data from the environment, its internal dynamics are constrained to invariant manifolds reflecting the structure of the data. In this framework, these reduced manifolds shape the qualia—the fundamental units of structured experience associated with the agent’s perception of reality.

Neurophenomenology, which seeks to bridge first-person experiential phenomena with third-person neural dynamics, can provide methods to validate this perspective. Structured experiences can be viewed as arising from the agent’s successful alignment of internal models with world data, as mediated by the Comparator mechanism. This suggests combining the characterization of brain states by the geometry and topology of the underlying neural manifolds, which reflect the agent’s interaction with its environment together with phenomenology methods [136] for the characterization of the structure of experience—see Figure 2.

The integration of psychophysics, phenomenology, and AI methods [137], such as natural language processing (NLP) [138,139,140], offers a powerful approach to characterizing the structure of experience. Psychophysics provides quantitative tools to link physical stimuli with subjective perception, while phenomenology delves into the qualitative aspects of experience, emphasizing first-person reports and the lived reality of consciousness. AI methods like NLP can bridge these domains by analyzing and modeling linguistic data, enabling the systematic study of subjective descriptions. This combination allows for a multi-dimensional exploration of structured experience, where psychophysics anchors the investigation in measurable variables, phenomenology enriches it with introspective depth, and AI scales and systematizes the analysis, revealing patterns and latent structures in experiential data. Together, these tools provide a framework for advancing our understanding of the dynamics and features of structured experience.

5.4. Discovering Structure

In this paper, we have discussed the implications of using compressive models of generative world data, but we have not touched on the fundamental issue of how these models are discovered. A potential insight in this direction is that recursion and compositionality are the key features of algorithmics [5].

Recent research indicates that deep neural networks inherently favor low-complexity, compositional data structures, enabling them to identify and represent underlying symmetries without explicit prior knowledge. This intrinsic bias allows for effective generalization across various domains, supporting the unification of diverse tasks within a single learning algorithm [141,142]. The hierarchical and compositional nature of deep networks facilitates efficient approximation of complex functions, mitigating the curse of dimensionality and enhancing performance on tasks with compositional structures [85,86,114,143]. These characteristics underscore the adaptability and efficiency of deep learning models in processing structured, real-world data.

Neural networks can learn to detect group structure (Lie algebras) from training data, and extracting invariances remains an active research area. Moskalev et al. [76] propose a two-step procedure: (i) train a network on a task requiring invariance, and (ii) extract the Lie algebra that underlies this learned symmetry. Their Lie Group Generators (LieGG) method builds a polarization matrix from the network’s input-output relationship, applies singular value decomposition, and identifies near-zero singular values that correspond to the basis of the learned Lie algebra. For instance, on rotation MNIST, LieGG recovers the rotation symmetry and uses metrics like “symmetry variance” and “symmetry bias” to assess how thoroughly the network has internalized this invariance.

Recent advancements have introduced additional methods for discovering symmetries and invariances in data using neural networks. For example, LieSD (Lie Symmetry Discovery) identifies continuous symmetries by analyzing the gradients and outputs of trained neural networks, demonstrating its utility in tasks like the two-body problem and top quark tagging [144]. Similarly, LaLiGAN (Latent LieGAN) learns a latent space representation where nonlinear symmetries become linear, enabling the discovery of intrinsic symmetries in high-dimensional systems [145]. Another approach, Lie Algebra Convolutional Networks (L-conv), uses Lie algebras instead of groups to construct group-equivariant architectures, facilitating symmetry discovery and connection to physical principles like conservation laws [146]. These methods extend the capabilities of LieGG, enabling the exploration of both geometric and more abstract symmetries in neural networks.

Another perspective is exploring the topology of Lie groups [147] (and, more generally, Lie pseudogroups), which provides an essential framework for understanding the ways in which agents track and encode structured data in their internal dynamics. In particular, based on topological data analysis, properties such as connectedness and the presence of non-trivial loops or higher-dimensional “holes” (often characterized by Betti numbers) can inform how a neural system might transition among configurations or maintain stable invariants in response to environmental inputs. Within neurophenomenology, such topological features of the agent’s internal manifold—mirroring the underlying compositional actions of a Lie group—can shape how perceptual states coalesce into coherent experiences.

5.5. Applications in AI and Neurosynthetic Agent Design

In artificial intelligence (AI) and computational neuroscience, symmetry constraints are pivotal in guiding model design by imposing structural limitations that reflect the symmetries inherent in World interaction tasks. These symmetry principles can be incorporated as regularization terms to reduce parameter degeneracy or structurally (e.g., convolutional networks), enhancing model robustness and efficiency. From the earliest developments of deep networks [65,70,71,148] to current advancements, symmetry-driven approaches continue to shape and fuel research into more adaptable and efficient models [66,68,76,81,149].

Whole-brain computational models and digital twins [150] are neurosynthetic neural networks characterized by a very large number of parameters that are difficult to adjust given the limited available neuroimaging data. Just as with artificial neural networks, enforcing symmetry and conserved quantities with low dimensionality, we can reduce the parameter space’s degeneracy when fitting models to data [151]. Additionally, when parameters are known, the system’s dynamics can be explored through its symmetry properties. Analyzing the number, dimension, structure, and stability of the system’s attractors (using eigenvalue and bifurcation analysis) provides insight into its long-term behavior [73,93] and computational capabilities, including multistability, metastability, and criticality [31,37,152,153].

For instance, whole-brain models need to reflect the spatial-temporal symmetries and sensory-motor contingencies inherent in the data they process [154]. This includes fundamental transformations like translations and rotations. Such considerations will become increasingly important as whole-brain models are developed to interact with external inputs and generate outputs, whether in classification tasks or motor actions.

An important part of this program is to characterize more general world symmetries and translate them into generative world models and Lie groups. An approach to attack this may be the use of deep learning models that can produce the appropriate latent spaces or derive the Lie group generators from networks that have been trained on world data [67,69,76]. The discovery of these attractors can be used to explore the mutual algorithmic information between data and dynamical attractors in the artificial system (or in the human brain).

Finally, symmetry considerations can serve as a regularization principle, instrumentalizing a bias towards simplicity. By demanding that models remain as symmetric as possible while fitting empirical data, we implement a form of Occam’s razor, promoting simpler, more generalizable models. This principle is particularly relevant for brain-inspired networks, where preserving computational simplicity can improve interpretability and robustness [73].

6. Conclusions

In past work, we put forth the idea that because agents possess and run models of the World, there is significant Mutual Algorithmic Information (MAI) between the agent and the world [6]. In this paper, this idea is made concrete in dynamical terms using the language of group theory: the agent, viewed as a dynamical system, encodes symmetries that reflect the structure of world data.

The historical trajectory of group theory—the mathematical theory of symmetry—is a testament to its fundamental role in shaping the landscape of modern mathematical and physical sciences. From its initial development by Évariste Galois and Paolo Ruffini [155,156,157]—who, to unveil the algebraic symmetries governing polynomial equations, created the seeds of group theory—to the works of Sophus Lie [73,93], who extended these concepts into the role of continuous groups in differential equations, group theory is rich in its fruits. The subsequent epoch, marked by Emmy Noether’s transformative theorem [63,64], cemented the profound connection between symmetries and conservation laws, further elucidating the inherent simplicity underlying complex natural phenomena and driving the development of General Relativity and the Standard Model in physics.

These historical milestones underscore the enduring quest for understanding through the lenses of simplicity and symmetry, to which we hope this paper contributes. As we have discussed here through the model of the algorithmic agent (KT), the convergent principles of algorithmic simplicity and symmetry can yield useful insights into the nature of artificial and natural intelligence and the deep connections between mathematics and the phenomenology of structured experience.