Decomposition-and-Verification Framework

• Decomposition-and-Verification Framework is a structured approach that splits complex verification tasks into modular subproblems, ensuring global correctness through recomposition.
• It employs formal mathematical foundations and tailored decomposition methods—such as hierarchical, scenario-based, and learning-driven strategies—to enhance scalability and precision.
• The framework leverages optimization techniques, including bilevel and reinforcement learning, to dynamically adapt subtask policies and improve verification confidence.

A decomposition-and-verification framework organizes a complex verification problem into modular subproblems through systematic decomposition, then verifies the correctness of each subproblem and reunites their results to establish a global guarantee. Across domains—factual claim verification, reinforcement learning, formal methods, program synthesis, and safety assurance—these frameworks balance scalability, modularity, and soundness by explicitly structuring problem division, subproblem interface definition, and evidence aggregation, sometimes leveraging formal optimization or learning-based components.

1. Mathematical Foundation and Problem Formalization

Decomposition-and-verification frameworks are underpinned by formal mathematical definitions that dictate both the decomposition process and the nature of verification.

• Factual Claim Verification: The decomposition process seeks a policy $\pi$ that breaks a complex claim $c$ into subclaims optimizing for downstream verifiability. The objective is formalized as a bilevel optimization:

$$\min_{\pi}\; \mathbb{E}_{(c,y)} \Bigl[ \mathcal{L}_{V}(V(\pi(c)), y) \Bigr] \quad\text{s.t.}\quad A(\pi(c)) \le \alpha$$

where $A(\cdot)$ is an atomicity metric—specifically $A(c) = \log_2 N(c)$ for $N(c)$ atomic facts in $c$—and $V(\cdot)$ is the verifier's output (Lu et al., 19 Mar 2025).

• Reinforcement Learning (Compositional Verification): The global specification (e.g., reach a goal set with probability at least $1-\delta$) is expressed in pMDP or POMDP terms, automatically allocated to subsystem probability budgets via a bilinear program (Neary et al., 2021):

$\min \sum_{c \in C} p_c$

subject to flow constraints, yielding local subtask specs and guaranteeing global compositionality.

• Model Checking: Systems are decomposed into LTS components (e.g., $C_1, \ldots, C_n$), then recomposed via a recomposition map for efficient verification, maintaining invariant relationships to properties of interest (Dardik et al., 2024).
• Neural Network Verification: The original monolithic constraint system is split into decoupled "blocks" (e.g., via Lagrangian decomposition), with consistency maintained via dual variables:

$$\mathcal{L}(z, \hat{z}, \lambda) = \hat z_{A,n} + \sum_{k=1}^{n-1} \lambda_k^\top (\hat z_{B,k} - \hat z_{A,k}) + \cdots$$

allowing parallelized and scalable verification while preserving bound tightness (Bunel et al., 2020, Palma et al., 2021).

2. Decomposition Methodologies and Mechanisms

Different frameworks operationalize decomposition using tailored approaches suited to their domain and verification objective.

• Hierarchical and Atomicity-Driven Decomposition: In claim verification, decomposition algorithms may be RL-trained to choose decomposition points adaptively, calibrating subclaim atomicity to maximize verifier performance while observing atomicity constraints (Lu et al., 19 Mar 2025).
• Policy/Program-Guided Subproblem Construction: Table-based fact verification employs weakly-supervised semantic parsing to synthesize operator programs, which are then used to guide the decomposition into subproblems. This pseudo-label-aided approach enables domain-specific control over subproblem boundaries and types (conjunction, comparative, etc.) (Yang et al., 2021).
• Scenario- and Module-Based Decomposition (Hardware/Firmware): Verification frameworks such as HIVE decompose large systems by test scenario, then further at the module level, using static and dynamic analysis to tailor verification hints and constraints for each scenario, thus scaling formal proofs (Jayasena et al., 2023).
• Proof and Code Decomposition: In code verification, complex methods containing nested loops are dissected via transient refactoring—extracting each loop into an auxiliary method—enabling isolated subproofs. The recomposition phase restores the original code and proof context (Wang et al., 29 Oct 2025).
• Abstract-State and Learning-Driven Decomposition: For infinite-state systems, abstractions (predicate abstraction) enable decomposition into abstract finite-state subsystems; automated assumption learning (e.g., L*) permits assume-guarantee verification compositions even where component behaviors are partially unknown (Giannakopoulou et al., 2013).

3. Interface Definition, Subproblem Verification, and Compositional Soundness

Sound interface definition—the contract between subproblems—is central to compositional correctness and modularity.

• Subtask Contracting: For RL, each subsystem $c$ is assigned a tuple $(I_c, F_c, T_c, \mu_c)$: entry condition, success set, time horizon, and policy. The main theorem states that if each subsystem meets its success probability budget, then the composed policy meets the overall task's target with formal compositionality (Neary et al., 2021, Neary et al., 2023).
• Verification Aggregation: In factuality frameworks, the final decision aggregates subclaim verifications, often via conjunction. However, advanced frameworks integrate additional contextualization (e.g., DnDScore— NewScore blends atomic and decontextualized judgments to mitigate ambiguity) (Wanner et al., 2024).
• Reconciliation Procedures: Distributed decision procedures reconcile solutions to loosely coupled partitions with Craig interpolation, iteratively refining global constraints until consistency or infeasibility is achieved (Hamadi et al., 2011).
• Proof Composition: In refactoring verification, each prime refactoring instance is proven against an operational semantics; correctness of the composite follows from the correct assembly of these verified blocks through established proof combinators (Horpácsi et al., 2017).

4. Optimization, Learning, and Adaptivity

State-of-the-art decomposition-and-verification frameworks frequently employ bilevel or RL-based optimization to adapt the decomposition process dynamically to the requirements of verification or system characteristics.

• Bilevel and RL Optimization for Decomposition: Dynamic decomposition learns a policy that maximally aligns subclaim atomicity to verifier preference using PPO, with RL rewards based on the verifier’s confidence. The overall bi-level optimization is strongly NP-hard, but tractable approximations with neural policy learning provide substantial empirical gains in both confidence and accuracy (+0.07/+0.12 improvements on synthetic datasets over static policies) (Lu et al., 19 Mar 2025).
• Automated Budget Revision and Adaptivity: RL decomposition frameworks iteratively revise subtask probability budgets and policies based on empirical training success, reallocating resources to harder or more important subtasks as needed, enabling robust performance even if some subtasks underperform (Neary et al., 2021, Neary et al., 2023).
• Heuristics and Portfolio Approaches: In compositional verification for model checking, multiple recomposition maps are heuristically generated and tried in parallel, with the fastest successful verification dictating the final result. Heuristics are informed by action alphabets, data-flow, and static reduction (Dardik et al., 2024).

5. Empirical Evaluation and Comparative Impact

Decomposition-and-verification frameworks are consistently shown to yield marked improvements in scalability, verification efficiency, and accuracy across diverse domains.

• Performance Benchmarks: RL compositional methods reduce sample complexity by an order of magnitude compared to monolithic RL (1.5M vs. 30M steps for gridworld/labyrinth); compositional predictions closely match empirical success rates (±2%) (Neary et al., 2021).
• Verification Confidence and Accuracy: RL-trained dynamic decomposition for claim verification improves downstream verification confidence and accuracy (average gains of 0.07 and 0.12, respectively, versus static decomposition) (Lu et al., 19 Mar 2025).
• State Explosion Mitigation in Hardware/Firmware: Scenario-based decomposition combined with automated hint extraction enables >80% state-space reduction, validating >70% of hints, and detecting complex bugs in SoCs in tractable time/memory settings (Jayasena et al., 2023).
• Empirical Superiority in Model Checking: Portfolio-driven recomposition reduces exploration time and state counts by 10–100× over monolithic runs in large distributed protocol benchmarks (Dardik et al., 2024).
• Verification of Modular Refactoring: Decomposition of code and proofs in program synthesis leverages modularity and LLM integration, verifying 86% of complex code tasks versus 68% for baseline approaches, with the most pronounced improvements on deep-nested or specification-misaligned methods (Wang et al., 29 Oct 2025).

6. Limitations, Open Problems, and Future Directions

Despite their successes, decomposition-and-verification frameworks are subject to key limitations and active areas for further research.

• Limited Scope of Subproblem Qualities: Current frameworks often optimize solely for atomicity, omitting other subclaim properties such as coverage or semantic cohesion. Future work may integrate multi-objective rewards and jointly optimize decomposer and verifier modules (Lu et al., 19 Mar 2025).
• Precision Tradeoffs: In logical system verification, decomposition can introduce over- and under-approximation errors, potentially reducing proof precision. Adaptive partitioning, selective inlining, and mixed fixed-point computations are deployed to mitigate, but not eliminate, these losses (Schrammel, 2016).
• Heuristic Dependence and Generalization: Many frameworks rely on heuristics for component selection or scenario definition. Robustness across domains and problem scales remains an open challenge (Dardik et al., 2024).
• Human Intervention Requirements: In proof-level decomposition with LLMs, user guidance is sometimes required to resolve difficult proof obligations or align proof search, and code restoration may involve minor manual mapping in complex transformations (Wang et al., 29 Oct 2025).
• Formalization Gaps in Assurance Cases: In assurance for complex AI systems (e.g., self-driving vehicles), decomposition supports traceability and coverage but lacks formal quantitative measures of confidence—prompting calls for integration with probabilistic assurance models (Chen et al., 30 Sep 2025).

Further investigation into automated, multi-objective, formally sound decomposition strategies stands to yield more robust, interpretable, and scalable verification workflows across a diversity of computational systems.