Minimal Cognitive Grid

Updated 4 July 2026

Minimal Cognitive Grid is a compact, structured coordinate system that defines cognitive spaces using sparse parameterizations and modular grid codes.
It is applied across dynamic navigation, behavioral simulation, neural coding, and architectural design to optimize prediction and planning while balancing accuracy and complexity.
Practical systems demonstrate that minimal grids expand only as needed, enabling efficient state mapping and cognitive control through mechanisms like free energy minimization and MDL.

Taken together, recent research uses “Minimal Cognitive Grid” to denote a compact, structured coordinate system for cognition that is only as large as needed for prediction, planning, evaluation, or explanation. In navigation, it can denote a dynamically expanding cognitive map over latent states and poses learned online under Active Inference; in behavioral modeling, a low-dimensional parameterization of a group’s cognitive state linked to a small behavioral outcome space; in architecture, a minimal set of modules sufficient for belief updating and decision-making; and in model assessment, a grid of dimensions for ranking cognitive plausibility (Tinguy et al., 2024, Bian et al., 2024, Gashler et al., 2015, Donvito et al., 2 May 2026). This suggests that the term is best treated as a family of related formalisms rather than a single canonical model. Across that family, “minimality” is operationalized through sparse parameterizations, modular grid codes, finite-state or low-bit control, parsimonious descriptive languages, and explicit accuracy–complexity trade-offs.

1. Conceptual scope

Recent work instantiates the idea in several technically distinct forms: a latent graph-like map over discrete states and poses under Active Inference, a $2$-dimensional cognitive parameter space $(K,T)$ linked to a $7$-bin outcome simplex, a minimal cognitive architecture with beliefs, perception, dynamics, valuation, and planning, a grid module that discretizes continuous variables into $0/1$ codes, a descriptive grid language over objects and masks optimized by the Minimum Description Length principle, and a three-dimensional evaluation framework over Functional/Structural Ratio, Generality, and Performance Match (Tinguy et al., 2024, Bian et al., 2024, Gashler et al., 2015, Feng et al., 2023, Ferré, 2021, Donvito et al., 2 May 2026).

Formulation	Minimal structure	Representative use
Dynamic cognitive map	States, poses, transitions, online expansion	Navigation
Cognitive simulator	$(K,T)$ and $7$-bin attempt distribution	Wordle group cognition
Minimal architecture	Belief, dynamics, observation, valuation, planner	General cognition
Grid-feedback unit	Binary encoding, comparison, correction	Cognitive learning
Descriptive grid model	Objects, masks, positions, MDL	ARC-like reasoning
Evaluation grid	FSR, Generality, Performance Match	Cognitive plausibility

What unifies these formulations is not a shared implementation, but a shared constraint. The representation must remain compact without becoming functionally trivial. In the navigation literature, that means a map that expands only when observations, poses, or predicted states force additional structure. In the simulation literature, it means representing group cognition with two interpretable parameters rather than a large feature set. In architectural work, it means reducing cognition to the smallest set of subsystems that still supports perception, prediction, valuation, and action. In evaluative work, it means locating a model in a low-dimensional space of theoretically motivated criteria rather than equating cognitive plausibility with benchmark accuracy alone.

A second recurrent feature is that the “grid” is often not merely spatial. It can be a pose lattice, a parameter manifold, a symbolic state space, an action-equivariant frequency lattice, or even an evaluative matrix over cognitive-theoretic constraints. This suggests that the term “grid” functions as shorthand for an organized coordinate system over cognitive states or model properties, while “minimal” denotes a pressure toward the smallest sufficient organization.

2. Dynamic cognitive maps and minimal control

A direct spatial formulation appears in “Learning Dynamic Cognitive Map with Autonomous Navigation” (Tinguy et al., 2024). There the cognitive map is a latent, graph-like model over discrete hidden state $s_t$ and hidden position $p_t$ , coupled to observations $o_t$ , actions $a_t$ , and policies $(K,T)$ 0. The generative model is written as

$(K,T)$ 1

with observation likelihood $(K,T)$ 2, position likelihood $(K,T)$ 3, state transition model $(K,T)$ 4, and deterministic position transition model $(K,T)$ 5. States are graph nodes, $(K,T)$ 6 defines edges, and poses provide an intrinsic spatial metric. The map is explicitly dynamic: it starts from a minimal $(K,T)$ 7 matrix with only $(K,T)$ 8 meaningful state, then grows only when new observations, new poses, or new states are needed.

The minimality mechanism is not only the small initial state space. It is enforced by the learning rules. Parameters are updated by minimizing augmented free energy with a complexity penalty $(K,T)$ 9, imagined transitions are learned at lower rates than experienced transitions, and impossible transitions receive negative updates. New states are created only when a predicted next pose $7$0 cannot be associated with any existing state with appreciable probability. Crucially, those new states may be created for predicted but not yet observed locations, so the map grows over imagined trajectories, not only over experienced ones. At the same time, aliased environments are handled by joint inference over state and pose and by a $7$1 certainty threshold: if state certainty drops below that threshold, the agent temporarily ignores pose and re-localizes from observations rather than immediately duplicating states.

The empirical result is a map that is both sparse and operational. On a $7$2 grid, the oracle requires $7$3 steps to visit each room once, the proposed model averages $7$4 steps, and CSCG variants require roughly $7$5–$7$6 steps. On a $7$7 grid, the model averages $7$8 steps, whereas CSCG variants require roughly $7$9–$0/1$0 steps. The same system also performs dynamic remapping in donut and Tolman mazes by weakening obsolete edges when new obstacle evidence arrives, then re-planning accordingly. A plausible implication is that the model realizes a “minimal sufficient cover” of the environment: a sparse graph of states anchored to an expanding pose lattice, with new nodes and edges paid for only when prediction and control fail without them.

A complementary notion of minimality appears in “Patrolling Grids with a Bit of Memory” (Amir et al., 2023). In that setting, cognition is proxied by finite internal state. The paper proves that some grid graphs cannot be patrolled with $0/1$1 bits of memory regardless of sensing range, gives an exact characterization of the grids that can be patrolled with $0/1$2 bits and sensing range $0/1$3, and shows that a $0/1$4-bit algorithm with $0/1$5 patrols any $0/1$6-dimensional grid graph, with every vertex visited within at most $0/1$7 steps. Here the minimal cognitive grid is not a learned map but a regular environment whose structure is rich enough that one bit of internal state suffices for persistent global coverage. The result sharpens the idea that minimal cognition can sometimes be achieved by offloading regularity into environmental structure rather than increasing internal model capacity.

3. Optimal grid codes and actionable spatial representations

A second major line of work interprets minimal cognitive grids as optimal neural codes. “A computational model for grid maps in neural populations” derives hexagonal grid-cell receptive fields from three assumptions: stationary second-order statistics of inputs, Oja’s rule, and minimal-variance encoding of position (Anselmi et al., 2019). The key result is that, in $0/1$8 dimensions, the optimal frequency vectors form an equiangular tight frame. With the minimal number of vectors, that frame is either an orthonormal pair, yielding a square grid, or the “Mercedes–Benz frame” of three equiangular vectors at $0/1$9, yielding a hexagonal grid. The resulting code is minimal in two senses stated in the paper: it minimizes the Cramér–Rao lower bound on positional variance and uses the smallest number of directions needed to achieve isotropic precision.

“Actionable Neural Representations: Grid Cells from Minimal Constraints” generalizes that idea by making actionability itself the central axiom (Dorrell et al., 2022). A representation $(K,T)$ 0 of position is actionable when there exists a linear operator $(K,T)$ 1 such that

$(K,T)$ 2

Representation theory then implies that actionable codes are linear mixtures of sinusoidal modes. When actionability is combined with non-negative firing, bounded neural activity, and a precise-coding objective, the optimal population code becomes a set of multiple modules of hexagonal grid cells. The paper identifies a “Goldilocks annulus” in frequency space, bounded below by the occupancy scale and above by the task’s required resolution; among lattices that can populate that annulus, the hexagonal lattice is optimal. Multiple modules arise because within-module harmonic structure helps non-negativity, while cross-module non-harmonicity improves discriminability.

“Complete coverage of space favors modularity of the grid system in the brain” adds a complementary coverage argument (Sanzeni et al., 2016). There the grid system must cover space completely, without gaps, with high probability. Variability in periodicity, orientation, and ellipticity within a module shrinks the correlation length $(K,T)$ 3, increasing the number of effectively independent spatial patches and reducing the range that can be covered without gaps. The paper derives a scaling relation between the number of neurons and the period of a module, and predicts that more neurons are required at smaller grid scales than at larger ones. Co-modularity in period, orientation, and ellipticity then receives a functional rationale: it is the efficient way to keep coverage reliable while limiting neuron count.

A computational extension of this line appears in “A Grid Cell-Inspired Structured Vector Algebra for Cognitive Maps” (Krausse et al., 11 Mar 2025). GC-VSA merges continuous attractor network intuitions with Vector Symbolic Architectures. Each component of the representational vector is replaced by a $(K,T)$ 4-dimensional neuronal module, with $(K,T)$ 5 explicitly chosen as the minimal number of sampled points that allows the shift using circular convolution. The model defines bundling $(K,T)$ 6, binding $(K,T)$ 7, unbinding $(K,T)$ 8, and rotation/permutation $(K,T)$ 9, and uses the same underlying code for path integration, spatio-temporal representation, and symbolic reasoning over family trees. This makes the minimal cognitive grid neither purely metric nor purely symbolic. It is a shared algebraic substrate in which spatial and abstract computations reuse the same module types and vector operations.

4. Compact architectures, local units, and descriptive models

Minimality also appears as architectural compression. “A Minimal Architecture for General Cognition” proposes MANIC, a system with three function approximating models and one state machine (Gashler et al., 2015). The learning system consists of a belief vector $7$0, a transition model $7$1, and an observation model $7$2 with inverse $7$3, while the decision-making system adds a scalar contentment model $7$4 and a planner that searches over action sequences. The core equations are

$7$5

The paper’s sufficiency claim is theoretical rather than empirical: with universal approximators, arbitrarily large belief vectors, and sufficiently large search, this architecture is argued to be functionally equivalent to richer cognitive architectures. In the language of minimal cognitive grids, MANIC reduces cognition to a small set of slots—representation, observation mapping, dynamics, valuation, and planning—while treating everything else as derivable or practically optional.

A more explicitly grid-mediated learning system is “Grid-SD2E: A General Grid-Feedback in a System for Cognitive Learning” (Feng et al., 2023). Its grid module converts continuous predictions and labels into $7$6 codes via recursive space division, and the SD2E machinery uses these codes for comparison and correction within a Bayesian learning loop. The paper explicitly extracts a “smallest computing unit” analogous to a neuron: an unsupervised decoder $7$7, a binary encoder $7$8, and an update rule $7$9 that either preserves the current prediction or applies a symmetric correction when the predicted bit and target bit disagree. Here the minimal cognitive grid is the smallest binary-coded partition that can close a local predictive-correction loop.

“First Steps of an Approach to the ARC Challenge based on Descriptive Grid Models and the Minimum Description Length Principle” pushes minimality into symbolic description (Ferré, 2021). A task model is

$s_t$ 0

where grids are constructed from backgrounds and layers of positioned objects, and shapes are points or rectangles with masks such as Full, Border, EvenCheckboard, OddCheckboard, PlusCross, TimesCross, or Bitmap. The system defines paired reading and writing operators for parsing and generating grids, and selects models by the two-part MDL score

$s_t$ 1

The claim is not that ARC can be solved by an unrestricted program search, but that a large class of ARC tasks can be compressed into object-based grid models plus a small arithmetic language over positions and sizes. Over one year of work, performance on the $s_t$ 2 training tasks increased from $s_t$ 3 to $s_t$ 4 solved tasks using $s_t$ 5 seconds per task, with the added feature that the resulting models and their incremental refinements remain intelligible. This is a strong instance of minimal cognitive grid design as compressive explanation.

5. Behavioral landscapes, benchmarks, and evaluative grids

In “CogSimulator: A Model for Simulating User Cognition & Behavior with Minimal Data for Tailored Cognitive Enhancement,” the grid is explicitly behavioral rather than spatial (Bian et al., 2024). Group cognition in Wordle is represented by just two tunable hyperparameters: $s_t$ 6, the “cognitive limit,” and $s_t$ 7, a frequency-scaling factor. Those two parameters define a low-dimensional cognitive state, while each word’s behavioral profile is a point in a $s_t$ 8-dimensional simplex over $s_t$ 9. Fitting minimizes the mean Wasserstein-1 distance

$p_t$ 0

using $p_t$ 1 daily Wordle solutions from Jan–Dec 2022 and word frequencies from the Google Books Ngram corpus. The reported accuracy is $p_t$ 2, and the paper states that CogSimulator outperforms most conventional machine learning models in mean Wasserstein-1 distance, mean squared error, and mean accuracy. A plausible implication is that minimal cognitive grids need not be spatial at all; they can be low-dimensional control spaces whose geometry is supplied by a task-specific discrepancy metric.

Benchmark work uses grids in yet another sense: not as internal representations, but as controlled environments for probing cognition. “GRASP: A Grid-Based Benchmark for Evaluating Commonsense Spatial Reasoning” defines $p_t$ 3 grid-based environments, each an $p_t$ 4 energy collection task with different energy distributions, starting positions, obstacle settings, and agent constraints (Tang et al., 2024). The benchmark compares random walk, greedy search, GPT-3.5-Turbo, and GPT-4o. The reported averages indicate that Greedy Search substantially outperforms the LLMs, while GPT-4o remains close to Random Walk and both advanced LLMs struggle to consistently achieve satisfactory solutions. The result is important because the environment is fully specified and evaluation is performed on executed plans rather than on verbal descriptions of space.

“KidGym: A 2D Grid-Based Reasoning Benchmark for MLLMs” makes the same move for multimodal systems, but with an explicit psychometric decomposition (Ye et al., 2 Mar 2026). It uses a $p_t$ 5 grid with a $p_t$ 6 active gameplay region and organizes $p_t$ 7 tasks around five core capabilities: Execution, Perception Reasoning, Learning, Memory, and Planning. For each task $p_t$ 8, the weighted success score is

$p_t$ 9

and capability scores average those task-level quantities. The benchmark shows that closed-source MLLMs outperform open-source models overall, but all current models exhibit pronounced weaknesses in abstract perception reasoning, counting, and composite tasks that require coordination across memory and planning.

A further shift occurs in “Structural Ranking of the Cognitive Plausibility of Computational Models of Analogy and Metaphors with the Minimal Cognitive Grid” (Donvito et al., 2 May 2026). Here the grid is an evaluative framework rather than a task environment. Cognitive plausibility is quantified along three dimensions: Functional/Structural Ratio, Generality, and Performance Match. The global score is

$o_t$ 0

Applied to SME, CogSketch, $o_t$ 1, and LLMs, the framework yields high structurality for SME and CogSketch, intermediate structurality for $o_t$ 2, and low structurality but high generality for LLMs. This use of “Minimal Cognitive Grid” is orthogonal to spatial maps and low-dimensional simulators, but it preserves the same organizational principle: cognition is analyzed in a small, explicit coordinate system whose dimensions are intended to be theoretically meaningful.

6. Philosophical disputes, limitations, and open directions

Minimality is not conceptually innocent. “Separating minimal from radical embodied cognitive neuroscience” argues that minimal embodiment remains staunchly internalist and neurocentric, whereas radical embodiment treats cognition as realized by brain–body–environment systems and rejects a strong isomorphism between neural and cognitive processes (Wit, 2024). In that vocabulary, many minimal cognitive grids are “body-in-brain” formalisms: they compress cognition into latent states, neural codes, or compact algorithms while leaving embodiment and environmental constitution outside the representational core. The commentary therefore raises a methodological challenge: whether minimal grids should be optimized inside the brain-like model alone, or across the wider coupled system that includes morphology and environmental structure.

The technical literature itself records several limits. The dynamic cognitive map model successfully uses horizons up to about $o_t$ 3 steps, but longer planning increases computational cost, and the model has no explicit state merging or global Bayesian model reduction over the learned structure (Tinguy et al., 2024). CogSimulator reduces cognition to $o_t$ 4, but the authors explicitly note that it tends to represent an “average” player and may fail where group data are multimodal (Bian et al., 2024). GC-VSA offers a unified spatial-symbolic algebra, yet its generators and role codes are hand-designed, it does not model learning, and its parameter choices are not optimized for minimality (Krausse et al., 11 Mar 2025). The ARC MDL approach is transparent, but it still lacks rotations, loops over collections, and richer invariances, all of which are needed for a broader fraction of ARC tasks (Ferré, 2021).

Benchmarking results sharpen these concerns from the opposite direction. GRASP shows that contemporary LLMs remain poor at global spatial planning even in small, fully specified grids (Tang et al., 2024). KidGym shows that MLLMs degrade sharply on abstract puzzles, counting, and tasks that require simultaneous perception, memory, and planning (Ye et al., 2 Mar 2026). These results do not refute minimal cognitive grids, but they do show that minimal environments are already sufficient to expose substantial deficits in present-day foundation models.

Across the literature, future work converges on three extensions. The first is richer structure without abandoning parsimony: hierarchical maps, chunking, state merging, rotations, loops, and better treatment of multimodality. The second is broader grounding: bringing minimal grids into contact with realistic sensory pipelines, embodied dynamics, and multi-agent settings. The third is tighter cross-domain unification: the same article family already spans active inference maps, vector-symbolic algebras, MDL object languages, few-shot behavioral simulators, and cognitive-plausibility metrics. This suggests that the enduring value of the minimal cognitive grid is not a single ontology, but a discipline of representation design: retain only the structure that is indispensable for the target cognitive phenomenon, and make the resulting coordinate system explicit enough to support analysis, intervention, and comparison.