Memory Evolutive Systems

• Memory Evolutive Systems (MES) is a framework modeling evolving, hierarchical cognitive networks using category theory to integrate symbolic and geometric memory representations.
• It employs poc-sets, median graphs, and cubical complexes to capture the dynamic evolution, efficient updating, and structural reorganization of memory.
• MES underpins applications from autonomous agent planning to language acquisition by bridging high-level abstraction with practical, connectionist methods.

Memory Evolutive Systems (MES) refers to a broad mathematical and conceptual framework for modeling memory and cognition as evolving, hierarchical networks organized according to category-theoretic principles. MES was originally developed in cognitive science and mathematical biology, but its influence has extended into areas such as machine learning, artificial intelligence, autonomous systems, and the formal modeling of neural and social networks. MES emphasizes memory not as a static storage device but as a dynamically evolving complex, whose architecture, logic, and efficiency reflect continuous structural transformations in response to experience.

1. Mathematical and Category-Theoretic Foundations

Memory Evolutive Systems are grounded in the language of category theory, specifically leveraging the notion that cognitive and memory systems are complexes of interacting subsystems whose transformations and reorganization over time can be represented as morphisms between objects in a category. Many MES instances combine symbolic (combinatorial, logical) modeling—such as poc-sets (partially ordered sets with complementation)—with geometric representations, e.g., dual median graphs or cubical complexes, to encode relational and spatial structure in memory (Guralnik, 2010).

Central constructs include:

• Complexes and Hierarchies: Higher-order memories or concepts ("complexes") emerge from the aggregation and composition of lower-level subsystems (e.g., neural assemblies, logical questions, or memory states), formalized using colimits in the category.
• Poc-Sets and Dual Graphs: Elementary perceptions or “questions” are encoded in poc-sets, capturing Boolean relations among atomic propositions. The dual realization as weighted median graphs (Γ(P)) or cubical complexes embeds logical, spatial, and hierarchical structure (Guralnik, 2010, Guralnik et al., 2015).
• Morphisms and Degeneration: Structural evolution is formalized as morphisms in the category, permitting retraction/pruning operations (e.g., selective forgetting or complexification) and maintaining coherence across evolving structures.

Mathematical formulas central to MES include the median operation in median graphs,

$$\mathrm{med}(u, v, w) = (u \cap v) \cup (v \cap w) \cup (u \cap w),$$

and the mapping of environment states to memory states,

$\pi_{(P, f)}(x) = \{ a \in P \mid x \in f(a) \}.$

2. Dynamics of Evolution, Updating, and Efficiency

MES models systems as inherently dynamic, responding to sensory inputs and experience through parallel processes of local update and large-scale structural reorganization (Guralnik, 2010, Guralnik et al., 2015). Key aspects include:

• Two-Level Memory Updates: Immediate, local updates modify the “current conjecture” (ε) about the world based on new sensory inputs, ensuring coherence with past data. Over longer spans, the underlying relational or logical structure P is itself evolved, e.g., by pruning low-excitation or rarely activated regions of the internal memory graph.
• Category-Theoretic Reorganization: Updating, forgetting, and abstraction correspond to categorical morphisms, degenerations, or retractions between poc-sets, preserving record of essential relationships while maintaining computational efficiency.
• Computational Efficiency: Many MES-driven models achieve linear or quadratic update and query complexity, where complexity bounds are in the cardinality of the sensorium or underlying poc-set (Guralnik, 2010, Guralnik et al., 2015). Local updates are often parallelizable, and the overall system adapts efficiently as the data stream or environment changes.

MES places particular emphasis on coherence in evolution: as the system complexifies or prunes its internal structure, it preserves functional and informational integrity of memory.

3. Symbolic and Geometric Memory Representations

A haLLMark feature of MES is the integration of discrete symbolic structures (e.g., poc-sets) with metric or geometric objects such as median graphs or cubical complexes (Guralnik, 2010, Guralnik et al., 2015).

• Symbolic Level: Logical relations, conceptual hierarchies, and inference rules are encoded algebraically. For instance, sensor implication relations are stored as a partial order in a weak poc-set, and updates tally observation co-occurrences to infer new logical dependencies.
• Geometric Level: Each poc-set or weak poc-set is dually represented as a non-positively curved cubical complex (CAT(0)), whose vertices correspond to globally coherent memory states. The convexity and projection properties of these complexes are used for efficient planning, recall, and nearest-point search, aligning with connectionist models of neural computation.

The duality between symbolic and geometric perspectives enables MES-based architectures to support high-level symbolic reasoning alongside resource-efficient, connectionist dynamics. This underpins powerful learning systems in both natural settings and artificial agents (Guralnik et al., 2015).

4. Applications: Learning, Reasoning, and Language Acquisition

MES has been applied to a wide range of problems in autonomous learning, planning, and reasoning. Notably:

• Language Learning: The deformation space of observers (i.e., internally evolving poc-sets reflecting language acquisition) is used to model how individual memory structures synchronize with population-level linguistic conventions. This provides a formal account for gradual community-wide convergence in shared meaning (Guralnik, 2010).
• Autonomous Agents and Planning: MES-driven architectures have been implemented in artificial agents capable of goal-directed reasoning, environment mapping, and real-time adaptation, with resource bounds typically quadratic in the number of sensors. The convexity theory of the geometric dual supports explicit computation of the nearest memory state and planning toward desired targets (Guralnik et al., 2015).
• Hierarchical and Multi-Scale Memory: Both in biological and engineered agents, MES predicts that memory structures will naturally evolve hierarchical representations, mirroring neurobiological results concerning multi-scale integration and modularity.

MES frameworks also provide a unified lens through which to analyze errors, temporary or permanent information loss, and recovery, yielding models that explain a broad range of observed memory phenomena (e.g., human reasoning errors, drift, and persistent forgetting) (Guralnik, 2010).

5. MES in Connectionist, Category-Theoretic, and Graph Models

MES is closely connected to both neurocognitive and computational models that employ category theory, distributed learning, and evolving graph-based architectures. Notable intersections include:

• Category-Theoretic Cognitive Models: MES’s use of objects, morphisms, and colimits parallels the modeling of neural assemblies as conceptual clusters, with morphisms representing synaptic or functional connections, and colimit diagrams formalizing the emergence of higher-order concepts (Mitavskiy et al., 2013).
• Hebbian Learning: Parallel dynamic algorithms inspired by the Geiringer theorem and implemented as agent graph traversals (bugs on weighted directed graphs) mirror the Hebbian “cells that fire together, wire together” principle found in both biological and artificial learning networks (Mitavskiy et al., 2013).
• Dynamic Graph and Mass-Based Models: MES-compatible architectures can be formulated as evolving, weighted, and sometimes stochastic graphs, where nodes represent memory propositions with continuous or mass-based significance, and edge weights evolve by update rules reflecting co-activation and reinforcement (Mollakazemiha et al., 2023). Such models enable the capture of both core, stable memories and peripheral, easily forgotten ones, further mirroring human-like behavior.

6. Extensions, Limitations, and Perspectives

While MES offers a powerful and general framework for modeling memory evolution, several technical and conceptual aspects are notable:

• Abstraction and Granularity: MES leverages high-level category-theoretic abstractions, sometimes resulting in a gap between mathematical formalism and biologically detailed modeling. Recent work seeks to bridge this via explicit mass-based graph models and dynamic equations that capture granular memory evolution features (Mollakazemiha et al., 2023).
• Scalability and Resource Cost: Formal models place emphasis on bounding storage and update costs (often quadratic) while maximizing expressivity—ensuring representation of all possible sensory equivalence classes and accurate topological modeling of the perceived environment (Guralnik et al., 2015).
• Integration with Modern AI: MES-influenced designs have been instantiated in hierarchical memory architectures for multi-agent systems (e.g., G-Memory (Zhang et al., 9 Jun 2025)), self-organizing memory frameworks, and hybrid symbolic-connectionist planners. These systems take advantage of category-theoretic structures (insight, query, interaction graphs) and dynamic updates to support continual learning and adaptive, robust behavior.
• Open Directions: Active research explores the formal linkages between MES and population-level knowledge acquisition, role-specific memory, and the evolution of neural and social memories as observed in both biological organisms and collaborative artificial agents.

7. Summary Table: Key Formalisms in MES

Formalism	Role in MES	Typical Mathematical Structure
Poc-set (P, ≤, *)	Logical relations, atomic perception units	Finite partially ordered set w/ involution
Median Graph Γ(P)	Geometric dual of poc-set, memory network	Graph, vertices = maximal coherent subsets
Cubical Complex Cube(P)	Symbolic-geometric memory representation	CAT(0) cubical complex
Category C of Memories	System evolution, abstraction, organization	Objects = memory states, morphisms = transitions
Mass-based Graph	Weighted and evolving memory representation	Node mass, edge weights, stochastic updates

The mathematical structures and categorical underpinnings of MES provide a rigorous, extensible foundation for modeling the evolution of memory in both biological and artificial adaptive systems. MES thus supplies a conceptual and technical bridge between high-level abstraction and implementable memory architectures, supporting both symbolic reasoning and low-level connectionist updating in dynamic, multiscale settings.