Dependency-Aware Statement Unfolding

• Dependency-aware statement unfolding is a methodology that decomposes complex statements into acyclic graphs to explicitly model dependencies and improve query processing.
• It uses topological sorting and recursive inlining techniques to ensure each subcomponent is processed only after its prerequisites are resolved, minimizing redundant computations.
• Applications in mathematical text processing, multi-hop QA, and concurrency analysis demonstrate enhanced factual consistency, retrieval effectiveness, and system scalability.

Dependency-aware statement unfolding is a class of algorithms and representations that systematically decompose complex statements or queries into interconnected subcomponents, explicitly modeling and leveraging their dependency structure. This paradigm has found applications in several domains, including mathematical text processing, multi-hop reasoning with LLMs, advanced retrieval-augmented generation frameworks, partial-order reduction in concurrency theory, and path-sensitive program analysis. Central to these methods is the construction of an acyclic dependency graph, followed by the recursive or policy-driven unfolding of statements or sub-tasks in topological order, enabling improved reasoning, retrieval efficiency, interpretability, and knowledge reuse.

1. Formal Models and Graph Representations

Dependency-aware statement unfolding is typically instantiated as a procedure over a directed acyclic graph (DAG) $G=(V,E)$, where each node $q_i \in V$ represents a statement, sub-question, or event, and each edge $(q_i, q_j) \in E$ indicates that $q_j$ depends on the answer or resolution of $q_i$.

• In mathematical corpora, such as in Matlas, $V = \{S_1, \dots, S_n\}$ where each $S_i$ is a statement (theorem, definition, lemma), each with an explicit dependency set $\mathit{Dep}_i$; edges are $S_j \to S_i$ for $S_j$ listed in $q_i \in V$0 (Ju et al., 19 Apr 2026).
• In multi-hop QA or retrieval-augmented generation (Dep-Search, PankRAG), $q_i \in V$1 comprises sub-questions produced by explicit decomposition; edges encode prerequisite relationships (Liu et al., 26 Jan 2026, Li et al., 7 Jun 2025).
• In concurrency analysis, the graph structure is instantiated as a labeled event structure (LES) or prime event structure to model causal and conflict relations among events (Rodríguez et al., 2015, Sousa et al., 2017).

The unfolding process involves a topological sort or transitive closure to ensure that no subcomponent is processed before its dependencies are satisfied.

2. Extraction and Construction of Dependency Graphs

In large-scale mathematical text mining, dependency graphs are constructed in a multi-stage pipeline:

• Convert the source document (PDF) to a plaintext/Markdown format.
• Use language-model-powered pattern recognition to localize statement environments (e.g., theorem, definition).
• Extract both the full LaTeX content and a first-order dependency list for each statement, mapping local labels to global identifiers.
• Assemble the graph $q_i \in V$2 iteratively: for each $q_i \in V$3, map every dependency label in $q_i \in V$4 to a $q_i \in V$5 already processed, emitting an edge $q_i \in V$6.
• The process yields a document-local DAG suitable for topological unfolding (Ju et al., 19 Apr 2026).

In retrieval-augmented generation, LLMs are prompted to perform hierarchical decomposition and dependency annotation. For example, PankRAG first splits the input into parallel and sequential sub-questions, constructs a DAG, then annotates ambiguous sub-questions for rewrite and propagates resolved dependencies for planning (Li et al., 7 Jun 2025). Policy-based interactive decomposition is used in Dep-Search, where each decomposition step declares explicit dependencies as part of reasoning trace construction (Liu et al., 26 Jan 2026).

3. Unfolding Algorithms and Execution

The unfolding step recursively rewrites statements, queries, or trajectories to make all first-order dependencies explicit, yielding more self-contained representations:

• Mathematical Texts (Matlas): The unfolding proceeds in topological order across statement layers $q_i \in V$7. At each $q_i \in V$8 in layer $q_i \in V$9, all dependencies $(q_i, q_j) \in E$0 have already been unfolded. The unfolded content $(q_i, q_j) \in E$1 is then produced by recursively inlining or merging $(q_i, q_j) \in E$2 into $(q_i, q_j) \in E$3, according to rules by type (definition, theorem, etc.). All references are resolved and deduplicated, and optional formatting/post-processing is applied to preserve readability (Ju et al., 19 Apr 2026).

$q_j$1

• Multi-hop Reasoning (Dep-Search): The system emits control tokens (〈Decompose〉, 〈Retrieve〉, 〈Memory〉, 〈Conclusion〉), iteratively unfolding the query into sub-questions and carrying out memory-augmented reasoning in DAG order. Each sub-question is only addressed after its prerequisite dependencies are resolved. Persistent memory buffers and retrieval ensure that redundancy is avoided and past facts are reused efficiently (Liu et al., 26 Jan 2026).
• Graph-RAG (PankRAG): The DAG of sub-questions is first constructed by global query analysis; unfolding then proceeds by executing sub-questions in topological order, with dependency-aware retrieval and reranking. Ambiguous queries are rewritten with previously resolved answers inlined, preventing polysemy and improving retrieval fidelity (Li et al., 7 Jun 2025).

4. Dependency-Aware Retrieval, Memory, and Reranking

A core advantage of dependency-aware statement unfolding in generative models and retrieval systems is the reduction of redundant computation and the improvement of factual consistency:

• Memory Buffering (Dep-Search): Facts are written to and read from a least-recently-used (LRU) memory buffer $(q_i, q_j) \in E$4, indexed by embeddings and recency, ensuring that each fact is only retrieved once per trajectory, and available for all dependents (Liu et al., 26 Jan 2026).
• Dependency-Aware Reranking (PankRAG): Retrieved passages are reranked not only by their original retrieval score but also by semantic similarity to the concatenated answers to all predecessor sub-questions, using a normalized score $(q_i, q_j) \in E$5, where $(q_i, q_j) \in E$6 is the cosine embedding similarity, and $(q_i, q_j) \in E$7 are tuned on held-out data (Li et al., 7 Jun 2025).
• This approach suppresses hallucinations, reduces the retrieval of irrelevant or contradictory content, and ensures the logical propagation of dependencies in generated outputs.

5. Applications: Mathematical Knowledge, Multi-Hop QA, and Concurrency

Mathematical Statement Unfolding

• Matlas: Unfolded statements make mathematical results more accessible to both human and algorithmic retrieval by rendering all necessary definitions, lemmas, and prerequisite results explicit within the statement body. Average statement length expands from ~120 to ~200 words post-unfolding. This produces robust, stand-alone representations suitable for semantic search and formal reasoning (Ju et al., 19 Apr 2026).

Multi-Hop Reasoning and QA

• Dep-Search: Dependency-aware statement unfolding in complex multi-step questions ensures that all required facts are available before answering each sub-question, minimizing retrievals and maximizing answer accuracy. For example, answering “What is the capital city of the country where the author of ‘1984’ was born?” is achieved by unfolding into author lookup, birthplace retrieval, and capital city lookup in strict dependency order, each step sharing information through memory (Liu et al., 26 Jan 2026).

Retrieval-Augmented Generation

• PankRAG: Improves context recall (+19.15%) and context precision (+18.57%) over non-dependency-aware baselines and reduces hallucination rates by +25.94% in multi-hop QA benchmarks. Faithfulness in generated answers increases from ∼74% to ∼96% on SCQ tasks (Li et al., 7 Jun 2025).

Concurrency and Program Analysis

• Partial Order Reduction and Abstract Interpretation: Dependency-aware unfolding is central to state-space explosion mitigation in concurrency, using prime event structures and independence relations to avoid redundant executions and conflicting transitions. Each configuration is only unfolded after all causal predecessors, and cutoffs based on state subsumption or smaller configuration coverage prune redundant exploration (Rodríguez et al., 2015, Sousa et al., 2017).

6. Complexity, Scaling, and Readability Considerations

• Graph Construction and Unfolding: In both mathematical and QA settings, complexity is $(q_i, q_j) \in E$8, where $(q_i, q_j) \in E$9 is the number of statements and $q_j$0 the dependency edges; the actual unfolding step may exhibit quadratic blow-up in degenerate chain cases, but shallow graphs in practice impose only a 1.5-2× size blow-up on average (Ju et al., 19 Apr 2026).
• Empirical Scaling: On a corpus of 8.07 million statements, construction and unfolding require ~72 hours on a 32-GPU cluster; the persistent memory buffer in multi-hop reasoning is kept at modest size by LRU truncation (Ju et al., 19 Apr 2026, Liu et al., 26 Jan 2026).
• Readability: Inlined statements may substantially enlarge the text. Effective unfolding strategies apply post-processing for readability, retain structured formatting, and suppress inlining of basic concepts where not essential, preserving human interpretability (Ju et al., 19 Apr 2026).

7. Comparative Analysis with Baseline Frameworks

Relative to prior approaches such as traditional RAG, chain-of-thought with unstructured retrieval, or linear decomposition without dependency tracking, dependency-aware statement unfolding offers several validated advantages:

• Reduces redundant lookups and computation by topologically enforcing that no subcomponent is processed multiple times (Liu et al., 26 Jan 2026, Li et al., 7 Jun 2025).
• Improves factual accuracy and consistency via memory reuse, dependency-aware reranking, and principled graph planning (Liu et al., 26 Jan 2026, Li et al., 7 Jun 2025).
• Minimizes hallucinations and irrelevant generation by semantically conditioning each answer on all resolved dependencies (Li et al., 7 Jun 2025).
• Enables scalable, path-sensitive analysis in program verification and abstract interpretation by only expanding relevant, non-redundant configurations, with sound termination guarantees via subsumption-based cutoffs (Rodríguez et al., 2015, Sousa et al., 2017).

In summary, dependency-aware statement unfolding defines a formal, algorithmically principled methodology for decomposing and reconstructing complex statements or queries in contexts where explicit dependency modeling is essential for correctness, efficiency, and interpretability. It underpins state-of-the-art toolchains in semantic search, knowledge extraction, advanced QA, and concurrency analysis (Ju et al., 19 Apr 2026, Liu et al., 26 Jan 2026, Li et al., 7 Jun 2025, Sousa et al., 2017, Rodríguez et al., 2015).