Decompositional Verification Framework

• The decompositional verification framework is an approach that divides verification tasks into smaller, manageable subproblems to ensure overall system correctness.
• It integrates component contracts, assume-guarantee reasoning, and automated toolchains to recompose local proofs into global guarantees across domains like refactoring, model checking, and reinforcement learning.
• While enhancing scalability and modularity, these frameworks balance fine-grained reasoning with global soundness, addressing trade-offs in precision and computational efficiency.

A decompositional verification framework is an approach to verification in which a verification task—whether program correctness, protocol safety, RL task satisfaction, or factuality assessment—is rigorously split into smaller subproblems, each of which can be analyzed, proven, or evaluated independently or semi-independently. Verification results for the components are then composed to yield system-level guarantees. Formal decompositional methods have been developed for code refactoring, model checking, program analysis, reinforcement learning, conformance testing, secure evaluation, and fact verification. Such frameworks balance fine-grained reasoning (to achieve tractability or manageability) against global soundness, often supporting formal or algorithmic guarantees relating component properties to system-level behavior.

1. Formal Foundations and Core Principles

Modern decompositional verification frameworks instantiate a set of core principles:

• Decomposition Operator: The system or problem is divided according to semantics of the domain—e.g., program control-flow, refactoring steps, Markov state-space, I/O transition systems, component graphs, or claim atomicity.
• Component Contracts: Each component is paired with a local specification or verification contract; in refactoring, a correctness obligation on rewrites; in model checking, assume-guarantee premises; in RL, subtask reachability or success thresholds; in I/O testing, a quotient specification.
• Compositionality Theorem: A formal result guarantees that satisfaction of all (or a subset) of component contracts entails satisfaction of the global property (possibly under side-conditions, e.g., composability, non-blocking, or assumption admissibility).
• Proof or Verification Toolchain: Semi-automatic or automatic methods generate, discharge, and/or compose proof obligations associated to the decomposition, such as symbolic execution, matching-logic reasoning, assumption learning, or simulation-based testing.

Refactoring frameworks such as that of (Horpácsi et al., 2017) model transformations as imperative sequences of "prime" refactorings, each instance of a parameterized refactoring scheme

$$S = (\mathrm{Sel}, \mathrm{Def}, \mathrm{Ref}, \mathrm{Cond})$$

with explicit element selection, rewrite rule, and side-conditions. Equivalence relations and operational semantics underpin the correctness contract, which is discharged for each step and then for the composite sequence.

Compositional model-checking frameworks, e.g., those based on assume-guarantee reasoning (Giannakopoulou et al., 2013), employ abstraction, automata learning, and compositional rules (such as ASym) to relate local and global properties.

Component-based conformance testing (Noroozi et al., 2013) introduces sufficient and strong decomposability criteria via quotient automata so that ioco-conformance at the component level lifts (and, under internal choice, characterizes) system-level correctness.

In RL, parametric MDP models coupled with Bellman-flow constraints support automatic decomposition of a global reachability property into subtask thresholds, with theorem-backed compositionality (Neary et al., 2021, Neary et al., 2023).

2. Decomposition Strategies and Architectural Patterns

Decompositional verification frameworks support a range of decomposition strategies:

• Syntactic Decomposition: In refactoring (Horpácsi et al., 2017), transformations decompose into sequential or iterative application of parameterized refactoring schemes, each matched via selectors over enriched ASTs or program graphs.
• Procedural Partitioning: Program verification frameworks decompose large formula encodings along procedural boundaries, translating cross-calling functions into call context and summary predicates for tractability (Schrammel, 2016).
• Assume-Guarantee Composition: Large models are decomposed into interacting subsystems whose behaviors are captured by learned or hypothesized assumptions (Giannakopoulou et al., 2013).
• Recomposition Map Construction: In model checking (Dardik et al., 2024), a fine-grained decomposition is recomposed into verification components and property components via a surjective map; portfolios of such maps are explored in parallel to optimize efficiency.
• Decomposition-Then-Verification in Fact Verification: In text factuality (Lu et al., 19 Mar 2025), claim decomposition via hierarchical policies or RL-augmented splitting routines is paired with atomicity metrics to align with verifier strengths.

Architectures commonly consist of a front-end for decomposition generation, abstract specification of subproblems (schemes, predicates, quotient automata, MDP subtasks), proof/genetic/learning engines for subproblem discharge, and combinatorial logic for result aggregation.

3. Formal Compositionality and Correctness Theorems

A defining feature of decompositional frameworks is the formal theorem ensuring that satisfaction of component contracts or subproblem obligations leads to satisfaction of the global specification.

• Refactoring Schemes: The correctness contract for a scheme $S$ is

$\forall \varepsilon.\;\mathrm{Cond}(p_1,p_2,\varepsilon)\;\Longrightarrow\; \bigl\langle \code{p_2},\,\env=\varepsilon\bigr\rangle \;\equiv\; \bigl\langle \code{\mathbf{begin}\;p_1\;;\;p_2\;\mathbf{end},\,\env=\varepsilon\bigr\rangle$

and sequential composition is proven via equivalence chaining (Horpácsi et al., 2017).

• Assume-Guarantee Soundness: The asymmetric rule (ASym)

$$\frac{\langle P \rangle A \langle R \rangle \quad \langle \text{true} \rangle B \langle P \rangle}{\langle \text{true} \rangle (A \|\ B) \langle R \rangle}$$

is both sound and, under P sufficiently strong, complete for finite/infinite-state systems (Giannakopoulou et al., 2013).

• Parameter Synthesis in RL: The compositionality theorem guarantees that for any policy $\pi_H$ in the high-level pMDP with parameters $\{p_c\}$, if each learned subpolicy $\pi_c$ achieves

$$\forall s \in I_c,\ \mathbb P^s_{M,\pi_c}(\Diamond_{\le T_c} F_c) \geq p_c,$$

then the overall policy in the original environment has at least the success probability predicted by the abstract model (Neary et al., 2021, Neary et al., 2023).

• ioco-Decomposability: Given an environment $E$ and system spec $S$, the sufficient condition for decomposability is $E \sqsubseteq S$ and quotient automaton $S\mathbin{/}E$ being valid and strongly non-blocking; for strong decomposability, internal-choice environment further yields necessity and sufficiency (Noroozi et al., 2013).

4. Toolchains and Automation

Semi-automatic and automatic toolchains implement decompositional verification frameworks:

• Strategic Rewriting Interpreters: For refactoring (Horpácsi et al., 2017), a static analyzer constructs enriched program graphs, prime and composite functions are executed on the graph, and, in parallel, a proof engine attempts to discharge per-step equivalence obligations via symbolic execution and matching-logic proof rules.
• Counterexample-Guided Synthesis: Second-order predicate synthesis in verification (Schrammel, 2016) uses counterexample-driven candidate refinement, partitioning global verification queries into local queries over abstract domains, with SMT backends for universal checking.
• Assumption Learning Loops: Abstraction and learning-based compositional verification (Giannakopoulou et al., 2013) interleaves predicate abstraction (may/must) with Angluin's L*-style automata learning to generate minimal assumptions, with membership/equivalence queries answered via model checking.
• Iterative RL Planning: In compositional RL (Neary et al., 2021, Neary et al., 2023), a bilinear program synthesizes minimal subtask requirements, which are iteratively re-solved as empirical estimates of subpolicy capabilities are obtained; meta-policies are derived from Bellman flow variables.
• Parallel Portfolio Model Checking: Recomposition (Dardik et al., 2024) builds a small portfolio of recomposition maps, schedules them in parallel, and employs static reduction and intermediate minimization to optimize the compositional reachability analysis.

5. Trade-offs, Limitations, and Empirical Insights

Decompositional approaches inevitably introduce trade-offs and practical limitations:

• Precision vs. Scalability: Finer decompositions increase tractability and often reduce resource usage (e.g., drop average solver time by an order of magnitude (Schrammel, 2016)), but induce over-approximation or loss of cross-component context, potentially weakening proofs or allowing spurious counterexamples. Larger subproblems preserve precision but may overwhelm solvers.
• Proof Obligation Discharge: Most frameworks discharge the majority of proof obligations mechanically but may require manual intervention for unproved obligations, rare corner cases, or semantic lemmas (as in (Horpácsi et al., 2017)).
• Component Selection Complexity: The space of recomposition maps is large, but heuristics (e.g., data-flow partial order) and parallel portfolios yield practical efficiency (Dardik et al., 2024).
• Upfront Specification Burden: In extensible frameworks (e.g., strategic refactoring), defining and proving new scheme contracts is a one-time but nontrivial cost; in conformance quotienting, subset-closure is exponential in state size (Noroozi et al., 2013).
• Domain Limitations: Most frameworks focus on deterministic, sequential models; concurrent or more complex side-effect models require further semantic machinery (Horpácsi et al., 2017).
• Empirical Results: Decomposition/abstraction has yielded orders-of-magnitude improvements in verification time and memory; fact verification and jailbreak scoring dramatically improved alignment with human evaluators by introducing decompositional atomicity (Lu et al., 19 Mar 2025, Chu et al., 28 Aug 2025).

6. Applications and Recent Developments

Decompositional verification is applied across domains:

• Program Refactoring: Automated and verified code transformations, composed from prime refactoring schemes, ensuring semantic equivalence in the presence of language features such as binding, scoping, and purity (Horpácsi et al., 2017).
• Compositional Model Checking: Verifying large-scale concurrent systems, protocols, and distributed algorithms by partitioning via assume-guarantee rules, learning minimal automata assumptions, or recomposing atomic components for reachability (Giannakopoulou et al., 2013, Dardik et al., 2024).
• Component Conformance Testing: Specification derivation for unknown or third-party components that, when passing ioco-tests for a quotient automaton, guarantee satisfaction of the overall system requirement when integrated into a known environment (Noroozi et al., 2013).
• Reinforcement Learning Task Decomposition: Synthesis and verification of RL subpolicies whose success, when composed according to a meta-planner, guarantees system-level reachability; iterative coarse-to-fine parameter tuning adapts to observed agent performance (Neary et al., 2021, Neary et al., 2023).
• Fact Verification and Evaluation: Adapting the decompositional paradigm to natural language, using dynamic, RL-based claim splitting with atomicity metrics to improve verifier confidence and accuracy, and scoring model outputs for complex tasks such as jailbreak success (Lu et al., 19 Mar 2025, Chu et al., 28 Aug 2025).

A plausible implication is that decompositional verification, while domain- and toolchain-specific, consistently yields greater scalability, modularity, and interpretability, subject to appropriate correctness theorems and contract management.

7. Future Directions

Notable research directions and open problems include:

• Dynamic Decomposition Automation: Automating property-driven or solver-aware repartitioning to balance precision and efficiency dynamically during proof search or learning, as proposed in (Schrammel, 2016).
• Broadened Semantic Coverage: Extending existing frameworks to concurrency, richer side-effect models, stochastic systems, and infinite-state settings, for example by integrating compositional fixed-point algorithms or advanced temporal logic quotienting (Horpácsi et al., 2017, Giannakopoulou et al., 2013).
• Online Assumption Generation: Integrating assumption synthesis/learning with quotienting and refinement procedures to adapt to unknown or evolving component behaviors (Noroozi et al., 2013).
• Universal and Interpretable Verification: Employing decompositional scoring and atomicity-guided splitting in high-stakes LLM evaluation and content verification to improve alignment with human assessment and support auditability (Lu et al., 19 Mar 2025, Chu et al., 28 Aug 2025).

The development and dissemination of extensible, modular, and formally justified decompositional verification frameworks remains a central thrust in both the theory and practice of scalable, trustworthy system analysis.