Verification through Spatial Assertions (ViSA)

• Verification through Spatial Assertions (ViSA) is a framework that encodes and verifies spatial relationships using formal mathematical assertions across vision-language and cyber-physical domains.
• It employs test-time claim generation with confidence scoring and simulation-based monitoring to integrate evidence for robust, safety-critical system validation.
• By linking formal proofs, simulation traces, and real-world experiments, ViSA provides interpretable and actionable reliability metrics for automated reasoning.

Verification through Spatial Assertions (ViSA) encompasses a family of methodologies that leverage localized, mathematically precise assertions about spatial relationships or properties within a given domain—from vision–language reasoning over scene-level egocentric imagery, to formal safety verification in cyber-physical and distributed systems, to evidence attribution in document understanding. The unifying principle of ViSA approaches is the explicit encoding of spatial relationships or property-claims (assertions), their algorithmic verification—often at test or run-time—and the systematic use of these verifiable assertions as reward, guarantee, or evidence signals, thereby yielding both interpretability and robustness in automated reasoning and assurance.

1. Mathematical Foundations: Spatial Assertions

At the core of ViSA are formal spatial assertions, which encode local or relational spatial facts in a property language. In cyber-physical system verification, such as for automated vehicles, assertions are built on traffic models:

• The spatial traffic model is defined as

$$M = (E, \mathrm{Pos}, \mathrm{Res}, \mathrm{Free}, \mathrm{Size}, \mathrm{Aut}, D, \mathrm{Regions})$$

where $E$ is the set of entities, $\mathrm{Pos}$ provides their $\mathbb{R}^2$ locations, $\mathrm{Res}$ their reserved region, $\mathrm{Free}$ the unoccupied space, $\mathrm{Size}$ the footprint, $\mathrm{Aut}$ the autonomy indicator, $D$ is the Euclidean distance, and $\mathrm{Regions}$ are named segments of the space (Schwammberger et al., 2022).

• The syntax of assertions $\varphi$ includes primitive predicates, spatial modalities (e.g., $\mathrm{near}(E_i,E_j,d)$ for proximity), spatial concatenation/chop ($\varphi_1 \mathbin{\large\bowtie} \varphi_2$ for adjacent regions), reservation, and more. The semantics is provided by a satisfaction relation $M \models \varphi$.

In vision–language reasoning with world models, assertions are formulated as natural-language micro-claims $c_{j,i}$ about localized spatial events or relationships, each anchored to a specific view or frame $f_j$ and the input context $x_0$ (Jha et al., 5 Dec 2025). Each claim is then assigned a verdict (ENTAILED, CONTRADICTED, INSUFFICIENT) and a confidence score.

In higher-order spatial verification, such as over simplicial complexes, spatial assertions are captured in logics with modalities for neighbourhood and transitive reachability, enabling reasoning over group-wise or topological spatial properties (Loreti et al., 2021).

2. Algorithmic Frameworks and Verification Procedures

ViSA methodologies involve the explicit generation, formalization, and checking of spatial assertions at various system or architectural levels.

2.1. Test-Time Spatial Verification in Vision–LLMs

For spatial reasoning in vision–LLMs (VLMs) augmented with world models (such as MindJourney), the ViSA framework operates as follows (Jha et al., 5 Dec 2025):

• Frame-wise micro-claim generation: For each sampled or imagined frame $f_j$ along a world-model trajectory, a claim generator produces a set of micro-claims $C_j = \{c_{j,1}, \ldots, c_{j,K}\}$ relevant to the posed question $q$ and change between $x_0$ and $f_j$.
• Claim verification and reward computation: Each micro-claim $c_{j,i}$ is automatically assigned a verdict $v_{j,i}$ and a confidence score $\mathrm{conf}_{j,i}$. Per frame, the aggregate Evidence Quality (EQ) score is

$$\mathrm{EQ}(f_j; q) = \mathrm{Coverage}_j \times \overline{\mathrm{Confidence}}_j$$

where $$\mathrm{Coverage}_j = \frac{1}{K} \sum_{i} 1[v_{j,i} = \mathrm{ENTAILED}]$$ and $$\overline{\mathrm{Confidence}}_j = \frac{1}{K} \sum_{i} \mathrm{conf}_{j,i}$$.

• Integration with beam search: The EQ score is used as a reward in the selection of plausible trajectories and evidence frames for downstream answer prediction.

This test-time assertional verification directly targets the resolution of spatial queries by grounding necessary reasoning in explicit frame-level evidence.

2.2. Simulation-Based and Formal Verification for Cyber-Physical Systems

In cyber-physical system settings (particularly automotive), ViSA supports a corroborative framework, spanning:

• Formal model checking of spatial traffic logics: Requirements (e.g., “do not cross until a gap of at least $\ell$ exists”) are encoded as temporal-spatial assertions and verified via timed-automaton frameworks.
• Simulation-based assertion checking: For each assertion, runtime monitors classify and check invariants, pre-/post-conditions, and execution events (e.g., INV($\varphi$), EXE($\varphi, t_0$), PRE($\varphi, t_0$), POST($\varphi, t_0$)) over simulation traces. Failures are systematically reported when violations are found (Schwammberger et al., 2022).
• Three-tier V&V architecture: The ViSA approach links evidence from formal proofs (Level 1), simulation trace validation (Level 2), and real-world experiment pass/fail results (Level 3), using abstraction/refinement functions and evidence relations to couple spatial assertions across all layers.

3. Empirical Performance and Applications

The effectiveness of ViSA methodologies has been demonstrated in a range of domains:

3.1. Spatial Reasoning with World Models

Empirical analysis on scene understanding tasks reveals that ViSA yields significant accuracy gains on spatial reasoning benchmarks:

Method	SAT-Real (Top-4, γ=1)	MMSI-Bench (K=1)
Baseline InternVL3-14B	41.33%	27.16%
MindJourney-MJ	64.67%	32.72%
Random	63.33%	33.33%
ViSA	72.67%	35.80%

ViSA not only increases absolute accuracy but ensures robust scaling with the amount of evidence (increasing $K$), and corrects exploration biases present in prior heuristics. Its answer distributions are more confident and better calibrated as a result of localized micro-claim checking (Jha et al., 5 Dec 2025).

However, when world models fail to generate views capturing sufficient fine-grained spatial cues (as in MMSI-Bench), ViSA's improvements saturate, indicating a fundamental bottleneck due to limited inductive support from generative models.

3.2. Formal and Simulation-Based System Assurance

In the automotive domain, ViSA establishes assurance by verifying that spatial assertions—proved in design-time models—are systematically checked at runtime in large-scale, systematically generated scenarios and ultimately in real-world tests. Each requirement's status is tracked via "coverage" metrics per verification tier, and combined into an overall "confidence score" (Schwammberger et al., 2022).

3.3. Declarative Spatial Assertions in Imaging

In medical image analysis, model checkers like VoxLogicA leverage spatial logics to encode ViSA-style specification and verification. Here, spatial assertions (e.g., “tumor boundary voxels are surrounded by healthy tissue within $r$ mm”) are declaratively model-checked across entire 3D volumes, yielding state-of-the-art accuracy in glioblastoma segmentation while providing interpretability and replicability (Belmonte et al., 2018).

4. Integrations, Methodological Distinctions, and Limitations

4.1. Architectures and Tooling

ViSA is modular: in world model spatial reasoning, it inserts separate claim-generation and claim-verification modules within existing pipelines, retaining model-based exploration machinery but changing only the reward and evidence aggregation functions (Jha et al., 5 Dec 2025).

In formal settings, ViSA is instantiated via property specification logics (spatial traffic logic, spatial logic for simplicial complexes), assertion databases, and runtime monitors, yielding a unified pipeline for multi-tiered verification (Schwammberger et al., 2022, Loreti et al., 2021).

4.2. Methodological Implications

• Exploration balance: ViSA discourages reward of visually novel but uninformative views, in contrast to entropy-reducing or global-saliency methods.
• Traceability and interpretability: Each decision (e.g., selected evidence frame, assertion outcome) is paired with explicit, human-readable justifications (micro-claims or logical formulas).
• Upper bounds from generative fidelity: ViSA’s effectiveness is fundamentally limited by the fidelity of underlying world or simulation models—i.e., if generative renderings lack crucial spatial cues, no verification scheme can fully restore correct reasoning (Jha et al., 5 Dec 2025).

4.3. Open Problems and Extensibility

Limitations include scalability to free-form or multi-modal evidence, challenges in multi-entity or higher-dimensional assertion verification, and automated generation of assertion templates in new domains.

Potential directions involve end-to-end co-training of generative and verifier modules for enhanced spatial sensitivity, use of geometric or topological priors, and extension to probabilistic or dynamic assertions as in the logic for simplicial models (Loreti et al., 2021).

5. Related Logics, Models, and Broader Connections

A diversity of spatial assertion logics underpin ViSA instantiations:

• Spatial Logic for Simplicial Models: Enabling property specification and verification over higher-order adjacency and connectivity—covering group-wise spatial relations beyond pairwise proximity, with efficient model checking algorithms ($O(|\varphi||\mathcal{K}|)$) (Loreti et al., 2021).
• Closure-space logics for image analysis: Used in VoxLogicA for pixel- and region-wise assertions, supporting a wide variety of spatial predicates and quantitative/statistical queries (Belmonte et al., 2018).
• Traffic/spatial logics for agent-based systems: Supporting assertions over occupancy, proximity, reservation, and region membership with inference rules enforcing system invariants or transition conditions (Schwammberger et al., 2022).

These logics facilitate the precise, property-driven style that characterizes ViSA frameworks.

6. Interpretability, Confidence, and Assurance Metrics

Assurance in ViSA is not limited to accuracy or falsification rates but incorporates structured evidence across verification tiers.

• Coverage metrics: Track the fraction of proof obligations (formal), assertion/trace pairs (simulation), and test-case executions (experiment).
• Composite confidence scores: Integrate coverage with assumptions, simulation fidelity, and experimental reliability,

$$\mathrm{Confidence}(ViSA) = 1 - \prod_{i=1}^3 (1 - c_i)$$

with $c_1, c_2, c_3$ representing formal, simulation, and experimental confidences, respectively (Schwammberger et al., 2022).

• Failure feedback: Violated assertions at any level mandate re-examination or withdrawal of formal proofs.

This framework ensures spatial assertions serve as unifying monitors and oracles across models, simulations, and physical deployments, yielding a robust, systematic assurance argument.