Primitive State Representations

• Primitive state representations are succinct, foundational encodings that capture essential system dynamics via discrete, compositional, or algebraic decompositions.
• They enable efficient dimensionality reduction in reinforcement learning, robotics, and 3D modeling by transforming high-dimensional states into tractable, low-dimensional forms.
• Their algebraic and automata-theoretic formulations support scalable system design, robust transfer, and interpretable architectures across diverse computational domains.

A primitive state representation is a succinct, foundational encoding of system state, typically constructed via discrete, compositional, or algebraic decompositions, with the aim of capturing essential dynamics without dependence on high-dimensional, opaque, or over-parameterized state spaces. The concept is realized across theoretical computer science, reinforcement learning, robotics, and geometric group theory, where minimal, interpretable, and transferable representations are favored for scalability, tractability, and invariance. This article surveys formal definitions, construction methodologies, practical instantiations, empirical benefits, and notable applications, ranging from RL-driven optimization to distributed systems, 3D perception, and low-dimensional hyperbolic geometry.

1. Foundational Formalisms and Definitions

Primitive state representations originate from the program of describing complex system behavior using algebraically minimal, recursive, or compositional constructs. The most canonical setting is the sequence-primitive recursive (s.p.r.) function formalism, which provides an alternative to the standard set-tuple-transition view of state machines:

Let $A$ be an input alphabet and $X$ an output set. A sequence map is a function $f: A^* \to X$, where $A^*$ denotes all finite words over $A$. The map $f$ is s.p.r. if there exists an auxiliary set $Y$ (typically representing “hidden” or “internal” states), an initial state $c \in Y$, a “step” function $g:Y \times A \to Y$, and an “output” map $h:Y \to X$, such that for all $w \in A^*$:

$$\begin{align*} f'(\varepsilon) &= c, \ f'(wa) &= g(f'(w), a), \ f(w) &= h(f'(w)). \end{align*}$$

This recursive structure coincides with the characterization of arbitrary Moore (or Mealy) machines. The algebraic perspective enables composition, product, and feedback constructions for distributed or concurrent systems, all within a primitive recursive schema (Yodaiken, 2016).

2. Construction Techniques in RL and Optimization

A key application of primitive state representations is in reinforcement learning for numerical optimization under computation and iteration constraints. In Sequential Update Selection (SUS) agents, the primitive state space $S'$ is defined as a low-dimensional, finite, discrete approximation of an otherwise high-dimensional environment state $S$:

$S' = \{1,2,\ldots,m_1\} \times \{1,2,\ldots,m_2\}$

where $m_1$ and $m_2$ are the numbers of bins (granularities) for two salient features:

• $s^1$: Discretized, normalized objective-value progress, given by $$s^1_k = \left\lfloor \xi(y_k) \cdot (m_1-1) \right\rfloor + 1$$ with $$\xi(y_k) = \min\{1, \max\{0, (y_k-\ell)/(u-\ell)\}\}$$ for bounds $\ell, u$.
• $s^2$: Discretized budget usage, $s^2_k = \left\lfloor k / (K/m_2)\right\rfloor + 1$.

This projection $\varphi : S \to S'$ dramatically reduces dimensionality, making tabular RL methods feasible and ensuring sample complexity scales as $O(|S'| \cdot |A|)$. Theoretically, this avoids the “curse of dimensionality” and enables efficient policy learning even when full state description is intractable (Sala, 2024).

3. Discrete, Symbolic, and Tokenized Primitives in Learning Architectures

Primitive state representations manifest as discrete codebooks or symbolic vocabularies in self-supervised sequence modeling. In motion analysis, the MoPFormer architecture tokenizes inertial sensor streams into discrete “motion primitives”:

• Raw time series $X = \{x_t\}_{t=1}^T$, $x_t \in \mathbb{R}^C$, are segmented into $S$ non-overlapping windows of length $L$.
• Each segment $S_i$ is mapped to its nearest prototype $c_k$ in a learned codebook $\mathcal{C} = \{c_k\}_{k=1}^K$ via $q(S_i) = \arg\min_k \| \hat S_i - c_k \|^2$.
• The sequence of primitive indices $q(S_i)$ forms an interpretable, fixed-size representation.

The model is pre-trained with masked prediction objectives (predicting missing token indices from context) and processes these discrete primitive sequences with attention-based encoders. Primitive states in this context serve as universal building blocks—shared, symbolic subunits that generalize across activities, subjects, and sensors, enabling robust transfer and diagnosis in human activity recognition (Zhang et al., 27 May 2025).

4. Explicit Primitive-Based 3D Representations

Primitive representations are central in explicit 3D modeling approaches for view synthesis and robotics. Instead of field-based (implicit) parameterizations, such as K-Planes, explicit primitive models use structured sets:

• Gaussian Splatting (GS): Objects are encoded as $K$ weighted anisotropic Gaussians $\{\mu_k, \Sigma_k, w_k, c_k\}$. The volumetric density is $$\rho(x) = \sum_{k} w_k \mathcal{N}(x|\mu_k, \Sigma_k)$$.
• Convex Splatting (CS): Objects are composed of $M$ convex polyhedra, each with vertices $\{v_{k, i}\}$ and color/alpha $(c_k, \alpha_k)$.

These primitive-based models achieve compact, interpretable, and clutter-free geometry representations critical for safety-critical applications like collision avoidance. Empirical results show that convex representations require orders of magnitude fewer parameters and achieve better true positive coverage (TPR=1.00) with no “floaters”—unattached, spurious geometry—compared to implicit or unstructured alternatives (Smijter et al., 12 Sep 2025).

Method	Primitives	IoU	TPR	FPR	Total Params
K-Planes	-	0.18	0.93	0.81	33.9M
Gaussian GS	2.1k–4.3k	0.48–0.53	0.90–0.96	0.13–0.15	2.1–4.3M
Convex CS	420–670	0.37–0.4	1.00	0.26–0.30	0.42–0.46M

Optimized use of appearance embeddings primarily reduces the number of primitives needed for photometric fidelity, but does not yield substantive geometric improvement (Smijter et al., 12 Sep 2025).

5. Algebraic and Automata-Theoretic Compositionality

Primitive state representations form the backbone of scalable and compositional system specifications, particularly in automata theory and distributed protocols. The s.p.r. sequence map framework allows the specification, verification, and composition of systems like the Paxos protocol, hierarchical counters, or parallel state machines. Internal state, input translation, and output generation remain within closed, recursive definitions—eschewing the need for higher-order logic, temporal operators, or fairness constraints. This enables both analytical tractability and direct correspondence to engineering implementations (Yodaiken, 2016).

6. Primitive-Stable Representations in Geometric Group Theory

In low-dimensional topology, primitive-stable representations refer to certain classes of free group homomorphisms into $\mathrm{PSL}(2, \mathbb{C})$ with geometric stability properties. For $F_n$ a free group and representation $\rho: F_n \to \mathrm{PSL}(2, \mathbb{C})$, the map is primitive-stable if the axis of every primitive element is mapped to a uniform quasi-geodesic in hyperbolic 3-space. This property is independent of generating set and persists under algebraic and geometric limits.

Main results include:

• For closed 3-manifolds $M = H_1 \cup_\Sigma H_2$ of sufficiently high Heegaard distance (with bounded subsurface Heegaard distance), the inclusions $$\pi_1(H_i) \to \pi_1(M) < \mathrm{PSL}(2, \mathbb{C})$$ are primitive-stable.
• Primitive-stable representations densely approximate boundary points of Schottky space, providing a bridge between combinatorial group presentations and hyperbolic manifold geometry.
• It is possible to construct sequences of primitive-stable representations $\rho^r: F_{n_r} \to \mathrm{PSL}(2,\mathbb{C})$ with $n_r \to \infty$ and bounded image covolume, demonstrating that domain rank is decoupled from geometric complexity (Kim et al., 2013, Minsky et al., 2010).

7. Interpretation, Generalization, and Practical Impact

Primitive state representations have concrete empirical and theoretical advantages:

• Efficient RL training and deployment via low-dimensional $S'$ (e.g., SUS agents achieve $30–40\%$ lower objective value than hyperparameter-optimized baselines).
• Substantial improvements in zero-shot and cross-domain transfer via interpretably tokenized primitives (e.g., MoPFormer exceeds prior methods by $3–6$ accuracy points and shows $13$-point drops without pretraining).
• Extreme parameter efficiency, interpretability, and geometric reliability in 3D modeling (e.g., convex splatting for robotics).
• Scalable specification, composition, and analysis of distributed and concurrent systems in software and protocol engineering.
• In group theory, the ability to realize stability, openness, and density properties in representation spaces, with applications to both theoretical and geometric problems.

A plausible implication is that further reductions in the granularity or choice of primitive features may enable new forms of invariance, modularity, and transferability in large-scale decision, modeling, and algebraic systems. This suggests that primitive state representations will continue to underlie scalable, robust, and interpretable architectures across scientific domains.