Evaluating Automated Theorem Proving Systems on Quantified Modal Logic Problems by Translating to Classical Logics

Core Concepts

Translating quantified modal logic problems to classical first-order and higher-order logic enables the use of state-of-the-art ATP systems as backend reasoners, which perform comparably or better than native modal logic ATP systems.

Abstract

The article evaluates the performance of automated theorem proving (ATP) systems on problems from the Quantified Modal Logics Theorem Proving (QMLTP) library. The key insights are:

The process of translating first-order modal logic problems into classical first-order and higher-order logic using a shallow embedding is reliable and successful when state-of-the-art ATP systems are used as backend reasoners.

The first-order and higher-order embeddings perform similarly in terms of proving theorems. Native modal logic ATP systems have comparable performance to classical systems using the embedding for proving theorems, but are outperformed by the embedding approach for disproving conjectures.

The embedding approach can cope with a wider range of modal logics than the native modal systems considered, which only support a subset of the modal logics in the modal cube.

The union of the higher-order and first-order embedding variants solves the most problems overall, proving slightly more theorems than the native modal logic systems and disproving significantly more conjectures.

Stats

The article provides detailed statistics on the number of problems solved (proved or disproved) by various ATP systems, including the native modal logic systems and the systems using the first-order and higher-order embeddings. These statistics are broken down by modal logic, domain restrictions, and whether the goal was to prove a theorem or disprove a conjecture.

Quotes

"The findings are that the embedding process is reliable and successful when state-of-the-art ATP systems are used as backend reasoners, The first-order and higher-order embeddings perform similarly, native modal logic ATP systems have comparable performance to classical systems using the embedding for proving theorems, native modal logic ATP systems are outperformed by the embedding approach for disproving conjectures, and the embedding approach can cope with a wider range of modal logics than the native modal systems considered."

Key Insights Distilled From

Solving Quantified Modal Logic Problems by Translation to Classical Logics

by Alex... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2212.09570.pdf

Deeper Inquiries

What other non-classical logics beyond modal logic could be tackled using the shallow embedding approach, and how would the performance compare to native reasoning systems

The shallow embedding approach used for modal logic problems could also be applied to other non-classical logics, such as temporal logic, epistemic logic, deontic logic, and intuitionistic logic. By translating these non-classical logics into classical logics like first-order or higher-order logic, the performance of the embedding approach could be compared to native reasoning systems for those logics.
In terms of performance comparison, the results may vary depending on the complexity and specific features of the non-classical logic being considered. For some logics, the embedding approach might outperform native systems due to the flexibility and adaptability of classical logic reasoning tools. However, for more specialized or intricate non-classical logics, native systems designed specifically for those logics may still have an edge in terms of efficiency and accuracy.

How could the embedding approach be further optimized to improve its performance advantage, especially for disproving conjectures

To optimize the embedding approach for improving its performance advantage, especially for disproving conjectures, several strategies can be considered:

Enhanced Translation Techniques: Develop more sophisticated translation techniques that capture the nuances and complexities of the non-classical logic being embedded. This could involve refining the mapping of connectives, quantifiers, and other logical constructs to ensure a more accurate representation in the classical logic.

Logic-Specific Embeddings: Tailor the embedding process to the specific features and requirements of different non-classical logics. By customizing the embedding approach for each logic, the performance for disproving conjectures could be enhanced.

Integration of Domain-Specific Knowledge: Incorporate domain-specific knowledge and heuristics into the embedding process to guide the reasoning systems towards more effective disproof strategies. This could involve leveraging insights from the specific domain of the non-classical logic to streamline the disproving process.

Optimized Reasoning Algorithms: Implement optimized reasoning algorithms within the classical logic systems used for the embedding approach. This could involve fine-tuning the inference mechanisms and search strategies to better handle the disproving of conjectures in non-classical logics.

What are the implications of the embedding approach's ability to handle a wider range of modal logics compared to native systems, and how could this be leveraged in practical applications

The ability of the embedding approach to handle a wider range of modal logics compared to native systems has significant implications for practical applications:

Versatility in Problem Solving: The embedding approach allows for a unified framework to tackle various modal logics, making it easier to address a diverse set of problems without the need for specialized reasoning systems for each logic. This versatility can be leveraged in applications requiring reasoning across multiple modal logics.

Scalability and Adaptability: The embedding approach's capability to handle a broader range of modal logics provides scalability and adaptability in problem-solving. It can accommodate new or customized modal logics without the need to develop dedicated reasoning systems, offering a more flexible and cost-effective solution.

Interoperability and Integration: The embedding approach facilitates the integration of modal logic reasoning into existing classical logic frameworks, enabling seamless interoperability between different types of logics. This interoperability can be beneficial for applications that require combined reasoning capabilities.

Customization and Tailoring: The embedding approach allows for customization and tailoring of the reasoning process to specific modal logic requirements. By adjusting the embedding techniques and logic specifications, the approach can be fine-tuned to meet the unique demands of different modal logics in practical applications.