Probabilistic Proof State Compression: Optimizing LLM-Guided Formal Verification

Authors: ali  rahim
noor  rahim
Venue: NeurIPS_cc_2024_Workshop_MATH-AI
Published: 2024-09-26 14:00:10

Abstract: Despite recent successes in LLM-guided formal proof search, scalability remains limited by the large search space. This paper introduces a novel approach that integrates off-the-shelf LLMs with conformal prediction-based compression to optimize proof search. By employing adaptive, probability-based binning informed by conformal prediction intervals, our method compresses the proof state space, reducing computational demands while retaining statistical proof guarantees. Preliminary results on the Lean miniF2F test set show similar success rates with 75% fewer passes, and on average 23% reduced wall clock time.

Main paper: 
 introduction: 
 Recent advances in large language models (LLMs) have significantly progressed the automation of formal proof generation. LLM-guided methods ( [1] , [2] , [3] ) demonstrate promising capabilities in navigating the complex search spaces of formal theorem proving, leveraging LLMs' pattern recognition and generalization strengths over traditional symbolic methods.LLM-based formal proving typically follows either proof-step or whole-proof generation strategies. While recent systems, such as DeepSeek-Prover-V1.5 [4] , have set state-of-the-art benchmarks using a truncate-and-resume approach, they still struggle to balance exploration and exploitation due to the sparse binary rewards of successful proofs, requiring extensive search with up to 2 17 typically.We propose a method that enhances LLM-guided formal proof search with conformal proof state space compression. Using open-weight LLMs ( [4] ) for generating candidate steps, our conformal prediction framework provides calibrated success probabilities. We introduce a rigorous compression algorithm that preserves the most promising proof paths, which allows efficient exploration.We evaluate our method on the MiniF2F [5] and ProofNet [6] benchmarks and demonstrate a 75% average reduction in the number of passes required compared to baseline open models. Additionally, our approach led to qualitatively simpler proofs on some examples. These results suggest promising directions for scaling automated theorem proving to more complex domains. 
proposed method: 
 We enhance LLM-guided formal proof search by introducing a novel proof state space compression technique. Our approach integrates three key components to efficiently navigate and compress the proof search space: the LLM-based Proof Step Generator, the Conformal Prediction Module [7] , and the Proof State Space Compression Module. Figure 1 illustrates the architecture of our system. We employ DeepSeek Prover V1.5 RL [4] , an open-weights large language model fine-tuned on Lean 4, to generate candidate proof steps. Given the current proof state s = (g, a, t) and goal g, the model produces a set of possible next steps {s ′ 1 , . . . , s ′ k } via repeated sampling.The Conformal Prediction Module provides calibrated probability intervals for the success of each candidate proof step, thereby guiding the proof search with reliable uncertainty estimates.We define a nonconformity measure A(z, z ′ ) for a labeled proof attempt z = (s, y) as:EQUATIONHere, P (success | s) estimates the probability of proof success given state s using the same model, and y is the binary outcome (success or failure).To address the dependency in proof states and uphold the exchangeability assumption required by conformal prediction, we partition the calibration set into strata using the Proof State Space Compression Module. For each candidate step s ′ , the module:1. Identify Stratum: Assign s ′ to a stratum Z j based on its similarity to existing proof states.Calculate A(z i , s ′ ) for all z i ∈ Z j .3. Calculate P-Values: For each outcome y ∈ {0, 1},EQUATION4. Construct Prediction Region:EQUATION5. Compute Probability Interval:[L(s ′ ), U (s ′ )] =    [0, 1] if Γ ε (s ′ ) = {0, 1} [1, 1] if Γ ε (s ′ ) = {1} [0, 0] if Γ ε (s ′ ) = {0}The Proof State Space Compression Module manages the search space by grouping similar proof states into strata, facilitating efficient stratified conformal prediction. C ← GENERATECANDIDATESTEPS(s, g) 4:for all s ′ i ∈ C do 5:Z j ← ASSIGNSTRATUM(s ′ i , C) 6: [L(s ′ i ), U (s ′ i )] ← COMPUTEPROBABILITYINTERVAL(s ′ i , Z j ) 7:end for = (G 1 , A 1 , T 1 ) and s 2 = (G 2 , A 2 , T 2 ), we define: sim(s 1 , s 2 ) = w G • GoalSim(G 1 , G 2 ) + w A • Jaccard(A 1 , A 2 ) + w T • LCS(T 1 , T 2 )where GoalSim(G 1 , G 2 ) is the binary goal similarity, Jaccard(A 1 , A 2 ) is the Jaccard similarity of assumptions, LCS(T 1 , T 2 ) is the normalized longest common subsequence of tactics, and w G , w A , and w T are weights summing to 1.We map probability intervals to bin indices using:b([l, u]) = n • max l, l + u 2where n is the number of bins. To prioritize high-probability regions, we use adaptive bin widths:w(p) = w 0 • exp(-αp)where w 0 is the base width and α controls adaptation rate.The compression process C maps a set of proof states to a set of representative states:C({s 1 , . . . , s k }) = {rep(B 1 ), . . . , rep(B m )}where B 1 , . . . , B m are non-empty bins.Our proof search algorithm integrates the three components to efficiently explore the proof space. Algorithm 1 outlines the procedure.Our method provides theoretical guarantees on the coverage of the true probability of proof success. Specifically, for a given significance level ε, we have:P (L(s) ≤ p * (s) ≤ U (s)) ≥ 1 -εwhere p * (s) is the true probability of a successful proof from state s. This guarantee ensures that our compression technique preserves the most promising proof paths with high probability.3 Experiments and Results 
datasets: 
 We perform a preliminary evaluation of our method on two benchmark datasets: MiniF2F [5] and ProofNet [6] . Specifically, we use the Lean 4 MiniF2F variant by Yang et al. [8] .We compare our method against several state-of-the-art models: GPT-4, DeepSeek-Prover-V1, and DeepSeek-Prover-V1.5.Model Pass Rate (%) @ Number of Passes GPT-4 23.0 @ 10 DeepSeek-Prover-V1 50.0 @ 32 DeepSeek-Prover-V1.5 60.2 @ 32 Our Method 63.5 @ 8 Table 1 Our method demonstrates a 75% reduction in passes, with 23% reduced wall clock time compared to DeepSeek-Prover-V1.5, This efficiency gain can be attributed to the effective pruning of the search space through our conformal prediction-based approach.These preliminary results demonstrate that our method significantly improves upon existing approaches in both proof success rate and efficiency. The integration of conformal prediction with LLM-guided search allows for more effective exploration of the proof space, leading to higher-quality proofs. While promising, the approach requires further refinement and empirical validation to fully realize its capabilities. Nonetheless, it represents a meaningful step towards bridging the gap between generative AI models and statistically robust decision-making processes in formal reasoning tasks.For a given significance level ε ∈ (0, 1), our stratified conformal prediction method for proof state evaluation satisfies the following coverage guarantee:P (L(S n+1 ) ≤ p * (S n+1 ) ≤ U (S n+1 )) ≥ 1 -ε where p * (S n+1 ) = P(Y n+1 = 1 | S n+1) is the true probability of a successful proof from state S n+1 , and [L(S n+1 ), U (S n+1 )] is the prediction interval produced by our method.For a given significance level ε ∈ (0, 1), our stratified conformal prediction method for proof state evaluation satisfies the following coverage guarantee:P (L(S n+1 ) ≤ p * (S n+1 ) ≤ U (S n+1 )) ≥ 1 -ε where p * (S n+1 ) = P(Y n+1 = 1 | S n+1) is the true probability of a successful proof from state S n+1 , and [L(S n+1 ), U (S n+1 )] is the prediction interval produced by our method.Proof. Consider a calibration set Z = {Z 1 , . . . , Z n }, where each Z i = (s i , y i ) consists of a proof state s i and its binary outcome y i ∈ {0, 1}. The calibration set is partitioned into m strata Z 1 , . . . , Z m using the Proof State Space Compressor C, ensuring that within each stratum Z j , the proof states are similar and exchangeable.For a new test proof state S n+1 = (g n+1 , a n+1 , t n+1 ), assign it to a stratum Z j based on its similarity to existing proof states using C. Given the exchangeability within each stratum Z j , the standard conformal prediction guarantee applies within Z j .Define the nonconformity score for a proof state Z i = (s i , y i ) as:A(Z i ) = |y i -P (Y i = 1 | s i )|where P (Y i = 1 | s i ) is the estimated probability of proof success given state s i .For the new proof state S n+1 , compute the nonconformity scores for both possible outcomes y ∈ {0, 1}:A((S n+1 , y)) = |y -P (Y n+1 = 1 | S n+1 )|The p-value for each outcome y is defined as:p(y | S n+1 ) = |{i ∈ Z j : A(Z i ) ≥ A((S n+1 , y))}| + 1 |Z j | + 1This ensures that p-values are properly calibrated within the stratum Z j .The prediction region is:Γ ε (S n+1 ) = {y ∈ {0, 1} : p(y | S n+1 ) > ε}Based on Γ ε (S n+1 ), derive the probability interval:[L(S n+1 ), U (S n+1 )] =    [0, 1] if Γ ε (S n+1 ) = {0, 1} [1, 1] if Γ ε (S n+1 ) = {1} [0, 0] if Γ ε (S n+1 ) = {0}By the properties of conformal prediction within each stratum Z j [7] , we have:P(Y n+1 ) ∈ Γ ε (S n+1 ) | S n+1 ∈ Z j ) ≥ 1 -εThis implies that the true outcome Y n+1 will lie within the prediction region Γ ε (S n+1 ) with probability at least 1 -ε.Since p * (S n+1 ) = P(Y n+1 = 1 | S n+1 ), observe the following:Y n+1 = 1 =⇒ p * (S n+1 ) ≥ L(S n+1 ) Y n+1 = 0 =⇒ p * (S n+1 ) ≤ U (S n+1 )Therefore, combining these two implications, we obtain: P (L(S n+1 ) ≤ p * (S n+1 ) ≤ U (S n+1 )) = P (Y n+1 ∈ Γ ε (S n+1 )) ≥ 1 -ε Thus, the coverage guarantee holds:P (L(S n+1 ) ≤ p * (S n+1 ) ≤ U (S n+1 )) ≥ 1 -εThis completes the proof.Implications for Theorem Proving:The theoretical guarantee ensures that our stratified conformal prediction method produces wellcalibrated probability intervals for proof success. Specifically, with probability at least 1 -ε, the true probability p * (S n+1 ) of successfully proving a theorem from state S n+1 lies within the interval [L(S n+1 ), U (S n+1 )]. This reliability enables the proof search algorithm to prioritize steps with higher upper bounds U (s ′ ), indicating a greater likelihood of success, while effectively managing uncertainty in steps with wider intervals. Consequently, the algorithm balances exploration and exploitation, enhancing both the efficiency and robustness of automated theorem proving. 
 
 Appendix:  


Decision: While the paper is clearly incomplete and rushed, the technique presented seems interesting and potentially interesting to workshop attendees. We recommend that the authors follow the comments of the reviewers to improve the presentation of the work and thoroughly revise the paper before the camera-ready version. In addition, the paper is above the 4-page limit.

Reviews: review: title: title: Probabilistic Proof State Compression: Optimizing LLM-Guided Formal Verification
main_review: main_review: This paper presents a novel approach that combines Large Language Models (LLMs) with conformal prediction techniques to optimize formal proof search. The core innovation is the introduction of a conformal proof state space compression algorithm that reduces the computational complexity of the proof search, while maintaining statistical guarantees on the success rate of proof discovery. The method is evaluated on two datasets, Lean 4 miniF2F and ProofNet, demonstrating a 23% reduction in proof search time and an improvement in the success rate over baseline models such as GPT-4 and DeepSeek-Prover-V1.5. The results suggest this method is effective at balancing exploration and exploitation in theorem proving while handling the exponential growth of proof search spaces.

### Review:

#### Originality:
- Strengths: The approach introduces a novel method of combining conformal prediction with LLM-guided proof search, which has not been explored before in formal verification. The adaptive binning and similarity measure between proof states are clever innovations that help in compressing the proof search space efficiently.
- Weaknesses: While the approach is innovative, the method largely builds on existing work, including the use of LLMs and conformal prediction techniques. The novelty mainly lies in the combination of these methods rather than in groundbreaking new algorithms or concepts.

#### Quality:
- Strengths: The paper is technically sound, and the method is rigorously described. The experimental results demonstrate the effectiveness of the approach, particularly in terms of proof search time reduction and pass rates on the datasets. The theoretical guarantees provided for the compression algorithm further strengthen the quality of the work.
- Weaknesses: The empirical evaluation could be expanded to include more diverse theorem proving benchmarks, beyond miniF2F and ProofNet. The current experiments, while promising, might not fully reflect the scalability of the approach to even more complex domains. Additionally, there is limited discussion on the limitations of conformal prediction in cases where the calibration set might be small or biased.

#### Clarity:
- Strengths: The paper is clearly written and well-structured, with a logical flow from the problem statement to the proposed solution and experimental results. The use of formulas and theoretical proofs is appropriate, and the explanation of the compression algorithm is detailed enough for an expert reader to reproduce the results.
- Weaknesses: Some sections, particularly those discussing the conformal prediction intervals, could benefit from additional examples or diagrams to aid understanding. The similarity measure introduced in the proof state space could be better explained with concrete examples from the experiments.

#### Significance:
- Strengths: The method addresses a critical issue in LLM-guided theorem proving—scalability. By significantly reducing proof search time and maintaining high proof success rates, this method could be a valuable contribution to the field of automated theorem proving and formal verification.
- Weaknesses: The significance of the work is limited by the scope of its evaluation. While the results are promising, they are based on two specific datasets. It remains to be seen whether this method will generalize well to broader, real-world formal verification problems.

#### Questions for Authors:
1. How does the method handle proof search spaces that are inherently noisy or where the proof states are highly dissimilar? Would the compression still be effective in such scenarios?
2. Can you provide more insights into the selection of hyperparameters, such as the binning granularity and the similarity measure weights? How sensitive are the results to these choices?
3. Have you considered any alternative statistical methods to conformal prediction for guiding the proof search, and how do you expect the performance to compare?

#### Limitations:
- The paper briefly mentions that scaling the method to particularly complex theorems remains a challenge, but this could be expanded. A separate limitations section would help discuss the bounds of the method, particularly in handling noisy data or scaling beyond the current datasets.
- There is no significant discussion about the impact of small or biased calibration sets on the accuracy of the conformal prediction intervals.
title: title: Strong rejection, incomplete and inaccurate.
main_review: main_review: This paper is clearly incomplete, even including several "TODO"s, missing citation references (rendering as ?), and the "Style" subsection from the NeurIPS formatting instructions. Regardless of presentation, the method seems to yield marginal results despite adding significant complexity, only improving by 3% on miniF2F over the baseline (60.2 -> 63.5).
title: title: Innovative Approach to Optimizing LLM-Guided Formal Verification
main_review: main_review: This paper presents a novel approach to enhancing Large Language Model (LLM) guided formal proof search by introducing a probabilistic proof state compression technique. The authors propose a method that combines LLMs with conformal prediction to guide and optimize formal proof search, addressing the challenge of scalability in automated theorem proving.

Pros:
1. Novelty: The integration of conformal prediction with LLM-guided proof search is an innovative approach to managing the complexity of formal verification.
2. Theoretical foundation: The paper provides a theoretical framework, including a proof of the coverage guarantee for their conformal prediction method.
3. Performance improvements: The authors report a 23% reduction in average proof time compared to state-of-the-art models, which is a significant efficiency gain.
4. Results: The authors test their method on established benchmarks (miniF2F and ProofNet), providing a clear comparison with existing approaches.
5. Detailed methodology: The paper offers a thorough explanation of the proof state representation, binning function, and compression mapping, allowing for potential replication and extension of the work.

Cons:
1. Limited experimental results: While the results are promising, they are described as "preliminary" and only cover a subset of the Lean 4 miniF2F test. More comprehensive testing would strengthen the paper's claims.
2. Computational resources: The paper doesn't provide detailed information about the computational resources required for the experiments, which is important for reproducibility and understanding the method's scalability.
3. Comparison with other compression techniques: While the method is compared with baseline models, there's no comparison with other proof state compression techniques that don't use conformal prediction.
4. Limitations discussion: The paper would benefit from a more in-depth discussion of the limitations of the proposed approach and potential failure cases. 
5. Insufficient error analysis: There's a lack of in-depth analysis of cases where the method fails or performs poorly, which would be valuable for understanding its limitations.
6.Unclear generalizability: The paper doesn't adequately address how well this method might generalize to other theorem-proving domains or more complex problems.
7. Broader impacts: The discussion of potential societal impacts in its applicability, both positive and negative, could be more 
comprehensive.

General comments:
- The paper presents a clear and well-structured argument for the proposed method. The introduction effectively sets up the problem and motivates the research. The methodology section is detailed and provides a good understanding of the proposed approach. The theoretical guarantees section adds credibility to the method by providing a formal proof of the coverage guarantee.
- The experimental results, while promising, could be expanded. The authors demonstrate improvements over state-of-the-art models on two benchmarks, but more extensive testing and analysis would strengthen the paper's claims. The discussion of the results is insightful, but could be deepened with more analysis of where and why the method performs well or fails.
- The paper's originality lies in its novel combination of LLM-guided proof search with conformal prediction and proof state compression. 
- While the approach is novel, the paper doesn't adequately situate this work within the broader context of research on proof state compression and management. A more comprehensive comparison with existing techniques would help clarify the true contribution of this method.
- In terms of clarity, the paper is generally well-written and logically structured. However, some sections, particularly in the methodology, are quite dense and could benefit from additional explanations or examples to improve accessibility for readers less familiar with the specific techniques used.
- Overall, this paper presents an innovative and promising approach to optimizing LLM-guided formal verification. While there are areas for improvement, particularly in the breadth of experimental results and analysis, the work represents a significant contribution to the field and opens up exciting avenues for future research.