Learning-Augmented Proof Synthesis

• The paper demonstrates a multi-phase architecture where neural models decompose complex proof obligations into sub-obligations that are verified using mechanical checking.
• Learning-Augmented Proof Synthesis employs retrieval-augmented generation with semantic embeddings to select in-context exemplars, enhancing syntactic correctness and domain fidelity.
• Empirical evaluations show that the approach improves proof synthesis for nontrivial obligations while highlighting challenges in handling global invariants and complex safety assertions.

Learning-Augmented Proof Synthesis is a research paradigm that integrates machine learning—especially neural models and LLMs—within formal methods and automated reasoning systems to support, accelerate, or automate the construction of formal proofs in mathematical logic, software verification, and theorem proving. This area combines statistical inference, retrieval-augmented generation, neuro-symbolic methods, reinforcement learning, and symbolic automation strategies to tackle varying levels of proof complexity, syntactic domain specificity, and data scarcity.

1. Foundational Principles and Two-Phase Architectures

Modern learning-augmented proof synthesis systems are commonly structured as multi-phase pipelines that decompose complex proof obligations into simpler components, and leverage context-aware generation for individual sub-obligations. In the context of TLAPS (TLA+ Proof System), a prominent architecture comprises:

• Sub-proof Obligation Generation: An LLM receives a structured prompt instructing it to break a complex obligation into sub-obligations $\{O_1, \ldots, O_k\}$ such that their conjunction implies the original goal: $\bigwedge_i O_i \implies G$ (Zhou, 6 Jan 2025).
• Retrieval-Augmented Generation for Proof Synthesis: For each $O_i$ not solvable by native automated tactics, the system performs semantic retrieval—utilizing cosine similarity in embedding space—to retrieve $k$ most similar, sentence-level TLAPS proof statements from a database. These statements serve as in-context exemplars guiding the LLM in constructing syntactically correct, domain-conforming proofs.
• Verification and Iterative Refinement: At every decomposition and generation step, TLAPS mechanically verifies both the structural adequacy of sub-obligations and the syntactic/logical validity of generated proofs. Upon failure, feedback including decomposition rationale and rejection reasons is directly fed back to the LLM for iterative scheme refinement.

This hierarchical, feedback-driven loop is designed to reinforce both syntactic correctness and logical entailment, exploiting the strengths of both LLM creativity and symbolic verification engines.

2. Retrieval-Augmented Generation and Semantic Embedding

Retrieval-augmented generation (RAG) addresses a crucial bottleneck in learning-augmented synthesis for proof assistants with rare, highly specialized syntax (such as TLAPS), for which LLMs lack significant training data. RAG works by:

• Building an archive of sentence-level, verified proof statements.
• Embedding proof obligations and database entries with a learned $f(\cdot)$ and computing cosine similarity to select in-context examples:

$$\operatorname{Sim}(\mathbf{v}_{O_i}, \mathbf{v}_{Q_j}) = \frac{\mathbf{v}_{O_i} \cdot \mathbf{v}_{Q_j}}{\|\mathbf{v}_{O_i}\|\|\mathbf{v}_{Q_j}\|}$$

• Prompting the LLM with retrieved fragments, explicit requirements for TLAPS syntax, and problem context.

Empirical findings show that RAG yields both higher syntactic correctness and improved domain fidelity, especially where vanilla LLM prompting produces plausible but formally unacceptable output. It crucially enables zero-shot domain adaptation for previously unencountered proof systems (Zhou, 6 Jan 2025).

3. Iterative Proof Search, Verification, and Recursion

The learning-augmented pipeline enforces robust proof search by recursively decomposing obligations and providing targeted feedback for subgoals that fail verification. Each sub-obligation undergoes:

1. Attempted solution via native automated provers (SMT, Isabelle, TLAPS 'OBVIOUS', etc.).
2. If unresolved, retrieval-augmented proof synthesis via LLM.
3. Formal verification by the proof assistant, with recursive decomposition applied to any failed subproofs.
4. Termination triggered upon either successful completion or bounded recursive depth.

This design allows LLMs to accommodate the hierarchical, block-structured reasoning style endemic to formal software proofs and to overcome the limitations of black-box, unconstrained generation.

4. Empirical Performance, Limitations, and Strengths

Experimental evaluations using obligations from the Boyer-Moore majority algorithm demonstrate the strengths and boundary conditions of learning-augmented synthesis:

• For simple and intermediate proof obligations, the retrieval-augmented system produces TLAPS-verified proofs, solving cases intractable for baseline provers or direct LLM prompting.
• RAG substantively improves adherence to TLAPS syntax, enabling mechanistic proof checking and reducing semantic drift.
• Systematic decomposition enables non-expert LLMs to emulate the human approach to TLAPS, succeeding on problems where non-hierarchical strategies fail.
• In contrast, complex properties involving global invariants or nontrivial safety assertions remain challenging; limitations arise in representing or decomposing domain-specific reasoning steps.

A typical trade-off is verbosity and possible over-expansion, reflected in LLM-generated proofs that are correct but not necessarily minimal or concise.

5. Implementation Details and Algorithmic Formalism

Implementation centers on constructing structured prompts and automated verification harnesses. Key algorithmic operations include:

• Embedding-based retrieval with explicit similarity functions and context window management.
• Structured LLM prompting for decomposition, justification, and machine-interpretable output.
• Recursive error-driven refinement, where feedback includes decomposition rationales and detailed failure reporting.
• Strict TLAPS-driven mechanical verification at all stages for both decomposition and generated proof scripts.

All synthesis is performed within a workflow that escalates from first applying automatic provers, then retreating to learning-augmented synthesis only for non-trivial or failed cases.

6. Significance, Implications, and Research Outlook

This paradigm demonstrates robust integration of LLMs into practical, mechanical proof development:

• LLMs augmented via semantic retrieval and iterative feedback can serve as creative engines in structured, evidence-driven formal verification.
• RAG mitigates domain data scarcity, establishing a methodology for learning-augmented synthesis in specialized verification environments.
• Future advances may lie in more robust decomposition strategies, theory-aware neural-symbolic modules, and pre-training on TLAPS-specific corpora to further optimize correctness and brevity.
• The methodology outlined calls for the development of standardized benchmarks and deeper coupling between provers and LLMs during synthesis—not just at verification boundaries.

A plausible implication is that scalable, retrieval-augmented, and feedback-driven learning architectures will become the backbone for next-generation automated formal verification, bridging the domain gap between expressive logical formalisms and neural generative capabilities.

7. Summary Table: Learning-Augmented Proof Synthesis Workflow

Step	Description	Mechanism
Decomposition	Split obligation $G$ into sub-obligations $\{O_i\}$	LLM + structured prompt
Automated Prover Attempt	Solve $O_i$ with built-in provers	OBVIOUS/AllProvers
Retrieval-Augmented Generation (RAG)	Retrieve top-$k$ similar proofs, augment LLM prompt for $O_i$	Embedding + cosine sim.
LLM Synthesis	Generate TLAPS-verifiable proof for $O_i$	LLM with exemplar context
Verification	TLAPS checks logical/syntactic validity of each sub-proof	Mechanical checking
Recursive Refinement	Decompose or revise failed sub-obligations	Iterative feedback
Termination	Success when all sub-obligations proved or failure at max depth	Attempt threshold

This paradigm positions learning-augmented proof synthesis as a scalable, extensible methodology for integrating LLMs into mechanical formal methods workflows, with strong empirical grounding and clear paths for future research in formal verification and automated reasoning.