Problem
Research questions and friction points this paper is trying to address.
theorem proving
proof search
proof quality
computational cost
AND/OR graphs
LEVER: Adaptive Cost-Aware Proof Search Over AND/OR Graphs
🤖 AI Summary
This study addresses the limitation of existing LLM-based theorem provers that prioritize correctness while neglecting proof conciseness and computational cost. We propose LEVER, an algorithm integrating large language models with Lean kernel verification and a prediction-based partial proof scoring mechanism. By evaluating partial proofs on AND/OR graphs, LEVER renders optimization objectives programmable, enabling joint dynamic optimization of proof length, purity, and computational cost during the search phase without requiring post-hoc reconstruction. Evaluated on PutnamBench, LEVER improves the solve rate to 96% while reducing computational costs by 34%, significantly outperforming baseline methods. Furthermore, it supports flexible trade-off adjustments between proof quality and resource expenditure, offering a principled approach to efficient and elegant automated theorem proving.
📝 Abstract
Mathematicians value proofs for more than correctness: among correct proofs, simplicity, purity and the computational cost of finding them vary widely. Yet LLM-powered theorem provers largely search for any correct proof, and improve its quality only after it is found. We propose LEVER, a proof search algorithm that makes the objective over correct proofs programmable and optimizes it during search. LEVER scores partial proofs over an AND/OR proof graph, combining realized objective values with predictions for open subgoals, so the objective guides search before a proof is complete. The same mechanism optimizes computational cost, proof length, topical impurity, and even their weighted combinations, while the Lean kernel enforces correctness. On PutnamBench in Lean 4, under matched budgets, LEVER costs 34% less than a strong single-conversation agent while raising the solve rate from 80% to 96%. On reducing topical impurity, i.e., how far a proof strays from its theorem's subject, it improves over post-hoc refactoring (42% reduction against 33%) at two-thirds of the cost and more reliably; on proof length, the metric refactoring is built for, it approaches refactoring. Varying the objective's weights traces a quality-cost trade-off curve, so the user can choose how much a better proof is worth. Overall, LEVER is a performant, cost-efficient and tunable proof search algorithm for navigating the space of correct proofs.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Proof Search
AND/OR Graphs
Cost-Aware Optimization
Programmable Objectives
Theorem Proving
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