Formalizing Agent Memory: Basis, Span, and Optimality Defined.

Hongyao Tang· August 13, 2026 View original

Key takeaways

  • Agent memory is formally defined as a basis, with knowledge as its span.
  • Optimal memory maximizes expected coverage within capacity constraints.
  • The framework addresses noise, distinguishing coverage from precision.
  • It provides a rigorous foundation for designing and evaluating agent memory systems.

Who benefits

AI ResearchSoftware DevelopmentRoboticsGamingAutonomous Systems

Summary

This paper proposes a formal definition of agent memory, conceptualizing it as a basis whose span represents knowledge and answerability as a coverage problem. It introduces the optimal memory as a capacity-constrained maximizer of expected coverage and formalizes the continual agent-memory problem.

Researchers are taking a foundational step towards a unified formal definition of agent memory, a concept widely used in large-model agents but lacking a consistent theoretical framework. The core idea presented is that memory functions as a "basis," with the "knowledge" an agent possesses being the "span" of this basis. The ability to answer a query is then framed as a "coverage problem," where a query is answerable if an item within the knowledge span covers it. The paper defines "optimal memory" as the capacity-constrained system that maximizes expected coverage, establishing a utility-capacity frontier for comparing different memory systems. It also addresses the issue of noise in memory, distinguishing between coverage and precision. Furthermore, it formalizes the continual agent-memory problem within a sequential Markov Decision Process (MDP), drawing parallels to biological memory formation and instantiating the framework with Homer's *Odyssey* to provide concrete metrics for memory quality and learning.

Why it matters

AI researchers and engineers can use this formal framework to design, evaluate, and optimize memory systems for large-model agents more rigorously, leading to more intelligent, efficient, and reliable AI systems.

How to implement this in your domain

  1. 1Apply the proposed formal definitions of memory basis and span to analyze existing agent memory architectures.
  2. 2Utilize the utility-capacity frontier concept to evaluate and compare the efficiency of different memory systems.
  3. 3Develop new memory write policies that explicitly address the trade-off between coverage and precision under noisy conditions.
  4. 4Frame the development of continual learning agent memory as a sequential MDP problem to guide optimization efforts.

Original post by Hongyao Tang

"arXiv:2608.11654v1 Announce Type: new Abstract: Despite the wide deployment of memory in large-model agents, there is no unified formal account of what a memory is or when it is optimal. This paper takes a first step toward this account. The central idea is that memory is a basis…"

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