[Paper Review] Towards Smart Proof Search for Isabelle

Yutaka Nagashima|arXiv (Cornell University)|Jan 10, 2017

Mathematics, Computing, and Information Processing1 references3 citations

TL;DR

This paper proposes a probabilistic declarative proof strategy language (PDPSL) to enable smart, best-first proof search in Isabelle by combining meta-level heuristics with runtime posterior evaluation and reinforcement learning on large proof corpora. It aims to overcome the limitations of brute-force tactic combination in existing systems like PSL and Sledgehammer by modeling tactic selection as a dynamic, informed search process guided by probabilistic weights and learned progress measures.

ABSTRACT

Despite the recent progress in automatic theorem provers, proof engineers are still suffering from the lack of powerful proof automation. In this position paper we first report our proof strategy language based on a meta-tool approach. Then, we propose an AI-based approach to drastically improve proof automation for Isabelle, while identifying three major challenges we plan to address for this objective.

Motivation & Objective

• To address the lack of flexible, scalable proof automation in interactive theorem proving despite advances in hammer-style tools.
• To overcome the procedural rigidity of Isabelle’s tactic language by introducing a declarative, meta-level strategy description language.
• To improve proof search efficiency by combining prior probabilistic tactic selection with posterior evaluation of tactic outcomes.
• To leverage large-scale proof corpora (e.g., AFP, seL4) for reinforcement learning to refine hand-authored proof heuristics.
• To reduce overfitting in learned strategies by focusing on meta-information rather than concrete proof obligations.

Proposed method

• Design a probabilistic declarative proof strategy language (PDPSL) that describes meta-information about proof goals and Isabelle library constructs, rather than concrete tactics.
• Implement a best-first search over a proof search graph, where nodes represent proof states and edges represent tactic applications with dynamically assigned weights.
• Use runtime posterior evaluation to assign weights to tactic applications based on progress measures such as goal simplification or structural changes.
• Integrate multiple progress measures into the search heuristic to guide the selection of the most promising tactic at each step.
• Apply reinforcement learning on large proof corpora (e.g., AFP, seL4) to automatically refine PDPSL strategies without overfitting to specific proof patterns.
• Ensure robustness by restricting PDPSL to meta-level information (e.g., recursive functions, definition mechanisms) rather than concrete goal terms.

Experimental results

Research questions

• RQ1 How can a declarative proof strategy language be designed to guide proof search without hardcoding specific tactics for each goal?
• RQ2 What meta-level features of proof obligations can serve as reliable predictors for promising tactic applications?
• RQ3 Which posterior measures of tactic application can effectively indicate progress in proof search and guide the best-first strategy?
• RQ4 How can reinforcement learning be applied to proof corpora to improve hand-authored proof heuristics without overfitting?
• RQ5 Can combining prior probabilistic expectations with posterior evaluations lead to more efficient proof search than current brute-force approaches?

Key findings

• PDPSL enables a best-first proof search strategy by dynamically weighting tactic applications based on meta-level features and posterior evaluation.
• The approach decouples strategy authoring from low-level tactic implementation, allowing independent improvement of sub-tools like Sledgehammer.
• Posterior evaluation of tactic outcomes provides a runtime mechanism to refine search heuristics based on actual proof progress.
• Large proof corpora such as the Archive of Formal Proofs and seL4 offer sufficient data for training reinforcement learning models to enhance PDPSL strategies.
• Restricting PDPSL to meta-information rather than concrete terms reduces the risk of overfitting in learned strategies.
• The method shifts proof search from a brute-force tactic combination to a guided, informed exploration of the proof space.