Abstraction Dynamics in System Analysis

• Abstraction dynamics is a framework for reducing complex systems to simpler, structured representations while preserving essential trajectories or observables.
• It integrates methods from continuous dynamics, neural network learning, reinforcement learning, and symbolic control to maintain behavioral equivalence.
• The approach balances exact reductions through quotienting with adaptive, data-driven abstractions, enabling scalable analysis and controller synthesis.

Searching arXiv for the provided topic and related papers. Abstraction dynamics denotes a family of research problems in which a complex system is replaced, during analysis or learning, by a smaller or more structured representation while preserving a specified notion of behavior. In the literature represented here, the term covers exact quotient constructions for ordinary differential equations, the time-evolution of shared concept directions in neural networks, reward-free state abstractions in reinforcement learning, pushdown abstractions of program control, and data-driven symbolic abstractions for black-box and perturbed systems. Across these settings, the common question is not only what an abstraction is, but how it is formed, updated, or made exact with respect to an underlying dynamics (Vandin, 2017).

1. Exact quotienting of continuous dynamical systems

In ordinary differential equation models, abstraction dynamics is formulated as reduction of a high-dimensional system

$\dot{x}_i = f_i(x_1,\dots,x_n), \quad i=1,\dots,n,$

to a lower-dimensional one that preserves a chosen class of observables or trajectories. A language-based perspective recasts this as finding an equivalence relation on variables, closely analogous to bisimulation and partition refinement in labelled transition systems (Vandin, 2017).

Two exact notions are central. Forward Differential Equivalence (FDE) aggregates variables by block sums. For a partition $H$ of variables, each block $B$ induces a macro-variable

$y_B(t) = \sum_{x_i \in B} x_i(t),$

and $H$ is an FDE when the derivative of each $y_B$ depends only on the block sums themselves. Backward Differential Equivalence (BDE) captures redundancy: if two variables in the same block start equal, then their trajectories remain equal for all time. FDE preserves aggregate dynamics, whereas BDE preserves representative trajectories exactly, subject to compatible initial conditions (Vandin, 2017).

This exactness is illustrated by the system

$$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$

With partition $H=\{\{x_1\},\{x_2,x_3\}\}$, FDE yields the reduced variable $x_{2,3}=x_2+x_3$ and the exact reduced dynamics

$$\dot{x}_1 = -x_1,\qquad \dot{x}_{2,3} = (k_1+k_2)x_1 - x_{2,3}.$$

If $H$0, the same partition is also a BDE, and one of $H$1 can be eliminated as dynamically redundant (Vandin, 2017).

The computational framework separates two cases. For general nonlinear ODEs, the paper introduces IDOL (Intermediate Drift-Oriented Language), where derivatives are expressions built from polynomials, rational functions, minima, maxima, and absolute values. For polynomial ODEs of degree at most $H$2, it introduces reaction networks (RNs) and defines forward and backward RN bisimulations. In the RN setting, a Paige–Tarjan-style partition refinement algorithm computes the coarsest bisimulation refining an initial partition in complexity $H$3, where $H$4 is the number of variables and $H$5 the number of monomials (Vandin, 2017).

This suggests a general interpretation of abstraction dynamics in continuous systems: the abstract system is itself a dynamical system obtained by quotienting, and the abstraction map either projects trajectories to exact aggregates or identifies variables whose trajectories are provably indistinguishable.

2. Learning-time formation of abstractions in neural networks

A distinct line of work studies abstraction as a time-dependent property of learned representations. In gated deep linear networks, the central claim is that learning can be reduced exactly to coupled one-dimensional dynamics along pathways and then, under mutual diagonalizability, to decoupled scalar ODEs for singular-value modes. This reduced picture is called a pathway race: pathways shared across many tasks receive larger gradient contributions and therefore learn faster (Saxe et al., 2022).

The formal reduction begins from path-specific correlation matrices

$H$6

and a gradient-flow equation over paths. Under a simultaneous diagonalization assumption, the weight dynamics reduce to scalar equations for each singular mode $H$7,

$H$8

In routing architectures with $H$9 input domains and $B$0 output domains, hidden pathways participate in all $B$1 trained paths, whereas input and output branches participate in only $B$2 paths each. The resulting theory predicts that hidden-layer singular values become larger by a factor $B$3 than input or output singular values, matching the reported simulations (Saxe et al., 2022).

In this framework, abstraction means emergence of domain-invariant hidden representations that encode shared content while gating encodes routing. The representational similarity matrix

$B$4

is used diagnostically: abstraction is present when examples from different input domains but corresponding to the same underlying item have high similarity at the shared hidden layer. The paper further reports that when at least approximately $B$5 of input–output combinations are trained per input domain, shared pathways win the race decisively and zero-shot generalization to untrained routes becomes near-perfect (Saxe et al., 2022).

A more recent theoretical treatment studies abstraction as alignment of concept directions during training. With two binary latent factors, class centroids $B$6 define context-specific concept directions, and the abstraction score is the cosine

$B$7

In the minimal linear setting, exact solutions show that data and target geometry jointly determine end-of-learning abstraction, abstraction improves with depth, and initialization scale controls the maximum abstraction reached during training. For deep linear networks, terminal abstraction satisfies

$B$8

so deeper networks interpolate more strongly toward target abstraction. In nonlinear infinite-width networks, both erf and leaky ReLU satisfy an attenuation law

$B$9

meaning activations are less abstract than preactivations. The paper reports evidence for this law in DINOv3 and Gemma 4, and applies local GELU ablation to improve linear probe generalization in large models (Yang et al., 9 Jul 2026).

3. Reinforcement learning: state, temporal, and attractor-based abstractions

In reinforcement learning, abstraction dynamics often refers to learning coarser state spaces from trajectories. Contrastive abstraction learning assumes that successive states in a trajectory usually belong to the same abstract state. It first learns continuous state representations by InfoNCE,

$y_B(t) = \sum_{x_i \in B} x_i(t),$0

where positive pairs are sampled using sequential proximity, and then maps those representations to fixed points of a modern Hopfield network

$y_B(t) = \sum_{x_i \in B} x_i(t),$1

Each fixed point is interpreted as an abstract state. The number and granularity of abstract states are controlled by the inverse temperature $y_B(t) = \sum_{x_i \in B} x_i(t),$2, with $y_B(t) = \sum_{x_i \in B} x_i(t),$3 collapsing all states to one mean fixed point and large $y_B(t) = \sum_{x_i \in B} x_i(t),$4 approaching one attractor per stored pattern (Patil et al., 2024).

The paper reports that Laplace temporal sampling yields representations “well suited for abstraction,” whereas Gaussian sampling “does not yield clear clusters.” It further reports planning success rates of $y_B(t) = \sum_{x_i \in B} x_i(t),$5 in CifarEnv, $y_B(t) = \sum_{x_i \in B} x_i(t),$6 in Maze2D-Large, and $y_B(t) = \sum_{x_i \in B} x_i(t),$7 in Maze2D-Medium using graphs over abstract states constructed from fixed points (Patil et al., 2024). A plausible implication is that temporal coherence and attractor dynamics provide a task-agnostic route from trajectories to symbolic state clusters.

A second RL line couples temporal abstraction and state abstraction through options and successor features. Here an abstract option is defined not by a specific policy but by a successor-feature vector $y_B(t) = \sum_{x_i \in B} x_i(t),$8, and grounding an option in a new environment becomes a feature-matching problem. State abstraction is then defined by a successor homomorphism, which groups states whose option-induced transition distributions and option successor features are close. The resulting abstract $y_B(t) = \sum_{x_i \in B} x_i(t),$9-SMDP preserves option-level behavior and supports near-optimal planning across tasks sharing the same feature representation (Han et al., 2021).

A third line derives task-independent state abstraction from causal dynamics. CDL learns a sparse causal dynamics model by testing conditional mutual informations

$H$0

and uses the resulting causal graph to classify state variables into controllable descendants of action, action-relevant ancestors of controllable variables, and action-irrelevant variables. The abstraction map keeps only $H$1 and discards $H$2. The paper reports that dense models degrade strongly on out-of-distribution states, whereas CDL’s performance is essentially unchanged, and that policies trained on the abstract state are more sample efficient on downstream tasks such as Lift and Stack (Wang et al., 2022).

Finally, contrastive spatiotemporal abstraction summarizes evolving agent behavior by jointly partitioning state space and training time to maximize Jensen–Shannon divergence between abstract transition distributions. Here abstraction dynamics is literal: the abstract Markov model changes across learned temporal windows, revealing phase structure in learning histories (Bewley et al., 2022).

4. Control-theoretic and symbolic abstractions from unknown or perturbed dynamics

In control, abstraction dynamics often means constructing a symbolic transition system whose dynamics mimic those of a continuous plant closely enough to support controller synthesis. A PAC framework for unknown deterministic systems defines an $H$3-PAC approximate alternating simulation relation, in which abstract transitions match concrete successors with accuracy at least $H$4 and confidence at least $H$5. For an abstraction with $H$6 discrete states and $H$7 inputs, the required number of successor samples per transition is

$H$8

The resulting theorem states that the learned symbolic model satisfies the PAC alternating simulation relation and therefore supports probabilistic refinement of controllers back to the concrete system (Devonport et al., 2021).

For black-box piecewise affine systems, a related data-driven method combines PWA system identification, symbolic abstraction, and active sampling. The main guarantee states that for any $H$9, the learned abstraction $y_B$0 satisfies

$y_B$1

with probability at least $y_B$2, where $y_B$3 is given explicitly in terms of the identification error bound and noise level (Chen et al., 2018). Here the abstraction evolves through a loop in which model identification, abstraction refinement, and active sampling constrain one another.

For perturbed nonlinear systems with rare disturbance spikes, abstraction is enriched by disturbance edges. The risk-aware abstraction is a bimodal system $y_B$4, where $y_B$5 contains nominal transitions and $y_B$6 contains transitions reachable only under high disturbances. Resilient reactive synthesis then computes the ordinal resilience $y_B$7, where a controller is $y_B$8-resilient if it tolerates at most $y_B$9 spikes, $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$0-resilient if it tolerates any finite number of spikes, and $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$1-resilient if it tolerates infinitely many spikes. Robot examples show that the resulting controller may prefer a wider passage over a nominally shorter one because the wider route has higher resilience against disturbance edges (Samuel et al., 2020).

A complementary, goal-specific direction constructs abstractions lazily along optimal trajectories using non-uniform ellipsoidal cells. Local affine feedback laws certify one-step transitions between cells, local costs are combined into a digraph, and an RRT*-style procedure incrementally refines the abstraction where it matters. This is explicitly a multi-scale abstraction: cell volume and shape are optimization variables, and the abstraction grows backward from the target until it covers the initial set (Calbert et al., 2024).

5. Program semantics and control-stack abstractions

In programming-language semantics, abstraction dynamics refers to systematic evolution of the semantic model itself. The abstracting abstract machines methodology begins from a concrete CEK or CESK$$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$2 machine, routes recursive structures through a store, and then varies what is bounded and what is left unbounded. Finite-state AAM bounds addresses and time, thereby merging return flow. The pushdown extension instead keeps the continuation unbounded while all non-stack components remain finite, yielding exact call–return matching without moving to automata theory (Johnson et al., 2013).

The paper treats closures, garbage collection, stack inspection, and first-class composable continuations by adjusting the machine and its abstraction. For stack-inspection and garbage-collection metafunctions, recursive stack traversal is itself abstractly executed. For shift/reset, continuations become first-class values, and the analysis introduces a continuation-closure mapping and flattening to preserve soundness and termination while sacrificing some precision at capture points (Johnson et al., 2013). This suggests an “abstraction dynamics” of semantic control: the abstraction changes as new dynamic language features require new choices about which structures remain exact and which are summarized.

A related line studies dynamically generated code. Static analysis of eval must interpret possibly infinite automata of strings-to-execute. The proposed code-abstraction model constructs abstract control-flow graphs from such automata and then refines them with a semantics-driven abstraction $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$3 so that abstract code is forward-complete with respect to the chosen abstract domain. Formally, for the completed abstraction $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$4,

$$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$5

This yields analyzable abstract code even when eval ranges over infinite families of strings (Mastroeni et al., 2021).

6. Completeness, exactness, and limits

A recurrent issue across the literature is the distinction between soundness, exactness, and completeness. In ODE reduction via FDE and BDE, the reductions are exact: aggregate sums or representative trajectories are preserved without approximation error (Vandin, 2017). In Lyapunov abstraction for continuous flows, however, completeness is much more restrictive. There, a timed automaton abstraction built from partitioning functions is complete only when trajectories starting on a level set traverse adjacent level sets in exactly the same time. The paper proves that there is no complete partitioning with transversal functions, even for systems with isolated critical points, and shows that if $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$6 generates a complete abstraction then stable or unstable manifolds must lie in critical level sets of $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$7 (Wisniewski et al., 2013).

This result sharply limits one common intuition: transversality is useful for sound over-approximation, but it is incompatible with exact timing equivalence in generic nonlinear systems. Analogous limitations appear elsewhere. For RN-based ODE reduction, forward RN bisimulation is only a sufficient condition for FDE, not a characterization; some exact forward lumpings may therefore be missed (Vandin, 2017). In contrastive abstraction learning, the learned fixed points depend strongly on the temperature $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$8, and a fixed $$\dot{x}_1 = -x_1,\qquad \dot{x}_2 = k_1 x_1 - x_2,\qquad \dot{x}_3 = k_2 x_1 - x_3.$$9 may yield many more fixed points than necessary; the paper addresses this with a learned $H=\{\{x_1\},\{x_2,x_3\}\}$0-network (Patil et al., 2024). In CDL, the abstraction depends on correct recovery of the causal graph and on assumptions such as factorized next-state conditionals and full observability (Wang et al., 2022).

A plausible implication is that abstraction dynamics has two persistent poles. One pole seeks exact correspondence, often through quotienting or formal simulation relations. The other seeks adaptive or data-driven compression, where the abstraction changes as model estimates, data coverage, or training dynamics evolve. The literature represented here treats both poles as legitimate, but it also makes clear that exactness is usually available only under strong structural conditions, whereas adaptive abstraction typically trades exactness for scalability, statistical guarantees, or broader task transfer (Devonport et al., 2021).