Solving Quantified Modal Logic Problems by Translation to Classical Logics

TL;DR

This work tackles automated reasoning for first-order quantified modal logic by translating problems from the QMLTP benchmark into classical first-order and higher-order logics via shallow embeddings, enabling solving with state-of-the-art ATPs and model finders. It systematically compares embedding-based backends (TFF/THF) against native modal ATP systems across mono- and multi-modal problems, revealing that embeddings are reliable and often superior for disproving conjectures while offering broader modal-logic coverage. The study highlights that higher-order embeddings (THF) generally outperform first-order embeddings (TFF) for disproof, and that the embedding approach can handle a wider range of logics, with notable S5 optimizations improving efficiency. These findings support a flexible, scalable toolchain for modal reasoning and point to practical directions for integrating non-classical logics into the TPTP ecosystem and ATP backends.

Abstract

This article describes an evaluation of Automated Theorem Proving (ATP) systems on problems taken from the QMLTP library of first-order modal logic problems. Principally, the problems are translated to both typed first-order and higher-order logic in the TPTP language using an embedding approach, and solved using first-order resp. higher-order logic ATP systems and model finders. Additionally, the results from native modal logic ATP systems are considered, and compared with the results from the embedding approach. 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.

Figures (2)

• Figure 1: Non-classical connective examples
• Figure 2: Embedding process (picture taken from Ste22)