Lassie: HOL4 Tactics by Example (2101.00930v1)

Abstract: Proof engineering efforts using interactive theorem proving have yielded several impressive projects in software systems and mathematics. A key obstacle to such efforts is the requirement that the domain expert is also an expert in the low-level details in constructing the proof in a theorem prover. In particular, the user needs to select a sequence of tactics that lead to a successful proof, a task that in general requires knowledge of the exact names and use of a large set of tactics. We present Lassie, a tactic framework for the HOL4 theorem prover that allows individual users to define their own tactic language by example and give frequently used tactics or tactic combinations easier-to-remember names. The core of Lassie is an extensible semantic parser, which allows the user to interactively extend the tactic language through a process of definitional generalization. Defining tactics in Lassie thus does not require any knowledge in implementing custom tactics, while proofs written in Lassie retain the correctness guarantees provided by the HOL4 system. We show through case studies how Lassie can be used in small and larger proofs by novice and more experienced interactive theorem prover users, and how we envision it to ease the learning curve in a HOL4 tutorial.

• The paper introduces a novel framework that enables users to define personalized tactic languages by example within the HOL4 environment.
• The study demonstrates that Lassie simplifies proof writing and reduces complexity, evidenced by the creation of 22 new tactics for Euclid's theorem and 42 for arithmetic proofs.
• The research shows that Lassie’s extensible semantic parser lowers entry barriers for both novices and experts, fostering reusable proofs and enhanced collaboration in formal methods.

Overview of Lassie: A Framework for Custom Tactic Programming in HOL4

The paper presents "Lassie," a novel tactic framework integrated into the HOL4 interactive theorem prover, addressing significant challenges in proof engineering, particularly for domain experts who may not be fully versed in low-level theorem prover tactics. Lassie allows users to define their personalized tactic language using a process of definitional generalization, thus simplifying the task of proof writing without compromising the correctness guarantees offered by HOL4.

Key Contributions

Lassie's core innovation is the use of an extensible semantic parser to enable custom tactic definitions "by example." This allows users to define new tactic sequences in plain language, mapping them to existing HOL4 tactics with precise arguments, which the system automatically generalizes for broader applicability. Importantly, Lassie maintains full compatibility with standard HOL4 proof constructs, and generated proofs remain portable across different projects and setups.

Evaluation and Methodology

The efficacy of Lassie is demonstrated through various case studies that involve proofs in logic and arithmetic domains. The studies indicate that both novice and expert users can benefit from using Lassie. Novices can use intuitive tactic names, speeding up their learning process, while experts can create reusable tactic sequences for frequently encountered proof constructs.

Several numerical results from these case studies are promising. The authors reported defining 22 new tactics for proving Euclid's theorem, a result that illustrates the potential reduction in proof complexity when using Lassie. Moreover, the paper on real and natural number theorems involved creating 42 new tactics, exhibiting how Lassie can facilitate developing comprehensive proof libraries efficiently.

Implications and Future Directions

The research has strong implications for enhancing interactive theorem proving usability, especially in academic and educational contexts. Lassie's ability to allow non-expert users to meaningfully interact with theorem proving tasks without needing in-depth system knowledge could significantly lower the entry barrier to formal methods. The framework supports custom libraries, which can define domain-specific languages, potentially fostering increased collaboration within interdisciplinary teams.

Looking forward, Lassie could be extended to integrate more complex decision-making capabilities, such as leveraging machine learning-based predictions for tactic applications, effectively intersecting with ongoing research in theorem proving, automated reasoning, and artificial intelligence. This would not only improve proof efficiency but could also lead to new insights in AI understanding of formal logic systems.

Conclusion

Lassie represents a substantial contribution to enhancing theorem prover accessibility and utility. By allowing personalized and intuitive interaction with HOL4 systems through a semantic parsing approach, this framework empowers users to engage more deeply with formal proofs. The extensibility and portability of Lassie proofs suggest it could serve as a template for future developments in interactive theorem proving environments, proposing a novel direction for reducing the complexity traditionally associated with formal proof development.