Interactive proofs of theorems often require auxiliary helper lemmas to prove the desired theorem. Existing approaches for automatically synthesizing helper lemmas fall into two broad categories. Some approaches are goal-directed, producing lemmas specifically to help a user make progress from a given proof state, but they have limited expressiveness in terms of the lemmas that can be produced. Other approaches are highly expressive, able to generate arbitrary lemmas from a given grammar, but they are completely undirected and hence not amenable to interactive usage. In this paper, we develop an approach to lemma synthesis that is both goal-directed and expressive. The key novelty is a technique for reducing lemma synthesis to a data-driven program synthesis problem, whereby examples for synthesis are generated from the current proof state. We also describe a technique to systematically introduce new variables for lemma synthesis, as well as techniques for filtering and ranking candidate lemmas for presentation to the user. We implement these ideas in a tool called lfind, which can be run as a Coq tactic. In an evaluation on four benchmark suites, lfind produces useful lemmas in 68% of the cases where a human prover used a lemma to make progress. In these cases lfind synthesizes a lemma that either enables a fully automated proof of the original goal or that matches the human-provided lemma.
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Program synthesis using conflict-driven learning
We propose a new conflict-driven program synthesis technique that is capable of learning from past mistakes. Given a spurious program that violates the desired specification, our synthesis algorithm identifies the root cause of the conflict and learns new lemmas that can prevent similar mistakes in the future. Specifically, we introduce the notion of equivalence modulo conflict and show how this idea can be used to learn useful lemmas that allow the synthesizer to prune large parts of the search space. We have implemented a general purpose CDCL-style program synthesizer called Neo and evaluate it in two different application domains, namely data wrangling in R and functional programming over lists. Our
experiments demonstrate the substantial benefits of conflict driven learning and show that Neo outperforms two state-of-the-art synthesis tools, Morpheus and DeepCoder, that
target these respective domains
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- Award ID(s):
- 1762363
- NSF-PAR ID:
- 10095674
- Date Published:
- Journal Name:
- ACM SIGPLAN Conference on Programming Language Design and Implementation
- Page Range / eLocation ID:
- 420 to 435
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Interactive proofs of theorems often require auxiliary helper lemmas to prove the desired theorem. Existing approaches for automatically synthesizing helper lemmas fall into two broad categories. Some approaches are goal-directed, producing lemmas specifically to help a user make progress from a given proof state, but they have limited expressiveness in terms of the lemmas that can be produced. Other approaches are highly expressive, able to generate arbitrary lemmas from a given grammar, but they are completely undirected and hence not amenable to interactive usage. In this paper, we develop an approach to lemma synthesis that is both goal-directed and expressive. The key novelty is a technique for reducing lemma synthesis to a data-driven program synthesis problem, whereby examples for synthesis are generated from the current proof state. We also describe a technique to systematically introduce new variables for lemma synthesis, as well as techniques for filtering and ranking candidate lemmas for presentation to the user. We implement these ideas in a tool called lfind, which can be run as a Coq tactic. In an evaluation on four benchmark suites, lfind produces useful lemmas in 68% of the cases where a human prover used a lemma to make progress. In these cases lfind synthesizes a lemma that either enables a fully automated proof of the original goal or that matches the human-provided lemma.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3192366.3192382
