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This paper establishes a formal framework for understanding agent memory by defining memory as a basis, knowledge as its span, and answerability as a coverage problem. It introduces a utility-capacity frontier to evaluate memory systems and addresses the challenges of noise and precision in memory storage through a continual agent-memory problem modeled as a sequential MDP. By applying this framework to Homer's *Odyssey*, the authors provide measurable insights into memory effectiveness and set the stage for future research on constructing and optimizing agent memory systems.
Memory systems can now be quantitatively assessed through a utility-capacity frontier, revealing how to optimize agent memory for better performance.
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, knowledge is its span, and answerability is a coverage problem: an agent stores events extracted from a material; a generation operator turns any event set into the knowledge it entails; and a query is answerable exactly when some single item in the span covers it. The optimal memory is then the capacity-constrained maximizer of expected coverage, and its value traces a utility--capacity frontier, the common yardstick on which memory systems can be compared. Next, we consider noise in the memory and discuss coverage versus precision under it: a memory may store false claims, so the write policy must infer the truth of what it stores. Drawing an analogy with biological memory, which is formed continuously through ongoing experience, we formalize the continual agent-memory problem in a sequential MDP that covers multiple levels, where memory is the state, writing is the action, and the utility settled at query time is the delayed reward that drives learning. To make the framework concrete, we instantiate it on Homer's \emph{Odyssey}, turning the frontier, the compression zone, and the divergence of coverage from precision into concrete numbers. Finally, we position existing systems within the framework, making ``how good is a memory''measurable and recasting the open problems of constructing and learning agent memory as concrete research questions.
