Memory-Driven Planning in Autonomous Systems

• Memory-driven planning is a paradigm that integrates past sensory experiences using self-organizing memory architectures for adaptive, goal-directed behavior.
• It constructs minimal internal models through a weak poc set and CAT(0) cubical complex, enabling efficient topological recovery and planning.
• The approach employs greedy update–execute cycles and convergence guarantees to achieve robust learning and precise environment representation without prior knowledge.

Memory-driven planning is a paradigm in autonomous systems and artificial agents wherein the planning and decision process is deeply integrated with the agent’s structured record of past experiences, sensor observations, and inferred environmental models. Rather than relying solely on reactive strategies or handcrafted models, memory-driven planning leverages self-organizing memory architectures, efficient recall mechanisms, and information-theoretic principles to facilitate robust learning, representation, and goal-directed behavior in previously unknown environments. The theoretical foundation and practical realizations of this paradigm have profound implications for the design of scalable, efficient, and adaptive autonomous machines, particularly in the absence of prior domain knowledge.

1. Self-Organizing Memory Architectures

A central concept in memory-driven planning is the agent’s construction of a compact yet expressive memory architecture for organizing and encoding sensory experiences. This is instantiated as a snapshot data structure over the agent’s sensorium, denoted $\mathcal{S}$, where every sensor $a \in \mathcal{S}$ (including its complement $a^*$) partitions the environmental state space $X$.

Each snapshot $S$ tracks for every proper sensor pair $(a, b)$ a binary state and a real-valued edge weight $w_{ab}$, maintaining consistency via threshold parameters $\tau_{ab}$ and enforcing partition relations $\rho(a^*) = \rho(a)^c$. The state update and consolidation into symbolic structure are defined as:

• Registration of sensory activations
• Incremental edge weight updates: $w^{(t)}_{ab} = \sum_{k=1}^t c^{k}_{ab}$, where $c^{k}_{ab}$ records the co-occurrence of $a$ and $b$ at time $k$
• Extraction of a weak partially ordered set (weak poc set) $(P, \leq, *)$ where $a \leq b$ iff $\rho(a) \subseteq \rho(b)$

This process is provably efficient, with quadratic bounds $O(n^2)$ in both space and time complexity for $n = |\mathcal{S}|$, and accommodates learning to arbitrary precision purely through random exploration cycles; error in the derived symbolic representation decays exponentially with time under ergodicity assumptions.

2. Minimal and Topologically Faithful Internal Models

Beyond raw sensory compression, the architecture derives an internal representation minimal among all structures explaining the agent’s sensory equivalence classes. The coherent projection operator

$$\operatorname{coh}(A) = \{p \in P : A \subseteq p\} \cap \{p \in P : A^* \subseteq p\}$$

maps sets of (possibly noisy or incomplete) sensor activations $A$ to clean, redundancy-free coordinates in the symbolic model.

Sageev–Roller duality transforms this weak poc set into a CAT(0) cubical complex, $\operatorname{Cube}(P)$, endowing the agent with a median graph convexity theory. The 0-skeleton of this complex comprises complete coherent selections, capturing all consistent sensor states and enabling robust inference of the system’s topology—even to the level of homotopy type recovery. The punctured subcomplex defined by states consistent with actual observations allows the agent to learn topologically crucial features such as obstacles, holes, and connectivity, which directly inform sub-goaling and complex path planning.

3. Efficient Update–Execute Cycles and Greedy Planning

The planning cycle exploits the duality between weak poc sets and cubical complexes. Observations update edge weights according to co-occurrence statistics, inducing an acyclic, directed poc graph. Planning the agent’s next action proceeds through greedy nearest-point projection in the cubical complex:

$$\operatorname{proj}_{h(T)}(u) = (u \cup \#_1 T) \setminus \#_1 T^*$$

where $u$ is the current sensor state selection and $T$ the target goal sensation set.

The update–execute cycle runs in $O(n^2)$ serial time, and may approach $O(h)$ with parallel network propagation (where $h$ is the implication graph height). Gate finding and coherent projection are performed by propagation algorithms over $\operatorname{Cube}(P)$, leveraging its rich convexity and contractibility properties for reliable action generation.

4. Learning Dynamics and Precision Guarantees

Learning progresses via adjusted edge weight increments subject to sensory event coincidence and learning thresholds. Empirical and discounted snapshot updates converge rapidly:

$w^{(t)}_{ab} = \sum_{k=1}^t c^{k}_{ab}$

where $c^{k}_{ab} = 1$ if $a$ and $b$ are active at time $k$, and zero otherwise. Under reversible random walks on the environment’s state space, the architecture’s discrepancy from true implication structure decays exponentially, yielding high precision models despite absence of prior knowledge. Both empirical and discounted variants of snapshot updates yield similar convergence guarantees.

5. Symbolic/Numeric Duality and Topological Planning

The symbolic structure (weak poc set) and its geometric dual (CAT(0) cubical complex) together form an intrinsic memory model that supports symbolic reasoning, topological inference, and dynamic planning. The duality is not a mere architectural artifact but is central to learning and acting: implications and subset inclusions learned empirically are transformed into actionable regions and gates in the planning space, supporting:

• Robust nearest-point projection to goal states
• Convex subgoal identification
• Recovery of global topology from sensory history

Convex geodesic propagation algorithms invoke the solved symbolic structure to drive actions while preserving topological constraints.

6. Theoretical Significance and Broader Implications

The universal architecture establishes that memory-driven planning can be made efficient, minimal in representational complexity, robust to sensory noise, and capable of mathematically precise recovery of the environment’s topology. These properties collectively address central limitations of both purely connectionist and entirely symbolic planners, supporting autonomous learning and planning solely from sensory data and action experience.

In particular, the quadratic complexity and proven convergence offer concrete implementational advantages for scalable, self-organizing machines. The design does not require prior environment models, task structure encodings, or handcrafted representations, making it broadly applicable to autonomous systems operating in unknown or dynamically changing domains.

Table: Key Features of Memory-Driven Planning (Guralnik et al., 2015)

Feature	Technical Realization	Impact/Significance
Self-organizing snapshot memory	Edge weights, thresholded learning, involutive sensors	Efficient compression and update
Minimal symbolic representation	Weak poc set, coherent projection	Information-theoretic optimality
Topological recovery	Dual CAT(0) cubical complex, punctured subcomplex	Global topology for sub-goaling
Efficient planning cycle	Greedy gate finding, median graph algorithms	Fast, parallelizable planning
Arbitrary precision learning	Empirical and discounted snapshot updates	Robust self-improvement

Memory-driven planning thus defines a rigorous paradigm for autonomous problem-solving, grounded in provable mathematical dualities and efficient empirical algorithms, enabling robust goal-directed behavior in unknown environments using only the agent’s sensory and action-driven experience.