Dependency Graph–Driven Formalization

• Dependency graph–driven formalization is a method that uses explicit, directed graphs to represent entities and their interdependencies across proofs, code, and language.
• It employs static and semantic analysis to extract, prune, and synthesize dependency graphs, ensuring modular reasoning and computational efficiency.
• This approach underpins automation in proof formalization, code generation, and package management, yielding measurable gains in compilation success and semantic accuracy.

A dependency graph–driven formalization is a methodology that places explicit, formalized dependency graphs at the center of representing, analyzing, and synthesizing mathematical, logical, linguistic, or computational objects. By making both the entities and their interdependencies explicit as directed graphs (often DAGs or hypergraphs, possibly labeled), this paradigm underpins a spectrum of advanced tools and workflows: proof autoformalization, code and system synthesis, natural language processing, knowledge visualization, package management, and adaptive machine learning. The formalism enables rigorous dependency tracking, modular reasoning, computational efficiency, and principled automation.

1. Core Models and Definitions of Dependency Graphs

Formalization by dependency graphs requires precise definitions of nodes, edges, and related structures, adapted to the domain in question:

• Term/Proof Dependency (Theorem-Proving). In Coq and similar systems, two canonical graphs emerge:
  • DG–1: The "lemma-to-lemma" (or "term") graph $G_1 = (V_1, E_1)$ where $V_1$ is the set of Coq objects invoked in proving a theorem $T$, and $(u \to v) \in E_1$ iff $u$ invokes $v$ in its proof. Built via proof-term analysis and transitive reduction, this graph captures immediate logical structure (Heras et al., 2014).
  • DG–2: The "file-to-file" or "library" import graph $G_2 = (V_2, E_2)$ with $V_2$ the set of source files and $(\ell_1 \to \ell_2) \in E_2$ iff "Require Import $\ell_2$" occurs in $\ell_1$.
• Mathematical/Proof Decomposition (LLM Formalization). In ProofFlow and Aria, a natural-language statement or proof is mapped to a DAG $G=(V,E)$:
  • $V$ is partitioned (ProofFlow) as $V_{TC}$ (Theorem Conditions), $V_D$ (Definitions), $V_L$ (Lemmas/intermediate steps), $V_{TS}$ (Theorem Solutions).
  • $E \subseteq V \times V$ encodes which statements (nodes) are required to prove others. Strict acyclicity and accompanied context constraints ensure logical faithfulness (Cabral et al., 13 Oct 2025, Wang et al., 6 Oct 2025).
• Component/Data-Flow Graphs (System Verification, UML).
  • Nodes correspond to components, channels, or data entities; edges to data-flow, dependency, or control relationships (Spichkova, 2014, Mao et al., 5 Aug 2025).
  • In software, nodes representing functions, interaction fragments, or APIs are enriched with data production/consumption sets, and edges become typed triples $(s,d,t)$ indicating $s$ produces and $t$ consumes data entity $d$.
• Linguistic/Semantic Dependency Graphs.
  • In NLP, sentences are parsed into graphs $G=(V,E)$ over tokens (nodes), dependencies (labeled arcs), with extended treatments for reentrancy and cycles (Ezquerro et al., 2024, Noravesh et al., 2 Jan 2025).
• Knowledge Visualization (Mathematical Exposition).
  • Statements, definitions, and theorems are nodes; explicit "uses" and "proves" relations become graph edges, rendered in visualizable formats (DOT, TikZ) (Uskuplu et al., 16 Dec 2025).
• Package Management.
  • Packages at specific versions are nodes in a directed hypergraph; edges are dependency relations with constraints (e.g., version sets, conflicts) (Gibb et al., 20 Feb 2026).

2. Dependency Graph Construction and Extraction Algorithms

The formalization pipeline typically proceeds through stages:

• Static Analysis and Parsing. Extract dependency relations from formal artifacts (proof terms, files, code) or annotated sources. In Coq, dpdgraph parses Gallina terms; in KnowTeX, environments, labels, and custom \uses/\proves commands are scanned from LaTeX (Heras et al., 2014, Uskuplu et al., 16 Dec 2025). In system verification, input/output/channel associations are explicitly mapped (Spichkova, 2014).
• Semantic Parsing of Informal or Semi-Formal Input. For proofs or requirements written in natural language or high-level notation, LLMs are tasked to:
  • Segment text into atomic steps or concepts.
  • Classify each segment (e.g., definition, lemma, hypothesis).
  • Extract or predict dependency edges (e.g., "this step uses...", or via explicit prompting and context enumeration) (Cabral et al., 13 Oct 2025, Wang et al., 6 Oct 2025, Mao et al., 5 Aug 2025).
• Context Pruning and Structural Constraints. To manage complexity, dependency graphs are pruned via reachability calculations, transitive reduction (only immediate dependencies retained; redundant edges eliminated), or minimal subgraph extraction for property checking (Spichkova, 2014, Mao et al., 5 Aug 2025).
• Modular Encodings and Efficient Representations.
  • Sequence labeling schemes encode graphs as token-level annotations amenable to efficient parsing models (e.g., bounded/unbounded bracketings, bitwise encodings for planarity and degree constraints) (Ezquerro et al., 2024).

3. Dependency-Driven Automation, Synthesis, and Reasoning

Direct use of dependency graphs enables modular and controlled automation in multiple domains:

• Proof Autoformalization. ProofFlow and Aria generate a dependency DAG from informal proof text or conjecture, then synthesize each node (definition, lemma, or theorem) into Lean code in topological order, strictly admitting only formalized dependencies into context. Bottom-up synthesis and context control ensure semantic fidelity and modular verification. Aria incorporates retrieval-augmented grounding (Mathlib search) and a semantic scoring system for true correctness (Cabral et al., 13 Oct 2025, Wang et al., 6 Oct 2025).
• Machine-Learning-Based Mining (ML4PG). Feature extraction transforms dependency graphs of terms/proofs into vector representations, with recurrent clustering yielding cluster IDs for subterms/types, enabling proof-pattern mining and auto-discovery of proof families. The statistical approach functions as a post-processor over the raw proof DAG, enhancing digestibility and pattern detection (Heras et al., 2014).
• Code Generation from Specifications. In UML2Dep, explicit data dependency graphs—extracted from enhanced sequence diagrams with integrated decision tables and API specs—drive code synthesis. Each node's data consumption/production ensures correct function and control flow wiring, with the inferred graph serving as a hard skeleton for LLM-based code completion and correctness (Mao et al., 5 Aug 2025).
• Package Management and Dependency Resolution. The Package Calculus provides a lingua franca for representing dependency graphs and cross-ecosystem translation. Dependency graphs here are (hyper)graphs capturing package/version constraints, conflicts, and virtual provisions, enabling uniform SAT-style resolution semantics (Gibb et al., 20 Feb 2026).
• Adaptive Structures in Time Series and NLP. Adaptive GNNs dynamically infer dependency graphs from data (using causal inference/statistical structure learning) to support accurate time-evolving graph construction, which underpins state-of-the-art spatiotemporal forecasting (Sriramulu et al., 2023). In biomedical NLP, full dependency parses (rather than SDP pruned graphs) fed to GCNs yield significant F₁ gains in event extraction (Noravesh et al., 2 Jan 2025).

4. Visualization, Knowledge Management, and Human-Computer Interaction

The explicit formalization of dependencies allows for advanced visualization, management, and educational tooling:

• Interactive Visualization. Tools such as KnowTeX and Lean Blueprint enable the extraction of dependency graphs from annotated mathematical texts, supporting both visual exploration (DOT/TikZ exports) and alignment with formal libraries. Key design features include: environment- and label-based parsing, conceptual/logical edge types, and pruning via transitive closure for interpretable layouts (Uskuplu et al., 16 Dec 2025).
• Error Reduction and Selective Importing. Visualization of explicit graphs reduces missing-dependency failures (~60% reduction in formalization error) and supports selective import of only those lemmas and axioms required for a target proof (Uskuplu et al., 16 Dec 2025).
• Reusability and Multilingual Interoperability. Name alignment between informal mathematical exposition (LaTeX labels) and formal library entries bridges the gap between narrative and formal content, directly feeding proof assistants for minimal, accurate importing (Uskuplu et al., 16 Dec 2025).

5. Evaluation Metrics and Empirical Results

Graph-driven formalization is subject to rigorous, multi-criteria evaluation:

• Faithfulness Metrics. In ProofFlow, ProofScore combines structural fidelity (matching dependency sets), syntactic correctness (successful compilation), and semantic faithfulness (fuzzy match to intended meaning) in a node-averaged metric. Substantial improvements over baseline autoformalization methods are reported (ProofScore 0.545 vs. 0.123 for holistic one-pass formalization) (Cabral et al., 13 Oct 2025).
• Compiler Success and Semantic Accuracy. Aria demonstrates a 91.6% compilation success rate and 68.5% final accuracy on undergraduate-level conjecture formalization, with large performance margins over state-of-the-art baselines on research-level algebraic conjectures (Wang et al., 6 Oct 2025).
• Task-Specific Metrics. For NLP tasks, sequence-labeling encodings yield near–state-of-the-art parsing accuracy with superior speed; biomedical event extraction models show 8–10 F₁-point jumps when incorporating full dependency graphs (Ezquerro et al., 2024, Noravesh et al., 2 Jan 2025).

6. Limitations, Open Problems, and Future Directions

Dependency graph–driven formalization, while robust, exhibits notable areas for further exploration:

• Semantic Alignment Bottlenecks. LLM-based formalization may still fail at semantic appropriateness or precise meaning preservation, even with perfect structural fidelity (Cabral et al., 13 Oct 2025, Wang et al., 6 Oct 2025).
• Graph Complexity and Scalability. Conjecture-level graphs in rich domains can become prohibitively large; dependency-cost estimation and automated pruning are necessary (Wang et al., 6 Oct 2025).
• Automated Generation and Annotation. Manual insertion of dependency annotations is labor-intensive. AI-assisted annotation, automatic extraction from natural text, and integration with interactive HTML viewers represent open development fronts (Uskuplu et al., 16 Dec 2025).
• Cross-Domain Generalization. Unification of formalisms (e.g., Package Calculus, data dependency graphs, proof DAGs) and translation between them is feasible via graph-theoretical reductions and encodings, paving the way for unified reasoning engines across disparate formal systems (Gibb et al., 20 Feb 2026).
• Adaptive, Dynamic Graphs. Integration of statistical and neural methods for dynamic structure learning expands applicability where static dependency graphs are unavailable, as in forecasting or rapidly changing codebases (Sriramulu et al., 2023).

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In summary, dependency graph–driven formalization unifies and mechanizes explicit, computationally tractable representations of dependencies in logic, mathematics, software, and language. Incorporating both rich structural semantics and precise workflow support, this paradigm is central to the next generation of mathematical reasoning, machine learning, and intelligent system synthesis (Heras et al., 2014, Spichkova, 2014, Sriramulu et al., 2023, Ezquerro et al., 2024, Noravesh et al., 2 Jan 2025, Mao et al., 5 Aug 2025, Wang et al., 6 Oct 2025, Cabral et al., 13 Oct 2025, Uskuplu et al., 16 Dec 2025, Gibb et al., 20 Feb 2026).