Box Reachability Analysis

• Box reachability is the study of ensuring that trajectories or runs remain within specified axis-aligned hyperrectangles, a critical concept in safety verification.
• It leverages diverse methodologies such as RKHS embeddings, Koopman operator lifting, and zonotope propagation to manage uncertainty and over-approximation.
• The framework bridges continuous and discrete systems by enforcing reach-avoid and stay-in-the-box constraints with provable guarantees across various control scenarios.

Box reachability denotes a family of reachability questions in which trajectories, states, or runs are constrained by axis-aligned hyperrectangles. In stochastic control, boxes appear as time-varying target or constraint tubes implemented as polyhedra or indicator functions; in nonlinear and neural verification they appear as initial sets, unsafe sets, error envelopes, and interval abstractions; in vector addition systems they define the requirement that a run from $\boldsymbol{0}$ to $\boldsymbol{v}$ remain inside $[0,\boldsymbol{v}]$; and in description-logic shape analysis they are simulated by reachability assertions plus concept inclusions over the reachable region (Thorpe et al., 2020, Bak et al., 2021, Tian et al., 2022, Almagor et al., 18 Aug 2025, Kotek et al., 2014).

1. Formal problem classes

In discrete-time stochastic control, box reachability is naturally expressed on a Markov control process $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$ over a finite horizon. A box $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$ is a special case of a convex polyhedron, so box-valued tubes fit directly into stochastic reachability formulations. Two canonical objectives are the terminal-hitting safety probability

$$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$

and the first-hitting, or reach-avoid, safety probability

$$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$

If the $\mathcal{T}_k$ or $\mathcal{K}_k$ are boxes, these become stay-in-the-box and reach-a-target-box-while-staying-in-a-constraint-box queries (Thorpe et al., 2020).

In continuous-time nonlinear verification, box reachability commonly takes the form of time-bounded safety from a box of initial conditions $$I=[x_{0l}^1,x_{0u}^1]\times\cdots\times[x_{0l}^n,x_{0u}^n]$$ against a box-like unsafe set $\boldsymbol{v}$0, under dynamics $\boldsymbol{v}$1. The Koopman-based workflow treats $\boldsymbol{v}$2 as a hyperrectangle in the original state space, lifts the dynamics to a linear observable space, and then addresses the resulting nonlinear initial-set constraints there (Bak et al., 2021).

In black-box reach-avoid for robotics, the state may be split into a workspace component $\boldsymbol{v}$3, a trajectory parameter $\boldsymbol{v}$4, and a disturbance signal $\boldsymbol{v}$5, with guarantees required over sets $\boldsymbol{v}$6. The backward reach-avoid set $\boldsymbol{v}$7 collects all initial-state and parameter pairs for which the terminal condition reaches the goal set $\boldsymbol{v}$8 and the entire trajectory avoids the obstacle set $\boldsymbol{v}$9 for all disturbances. Narrow-gap motion planning makes the box interpretation operationally important, because over-approximation error is often encoded as hyperrectangles that inflate obstacles or deflate goals (Chung et al., 2024).

These formulations share a common logical core: a target or safe region is represented by a Cartesian product of intervals, and the reachability question asks whether a trajectory, flowpipe, or run remains inside that rectangular envelope, reaches it, or avoids leaving it over a specified horizon.

2. Set representations and box operations

Across the literature, box reachability is rarely isolated as a single data structure. Instead, boxes coexist with polyhedra, zonotopes, and function-based sets. The resulting methods differ mainly in how they exploit the simplicity of hyperrectangles while controlling conservatism.

Representation	Box-related object	Role
Polyhedral tubes	Hyper-rectangles as special convex polyhedra	Constraint and target tubes
H-/AH-polytopes	Boxes as special H-polytopes	Goal, obstacle, and parameter-set computations
Interval boxes	$[0,\boldsymbol{v}]$0	Abstract states and segmentation cells
Zonotopes	Diagonal-generator interval zonotopes	Reachable-set over-approximations
$[0,\boldsymbol{v}]$1-bounding boxes	$[0,\boldsymbol{v}]$2	Tight coordinate-wise reachability bounds

In stochastic reachability, SReachTools supports polyhedral tubes and function-based tubes, so a box can be supplied either as a hyper-rectangle or through its indicator function $[0,\boldsymbol{v}]$3 (Thorpe et al., 2020). In NeuralPARC, the basic polyhedral objects are H-polytopes $[0,\boldsymbol{v}]$4 and AH-polytopes $[0,\boldsymbol{v}]$5, with boxes appearing both as special H-polytopes and as explicit modeling-error sets. For a hyperrectangle $[0,\boldsymbol{v}]$6, the paper gives closed-form Minkowski-sum and Pontryagin-difference formulas,

$[0,\boldsymbol{v}]$7

which are central to robust reach-avoid with buffered goals and obstacles (Chung et al., 2024).

In black-box RL safety layers, boxes appear as interval zonotopes. A zonotope is written $[0,\boldsymbol{v}]$8, and an axis-aligned box $[0,\boldsymbol{v}]$9 is encoded by

$\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$0

This allows box uncertainty, process noise, and Lipschitz residuals to be added by standard zonotope arithmetic while preserving a compact convex-set representation (Selim et al., 2022).

In abstraction-based neural control, the box itself is the state abstraction. Each abstract state is the interval box $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$1, obtained by discretizing each state coordinate according to a granularity vector $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$2. The reachability algorithm then propagates finite sets of such boxes and aggregates adjacent ones to control combinatorial growth (Tian et al., 2022).

In verification-friendly TLL-controlled LTI systems, the box is the output object of interest: a one-step $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$3-bounding box $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$4 must contain the exact reachable set and approximate its coordinate-wise extrema within $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$5 (Ferlez et al., 2022).

3. Data-driven and black-box system methods

A major contemporary direction is to compute box-reachability quantities without an explicit analytic model. In the stochastic setting, SReachTools augments dynamic programming with kernel mean embeddings of conditional distributions. The transition kernel $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$6 is embedded in an RKHS, the conditional embedding is learned from i.i.d. one-step samples $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$7 by a regularized least-squares problem, and the resulting reachability recursion reduces to linear operations on sample-based evaluations. Box membership enters only through indicator functions such as $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$8 and $\mathcal{H}=(\mathcal{X},\mathcal{U},Q)$9. The method admits finite sample bounds and has known convergence in probability, and a Random Fourier Features variant is used for scalability (Thorpe et al., 2020).

For nonlinear black-box systems, Koopman operator linearization converts box reachability into reachability for a linear system with nonlinear initial constraints. A box initial set $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$0 in the original state space is mapped through a dictionary of observables $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$1 to a generally nonconvex set $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$2 in observable space. The paper addresses this mismatch through a family of algorithms: Direct SMT-based encoding, Interval-based over-approximation, Hyperplane Backpropagation, and Zonotope Domain Splitting. The interval method constructs an observable-space box

$B=[a_1,b_1]\times\cdots\times[a_n,b_n]$3

that over-approximates $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$4; hyperplane backpropagation pulls unsafe linear constraints back through Koopman dynamics; and dReal is used only when the linear over-approximation is inconclusive (Bak et al., 2021).

A related black-box reachability direction learns the Hamiltonian rather than a box enclosure. Given simulator queries, one approximates $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$5 by finite differences, samples controls, forms an empirical Hamiltonian $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$6, and trains a neural Hamiltonian $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$7. The learned Hamiltonian is then inserted into an HJ solver or DeepReach. This is not organized around axis-aligned boxes, but it extends black-box reachability to reachable sets and safe controllers within the HJ framework and complements box-based abstractions in the broader verification landscape (Chilakamarri et al., 2024).

Taken together, these methods show that box reachability is no longer tied to gridding the state space. It can be expressed through RKHS expectations, Koopman-lifted linear dynamics, or learned HJ Hamiltonians, with boxes retained where they are structurally advantageous: as safe tubes, initial sets, or coarse but certifiable enclosures.

4. Neural-network controllers and verification-oriented abstractions

One line of work makes box reachability tractable by changing the controller architecture itself. In abstraction-based training for DNN-controlled systems, an abstraction layer is inserted between the input and the first hidden layer. For each coordinate,

$B=[a_1,b_1]\times\cdots\times[a_n,b_n]$8

so every concrete state is mapped to an interval box. The policy therefore becomes constant on each box. Reachability proceeds by segmenting each reachable box into abstraction-consistent boxes, querying the ANN once per box, applying a Taylor-model post operator under the fixed action, and aggregating adjacent successor boxes. The method is sound, avoids over-approximating the DNN itself, and is reported to yield time speedups up to $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$9 against Verisig 2.0 or Polar and up to $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$0–$$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$1 against ReachNN* in some instances (Tian et al., 2022).

NeuralPARC takes a different route. A ReLU trajectory model is trained from parameterized trajectory data, exact piecewise-affine regions are extracted with Reachable Polyhedral Marching, and robust reach-avoid is enforced by error hyperrectangles: $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$2 Goals are deflated by $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$3, obstacles are inflated by $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$4, and backward reach and avoid sets are computed as polyhedra in $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$5-space. The method is explicitly designed for narrow gaps, where polyhedral precision is needed but boxes remain the preferred vehicle for modeling error and parameter partitioning (Chung et al., 2024).

BRSL uses box-related reachability online for safe RL. The black-box system is modeled as $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$6, offline data are used to estimate Lipschitz constants and local affine models, and the forward reachable set is propagated as a zonotope by adding process noise, a residual zonotope, and a Lipschitz box

$$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$7

Collision checking reduces to constrained-zonotope intersection and a linear program whose optimum determines whether the intersection is empty. Because the LP is differentiated with respect to actions, the safety layer can correct unsafe RL actions while keeping the robot outside obstacle sets (Selim et al., 2022).

For LTI systems with ReLU Two-Level Lattice controllers, box reachability becomes unusually sharp. The exact one-step reachable set can be computed in polynomial time in the size of the TLL controller for fixed state dimension, the exact one-step bounding box is also computable in polynomial time, and the adaptive L-TLLBox algorithm combines semi-exact region reasoning with approximate box propagation. In the reported experiments, L-TLLBox completed reachability analysis as much as $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$8 faster than a state-of-the-art NN controller reachability tool, while producing reach boxes that were from $$\hat{p}^{\pi}_{x_0}(\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \Biggl(\prod_{i=0}^{N-1}\boldsymbol{1}_{\mathcal{T}_i}(x_i)\Biggr)\boldsymbol{1}_{\mathcal{T}_N}(x_N) \Biggr],$$9 to $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$0 times the area (Ferlez et al., 2022).

These works collectively dispel a common simplification: in neural verification, box reachability is not merely interval bound propagation. It may instead be enforced by architecture-level invariance, exact PWA region extraction, zonotopic safety layers, or specialized controller classes for which exact box bounds are tractable.

5. Discrete and logical formulations

In vector addition systems, box reachability has a precise combinatorial meaning. A $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$1-VAS is a finite set $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$2, standard reachability asks whether $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$3 as an $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$4-run, and box reachability strengthens this by requiring every prefix effect to stay below the target: $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$5 The main structural theorem states that for every $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$6-dimensional VAS $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$7, there exists an effectively computable threshold $$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$8 such that

$$p^{\pi}_{x_0}(\mathcal{K},\mathcal{T}) = \mathbb{E}_{x_0}^{\pi}\!\Biggl[ \sum_{j=0}^{N} \Biggl(\prod_{i=0}^{j-1}\boldsymbol{1}_{\mathcal{K}_i\setminus\mathcal{T}_i}(x_i)\Biggr) \boldsymbol{1}_{\mathcal{T}_j}(x_j) \Biggr].$$9

Thus, in two dimensions, ordinary reachability and box reachability differ only on finitely many points near the axes. The paper also proves that $\mathcal{T}_k$0 is semilinear in dimension $\mathcal{T}_k$1, gives strict-separation examples where a vector is reachable but not box-reachable, and shows that the threshold phenomenon fails in $\mathcal{T}_k$2-dimensional VAS and in VASS (Almagor et al., 18 Aug 2025).

The proof strategy is geometric. It relies on cones, integer cones, lattices, a deep-in-the-cone lemma, and Steinitz reordering to show that sufficiently large reachable targets admit runs that stay in the corridor between $\mathcal{T}_k$3 and the target. The reduction

$\mathcal{T}_k$4

for an associated $\mathcal{T}_k$5-dimensional VAS $\mathcal{T}_k$6 further links box reachability to ordinary semilinear reachability theory (Almagor et al., 18 Aug 2025).

In description logic, the term denotes a different but structurally analogous idea. The logic $\mathcal{T}_k$7 extends $\mathcal{T}_k$8 with reachability assertions $\mathcal{T}_k$9, where $\mathcal{K}_k$0 is a root concept, $\mathcal{K}_k$1 is the reachable region, and $\mathcal{K}_k$2 is a set of functional roles. Semantically, every element of $\mathcal{K}_k$3 must be reachable from some element of $\mathcal{K}_k$4 in the graph induced by $\mathcal{K}_k$5 on $\mathcal{K}_k$6. The paper explains that box-style properties can be simulated by combining such a reachability assertion with a concept inclusion $\mathcal{K}_k$7, thereby expressing that all reachable nodes satisfy property $\mathcal{K}_k$8. Finite satisfiability and finite implication for $\mathcal{K}_k$9 are polynomial-time reducible to finite satisfiability of $$I=[x_{0l}^1,x_{0u}^1]\times\cdots\times[x_{0l}^n,x_{0u}^n]$$0, and are therefore NEXPTIME-complete (Kotek et al., 2014).

These discrete and logical formulations emphasize that box reachability is not confined to Euclidean flowpipes. It also captures bounded-prefix behavior in counter systems and universal properties over reachable heap regions in shape analysis.

6. Guarantees, trade-offs, and recurrent distinctions

The strongest theoretical guarantees vary by model class. Kernel-embedding stochastic reachability admits finite sample bounds and convergence in probability for the learned conditional embedding and the resulting safety probabilities (Thorpe et al., 2020). Abstraction-based neural reachability provides soundness through the soundness of the post operator and the soundness of box aggregation (Tian et al., 2022). TLL-controlled LTI systems admit exact one-step reachability and exact $$I=[x_{0l}^1,x_{0u}^1]\times\cdots\times[x_{0l}^n,x_{0u}^n]$$1 bounding boxes in polynomial time for fixed state dimension (Ferlez et al., 2022). In black-box HJ reachability, the learned Hamiltonian is paired with scenario optimization and conformal prediction to obtain a probabilistically verified safe set $$I=[x_{0l}^1,x_{0u}^1]\times\cdots\times[x_{0l}^n,x_{0u}^n]$$2 (Chilakamarri et al., 2024). In 2D VAS, the threshold theorem and semilinearity provide a structural characterization rather than a local enclosure bound (Almagor et al., 18 Aug 2025).

Several distinctions recur across the literature. First, box reachability does not imply that reachable sets are propagated only as boxes. NeuralPARC uses H-/AH-polytopes for backward reach and avoid sets, while reserving hyperrectangles for modeling error and parameter partitions (Chung et al., 2024). BRSL propagates zonotopes and constrained zonotopes, with interval boxes appearing as special cases (Selim et al., 2022). Koopman-based methods similarly start from box initial and unsafe sets, but the observable-space reachable sets are handled through intervals, zonotopes, or SMT constraints rather than a pure box calculus (Bak et al., 2021). Second, box reachability need not be a primitive language feature: in $$I=[x_{0l}^1,x_{0u}^1]\times\cdots\times[x_{0l}^n,x_{0u}^n]$$3 it is expressed compositionally through reachability assertions and concept inclusions rather than by a dedicated modal operator (Kotek et al., 2014).

The main limitations are equally domain-specific. Kernel-embedding reachability has cubic sample-size scaling for the full Gram-system solve and depends on kernel bandwidth and regularization choice (Thorpe et al., 2020). Koopman-based verification inherits approximation error from the observable dictionary and may require nonlinear SMT solving, with transcendental observables leading to undecidable theories handled only through $$I=[x_{0l}^1,x_{0u}^1]\times\cdots\times[x_{0l}^n,x_{0u}^n]$$4-decidability (Bak et al., 2021). NeuralPARC relies on translation invariance in workspace and faces combinatorial growth in the number of PWA regions (Chung et al., 2024). Abstraction-based box propagation can still suffer state explosion when the abstraction is too fine or the horizon too long (Tian et al., 2022). In TLL reachability, the polynomial guarantees hold for fixed state dimension, so high-dimensional scaling remains difficult even though the controller complexity dependence is favorable (Ferlez et al., 2022).

A broad synthesis is that box reachability functions as a unifying constraint pattern rather than a single algorithm. It may mean staying inside a box tube with stochastic dynamics, verifying a nonlinear flow from a hyperrectangular initial set, robustifying learned reach-avoid with error hyperrectangles, propagating interval abstractions through closed-loop ODEs, or constraining VAS runs to the target’s own orthant box. What changes from one domain to another is the surrounding machinery—RKHS regression, Koopman lifting, PWA neural extraction, zonotopes, HJ PDEs, semilinear geometry, or description-logic reduction—while the central object remains the same: a reachability question sharpened by rectangular bounds.