Interval-Based Reachability Techniques

• Interval-based reachability techniques are methods that encode reachable configurations using intervals, offering either exact or over-approximated representations.
• They employ tools such as axis-aligned boxes, mixed monotonicity embeddings, and greedy dynamic programming to trade geometric precision for scalability.
• Applications span graph indexing, neural network verification, dynamical system analysis, and probabilistic models, enabling rapid query and robust certification.

Searching arXiv for papers on interval-based reachability techniques to ground the article. Interval-based reachability techniques are a family of methods that use intervals to encode either reachable configurations directly or certified over-approximations of them. In the literature, the interval object depends on the problem class: in directed graphs it can be a set of postorder identifier ranges for reachable vertices; in nonlinear dynamics and neural verification it is typically a hyper-rectangle $[\underline{x},\overline{x}]$ together with an inclusion function; in probabilistic models it is an interval-valued transition probability; and in software analysis it can be a statically inferred range for inputs or variables (Seufert et al., 2012, Meyer et al., 2019, Sproston, 2018). Taken together, these techniques trade representational simplicity and scalability against the loss of correlations that richer set representations or exact transitive-closure encodings would preserve. That trade-off is sometimes deliberate: several of the cited works explicitly target microsecond query processing, polynomial-time qualitative reasoning, differentiable toolchains, or extremely high-dimensional computation rather than maximal geometric precision (Seufert et al., 2012, Devonport et al., 2020, Harapanahalli et al., 2024).

1. Definitions and recurring abstractions

A common continuous-state formulation models uncertain sets as axis-aligned boxes. For $x,\hat x\in\mathbb{R}^n$, an interval is $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$, and an inclusion function for $f:\mathbb{R}^n\to\mathbb{R}^m$ is a map $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$ such that $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$ (Davydov et al., 2022, Harapanahalli et al., 2024). In graph reachability, the same word “interval” refers instead to contiguous ranges of node identifiers under a DFS or postorder numbering; in trees, a single interval can encode the entire reachable subtree exactly, while in DAGs a set of disjoint intervals can encode the transitive closure (Seufert et al., 2012).

The literature repeatedly distinguishes exact from approximate interval semantics. In graph indexing, exact intervals certify reachability immediately, whereas approximate intervals produce over-approximation and require recursive search (Seufert et al., 2012). In nonlinear systems and neural verification, interval reachability is also ordinarily over-approximate: the computed box is guaranteed to contain the true reachable set but may include states that are not reachable (Meyer et al., 2019, Rauh et al., 2021). In probabilistic interval models, a further distinction appears between attainability and arbitrarily close approximation: open and half-open probability intervals may allow extremal values $0$ or $1$ to be approached without being attained, and qualitative algorithms must preserve that distinction (Sproston, 2018).

A second recurring abstraction is mixed monotonicity. Several works build an embedding system on doubled variables $(\underline x,\overline x)$ or $(x,\hat x)$ so that one trajectory of the embedding yields interval bounds for an entire family of original trajectories (Davydov et al., 2022, Jafarpour et al., 2023, Harapanahalli et al., 2024). This embedding viewpoint unifies interval propagation, decomposition functions, and monotone-system reasoning.

Setting	Interval object	Representative role
DAG reachability	postorder ID ranges $x,\hat x\in\mathbb{R}^n$0	exact or approximate encoding of $x,\hat x\in\mathbb{R}^n$1
Nonlinear dynamics	state box $x,\hat x\in\mathbb{R}^n$2	reachable-set over-approximation
Neural verification	inclusion function output box	certified output/logit bounds
IMCs	interval transition probabilities	uncertainty over admissible distributions

This breadth is important because “interval-based reachability” does not name a single algorithmic lineage. It names a representational strategy that reappears in graph databases, ODE reachability, robust ML, probabilistic verification, and reactive software analysis.

2. Interval encodings for reachability in directed graphs

For finite graphs, interval-based reachability appears as a transitive-closure compression scheme. FERRARI starts from SCC condensation: for a directed graph $x,\hat x\in\mathbb{R}^n$3, one computes the condensed graph $x,\hat x\in\mathbb{R}^n$4, where $x,\hat x\in\mathbb{R}^n$5 and $x,\hat x\in\mathbb{R}^n$6; $x,\hat x\in\mathbb{R}^n$7 is a DAG, and reachability in $x,\hat x\in\mathbb{R}^n$8 and $x,\hat x\in\mathbb{R}^n$9 is equivalent (Seufert et al., 2012). A tree cover $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$0 is then built heuristically, a DFS assigns post-order numbers $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$1, and for trees the reachable set of a node is exactly one contiguous interval

$$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$2

For a general DAG, each node $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$3 receives a sorted disjoint interval set $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$4 such that

$$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$5

The difficulty is space. Exact interval labeling can require $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$6 space in dense graphs, so FERRARI formulates a $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$7-interval cover problem under an explicit budget $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$8, with $$[x,\hat x] := \{z\in\mathbb{R}^n \mid x \le \hat x\}$$9 (Seufert et al., 2012). The goal is to merge adjacent exact intervals into a cover of at most $f:\mathbb{R}^n\to\mathbb{R}^m$0 intervals while minimizing the amount of approximate content, measured as the number of identifiers lying inside approximate intervals. This creates a mixed label $f:\mathbb{R}^n\to\mathbb{R}^m$1 with exact and approximate entries. Query semantics are three-valued at the label level: if $f:\mathbb{R}^n\to\mathbb{R}^m$2 is in no interval, the answer is definitely negative; if it is in an exact interval, the answer is definitely positive; if it is in an approximate interval, the answer is unresolved and recursive exploration of successors is required.

The paper gives both an optimal dynamic programming formulation and a greedy gap-preservation heuristic. The key combinatorial reformulation treats the spaces between exact intervals as gaps $f:\mathbb{R}^n\to\mathbb{R}^m$3; preserving at most $f:\mathbb{R}^n\to\mathbb{R}^m$4 gaps induces a $f:\mathbb{R}^n\to\mathbb{R}^m$5-interval cover, and the optimization minimizes the approximate mass of the resulting merged intervals (Seufert et al., 2012). In practice, the implementation uses the greedy variant. Two budget policies are distinguished: FERRARI-L enforces $f:\mathbb{R}^n\to\mathbb{R}^m$6 at every node, while FERRARI-G enforces only the global constraint $f:\mathbb{R}^n\to\mathbb{R}^m$7 via deferred merging and a heap.

The querying side mirrors the index semantics. A top-level label probe is $f:\mathbb{R}^n\to\mathbb{R}^m$8 via binary search, followed when necessary by a DFS-like recursive search guided by interval exclusion, topological-order pruning $f:\mathbb{R}^n\to\mathbb{R}^m$9, effective topological levels, and seed-based pruning (Seufert et al., 2012). Empirically, the method reaches microsecond-level query times on large graphs, including web-scale datasets such as Twitter and Web-UK, and FERRARI-G is consistently among the fastest methods on large and web graphs while remaining smaller or more scalable than several competitors (Seufert et al., 2012).

A common misconception is that interval labeling in graphs is inherently exact. The graph-indexing literature shows the opposite: exact interval sets are often too large, and the practical problem is to decide which exact intervals should be preserved and which should be merged into approximate ranges under a memory budget (Seufert et al., 2012).

3. Boxes, growth bounds, and input–output intervals in dynamical systems

For nonlinear dynamical systems, interval-based reachability usually means computing a box that contains all states or outputs reachable from an initial box and an input box. TIRA formalizes this for both continuous-time systems $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$0 and discrete-time systems $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$1, with reachable sets $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$2 and interval over-approximations $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$3 (Meyer et al., 2019). The current implementation contains four methods: a continuous-time contraction/growth bound method, continuous-time mixed-monotonicity, sampled-data mixed-monotonicity for continuous-time systems, and discrete-time mixed-monotonicity. The paper emphasizes that interval methods prioritize simplicity and scalability, not the geometric fidelity of richer set representations such as zonotopes, support functions, Taylor models, or polytopes (Meyer et al., 2019).

The contraction/growth bound method is representative of one major design pattern. For additive-input systems $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$4, one chooses the center $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$5 and half-width $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$6, simulates the center trajectory $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$7, and combines it with a growth bound $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$8 derived from a componentwise contraction/growth matrix $$F=\begin{bmatrix}\underline F\ \overline F\end{bmatrix}$$9 to obtain

$$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$0

Mixed-monotonicity methods instead build a duplicated system in $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$1 dimensions using a decomposition function and then simulate it once from $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$2 to obtain lower and upper bounds (Meyer et al., 2019).

PIRK pushes this interval perspective toward extreme dimensions. It considers continuous-time ODEs $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$3 and implements three interval-based simulation methods: contraction/growth bounds, mixed monotonicity, and Monte Carlo sampling with probabilistic guarantees (Devonport et al., 2020). The main computational claim is scalability: PIRK has been tested on systems with state dimensions ranging from ten up to 4 billion, and the reported experiments include a traffic network with 4 billion states on an AWS c5.24xlarge in approximately 44.7 minutes, a quadrotor swarm with 3.6 billion states in 32 minutes, and a $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$4 heat-diffusion discretization with $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$5 states in approximately 350.2 minutes (Devonport et al., 2020). The method’s simplicity is structural: intervals require only lower and upper bound vectors, and the RK4 implementation is parallelized across state dimensions via OpenCL and pFaces.

A related but distinct interval construction appears in input–output reachability via Chen–Fliess series. For a Chen–Fliess operator $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$6 and an input constraint set $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$7, the reachable output set at time $$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$8 is

$$f([x,\hat x]) \subseteq [\underline F(x,\hat x),\overline F(x,\hat x)]$$9

and the paper derives a closed-form interval over-approximation by bounding each iterated integral $0$0 through interval arithmetic (Avellaneda et al., 2023). The central theorem states that if $0$1 takes values in $0$2, then

$0$3

where $0$4 and $0$5 are new formal power series obtained by interval arithmetic on the original coefficients and the interval power $0$6 (Avellaneda et al., 2023). This is notable because it avoids simulation of an embedding system or optimization over trajectories.

Fractional-order systems require yet another interval specialization. For commensurate Caputo systems of order $0$7, the paper on fractional-order reachability uses a Mittag-Leffler state enclosure

$0$8

and an iterative update of interval parameters $0$9 to enclose the state, together with interval extensions of the Mittag-Leffler function and explicit treatment of temporal truncation errors induced by the infinite-memory character of fractional derivatives (Rauh et al., 2021). The method is demonstrated on a nonlinear scalar example and a simplified fractional-order battery model with uncertain initial states and parameters (Rauh et al., 2021).

These works collectively show that interval reachability in continuous dynamics is not limited to one-step interval arithmetic. It includes growth-bound enclosures, decomposition-based embeddings, intervalized series expansions, and special-function enclosures for memoryful dynamics.

4. Sensitivity bounds, mixed monotone embeddings, and computational toolchains

A major strand of interval-based reachability replaces direct propagation of boxes through the original system by propagation through derivative information or auxiliary embeddings. One route uses sensitivities. For $1$0, the first-order sensitivity $1$1 satisfies $1$2, while the second-order sensitivity $1$3 satisfies

$1$4

with $1$5 and $1$6 (Meyer et al., 2019). The second-order sensitivity paper first encloses $1$7 and $1$8 by interval arithmetic, then uses a finite sample set $1$9 and a dispersion-dependent error term

$(\underline x,\overline x)$0

to turn sampled values of $(\underline x,\overline x)$1 into a guaranteed interval enclosure of the full sensitivity set (Meyer et al., 2019). The paper states that, unlike existing methods relying only on the first-order sensitivity matrix, this approach provides guaranteed over-approximations of the first-order sensitivity and can also provide such over-approximations with an arbitrary precision by increasing the number of samples (Meyer et al., 2019). It then plugs these sensitivity bounds into a mixed-monotone decomposition to over-approximate the reachable set.

A second route uses inclusion functions and mixed monotone embeddings directly at the vector-field level. immrax formulates interval analysis and mixed monotone interval reachability as JAX function transforms. It defines natural inclusion functions, Jacobian-based inclusion functions, and a mixed Jacobian-based inclusion function based on a permutation $(\underline x,\overline x)$2 and partial centering, then constructs an embedding system $(\underline x,\overline x)$3 on doubled state variables $(\underline x,\overline x)$4 so that any original trajectory starting in $(\underline x,\overline x)$5 remains inside the simulated interval $(\underline x,\overline x)$6 (Harapanahalli et al., 2024). The implementation detail matters: Interval is a JAX pytree, transforms such as natif, jacif, mjacif, ifemb, natemb, jacemb, and mjacemb are composable with jit, vmap, and autodiff, and the toolbox is fully parallelizable and differentiable (Harapanahalli et al., 2024). In the vehicle case study with 625 partitions, the reported runtimes were 27.1 s for ReachMM, 4.6 s for immrax Euler CPU, 0.095 s for immrax Euler GPU, 27.9 s for immrax Tsit5 CPU, and 0.588 s for immrax Tsit5 GPU (Harapanahalli et al., 2024).

The same embedding logic appears in neural-control verification. For nonlinear continuous-time systems with neural network controllers, one starts from an open-loop decomposition function $(\underline x,\overline x)$7 and an inclusion function for the controller $(\underline x,\overline x)$8, then defines a closed-loop embedding in $(\underline x,\overline x)$9 whose solution $(x,\hat x)$0 over-approximates the reachable set of the neural closed loop (Jafarpour et al., 2023). The paper distinguishes global, local, and hybrid embeddings, and for ReLU networks proves the ordering

$(x,\hat x)$1

so local embeddings are tightest and global embeddings are fastest (Jafarpour et al., 2023). This is an explicit example of how interval tightness can be tuned by choosing how much local information is injected into the inclusion function.

A plausible implication is that interval reachability has become increasingly modular. Several of these papers separate the problem into a plant-side abstraction, a controller- or nonlinearity-side inclusion function, and an execution substrate that can be parallelized or differentiated (Jafarpour et al., 2023, Harapanahalli et al., 2024). That modularity is one reason intervals remain attractive despite their geometric conservatism.

5. Neural robustness, implicit networks, and optimization-based safe control

In neural verification, interval reachability is often phrased as output certification over an input box. For a neural network $(x,\hat x)$2 and box $(x,\hat x)$3, interval reachability asks for a box $(x,\hat x)$4 such that $(x,\hat x)$5 (Davydov et al., 2022). The paper on implicit neural networks makes the comparison to Lipschitz-based certification explicit: if $(x,\hat x)$6 is the tight inclusion function of a continuous $(x,\hat x)$7, then

$(x,\hat x)$8

so tight inclusion functions provide at least as tight, and typically much tighter, robustness information than any Lipschitz bound on the same region (Davydov et al., 2022). The paper then develops a mixed-monotone plus contraction-based inclusion function for implicit neural networks of the form

$(x,\hat x)$9

under the contraction condition $x,\hat x\in\mathbb{R}^n$00, yielding an inclusion function

$x,\hat x\in\mathbb{R}^n$01

for the input–output map $x,\hat x\in\mathbb{R}^n$02 (Davydov et al., 2022). The paper also proves that, for FFNNs encoded as INNs, the construction reduces exactly to standard IBP, while for weight-tied infinite-depth models it is sharper than IBP (Davydov et al., 2022).

The same paper integrates interval reachability directly into robust training. For MNIST with $x,\hat x\in\mathbb{R}^n$03, it reports certified robust accuracy of 83.13% for INNs trained with mixed monotonicity–based robust training and 79.26% for 5-layer FFNNs trained with IBP-based robust training; for empirical robustness under PGD attack, it reports 8.04% for INNs without MM and 85.84% for INNs with MM (Davydov et al., 2022). These are not generic properties of interval methods, but they show that interval reachability can serve not only as a post hoc certificate but also as a training primitive.

A closely related development concerns systems with optimization-based controllers. For a control-affine plant

$x,\hat x\in\mathbb{R}^n$04

the CBF-based safety filter is defined by a strongly convex parametric optimization problem that computes a corrective input $x,\hat x\in\mathbb{R}^n$05 subject to performance constraints $x,\hat x\in\mathbb{R}^n$06 and CBF inequalities (Ajeyemi et al., 11 Apr 2025). The interval reachability method proposed there first approximates $x,\hat x\in\mathbb{R}^n$07 by a feedforward neural network $x,\hat x\in\mathbb{R}^n$08, then uses NN verification bounds of CROWN type,

$x,\hat x\in\mathbb{R}^n$09

inside a mixed-monotone embedding for the NN-controlled system (Ajeyemi et al., 11 Apr 2025). A “closeness of solutions” theorem bounds the distance between trajectories of the optimization-based closed loop and its NN surrogate; a final Minkowski-sum enlargement then yields an over-approximation of the original filtered system: $x,\hat x\in\mathbb{R}^n$10 In the reported experiments, reachable sets for 12 initial boxes in a 2D integrator example were computed in about 2.15 s, with Minkowski expansions taking about 0.02 s, and for 9 initial boxes in a double-integrator example the interval trajectories took about 2.7 s, with expansions about 0.015 s (Ajeyemi et al., 11 Apr 2025).

An objective reading of these neural and controller-oriented works shows a consistent pattern: interval reachability becomes especially useful when exact closed-loop analysis is obstructed by implicit fixed points, highly nonlinear controllers, or online optimization, and the interval enclosure serves as the certifying interface between a hard inner component and a tractable outer analysis (Davydov et al., 2022, Ajeyemi et al., 11 Apr 2025).

6. Probabilistic, symbolic, and software-oriented interval reachability

Interval-based reachability also appears in discrete probabilistic verification. An interval Markov chain is $x,\hat x\in\mathbb{R}^n$11, where $x,\hat x\in\mathbb{R}^n$12 assigns open, closed, or half-open intervals in $x,\hat x\in\mathbb{R}^n$13 to transition probabilities (Sproston, 2018). The paper studies qualitative reachability problems asking whether the optimal minimum or maximum probability of reaching a target set is exactly 0 or 1, under both UMC semantics, where a concrete DTMC is chosen once and for all, and IMDP semantics, where a scheduler may pick a new admissible distribution at every visit (Sproston, 2018). The core technical point is that open intervals matter. The methods do not rely on the closure of open intervals, and they can characterize situations in which probability 0 or 1 can be attained not exactly but arbitrarily closely (Sproston, 2018). Polynomial-time algorithms are obtained by symbolic predecessor operators based on valid edge sets, largeness conditions, and, for IMDP almost-sure reasoning, IMC-level end components.

A different symbolic use of intervals appears in one-dimensional piecewise affine maps. The PAM paper studies reachability on $x,\hat x\in\mathbb{R}^n$14 or $x,\hat x\in\mathbb{R}^n$15 under maps $x,\hat x\in\mathbb{R}^n$16 on rational subintervals $x,\hat x\in\mathbb{R}^n$17 (Kurganskyy et al., 2015). The general one-dimensional point-to-point reachability problem remains open even for two intervals, but the paper proves decidability for two subclasses by combining interval partitions with $x,\hat x\in\mathbb{R}^n$18-adic norms and invariant-density arguments (Kurganskyy et al., 2015). The interval structure does not over-approximate a continuous state tube here; rather, it determines the symbolic branch sequence that governs arithmetic growth of orbit denominators. This is another reminder that interval-based reachability is not synonymous with hyper-rectangular set propagation.

Reactive software analysis uses yet another variant. RERS-Fuzz employs LLVM interval analysis to identify ranges of input consumed by RERS to go into deeper program segments and then passes those ranges to a parametric fuzz engine, which performs crossover and mutation over those restricted ranges (Chowdhury, 2019). The method targets error reachability in sequential and industrial RERS 2019 benchmarks, combines branch-pair coverage with state coverage over global-variable valuations, and reports counterexamples for 217 academic reachability benchmarks and 2038 points in the industrial reachability track (Chowdhury, 2019). The interval component is deliberately lightweight: it does not prove unreachability, but biases greybox search toward deeper control-flow regions.

Taken together with the Chen–Fliess input–output method (Avellaneda et al., 2023), these works show that interval-based reachability extends well beyond state-space ODE verification. It includes qualitative probabilistic reachability, arithmetic orbit analysis, and software test-generation heuristics, with the meaning of “interval” shifting from box geometry to probability uncertainty, symbolic branch domains, or feasible input ranges.

7. Limitations, misconceptions, and directions of development

A recurrent limitation is conservatism. TIRA states directly that interval-based methods trade off the accuracy of the over-approximations for the simplicity and scalability of the reachability analysis (Meyer et al., 2019). PIRK makes the same trade-off in exchange for extreme dimensional scalability (Devonport et al., 2020). For neural and control settings, the box representation ignores anisotropy and variable correlations; this is why partitioning, local embeddings, mixed Jacobian constructions, or verifier-generated affine bounds are repeatedly introduced to sharpen intervals (Jafarpour et al., 2023, Harapanahalli et al., 2024, Ajeyemi et al., 11 Apr 2025).

Another limitation is that several methods are static or offline. FERRARI is designed as a static index; graph changes require partial or full recomputation (Seufert et al., 2012). The fractional-order method requires interval bounds on Jacobians, second derivatives, and Mittag-Leffler evaluations, and its reported examples are low-dimensional (Rauh et al., 2021). The second-order sensitivity method requires bounds on $x,\hat x\in\mathbb{R}^n$19 and $x,\hat x\in\mathbb{R}^n$20, and the sampling budget needed for small dispersion grows rapidly with dimension (Meyer et al., 2019). RERS-Fuzz explicitly cannot generate proofs for unreachability of error locations (Chowdhury, 2019).

Several common misconceptions are corrected by the literature. First, interval methods are not invariably crude: for monotone or sign-stable mixed-monotone systems, TIRA notes that some methods return the unique tight interval over-approximation (Meyer et al., 2019); for graph reachability, exact interval sets exist but may be too large (Seufert et al., 2012). Second, open interval endpoints are not a notational detail: in open interval Markov chains they can change whether probability 0 or 1 is attainable, and replacing an open IMC by its closure can change the qualitative answer (Sproston, 2018). Third, interval reachability is not restricted to feedforward layerwise propagation: it now includes contraction-based fixed-point embeddings for INNs, differentiable JAX transforms, and NN-assisted analysis of optimization-based safe controllers (Davydov et al., 2022, Harapanahalli et al., 2024, Ajeyemi et al., 11 Apr 2025).

The research directions named in the sources are correspondingly diverse. FERRARI suggests better tree-cover construction and extra compression on interval lists (Seufert et al., 2012). The fractional-order work points to optimal step-size control, oscillatory systems, complex-valued transformations, and state estimation (Rauh et al., 2021). immrax indicates a broader agenda of learning-enabled control, GPU-accelerated partitioning, and gradient-based design through reachability (Harapanahalli et al., 2024). The CBF-oriented neural method suggests disturbances, controller design using reachability and CBFs together, and higher-dimensional scaling (Ajeyemi et al., 11 Apr 2025). This suggests that interval-based reachability is best understood not as a finished technique but as a durable design principle: reduce a hard reachability object to interval data structures that admit exact semantics where possible, certified over-approximation where necessary, and aggressive computational acceleration wherever the geometry would otherwise be intractable.