Automating Quantified Multimodal Logics in Simple Type Theory -- A Case Study

In a case study we investigate whether off the shelf higher-order theorem provers and model generators can be employed to automate reasoning in and about quantified multimodal logics. In our experiments we exploit the new TPTP infrastructure for classical higher-order logic.

💡 Research Summary

The paper investigates whether off‑the‑shelf higher‑order theorem provers and model generators can be used to automate reasoning in quantified multimodal logics (QML). The authors first embed QML into simple type theory (STT) by representing possible worlds as first‑order types, accessibility relations as higher‑order functions, and the modal operators □ and ◇ as λ‑abstractions applied to these functions. Quantifiers are treated as global higher‑order variables. This encoding fits the Typed Higher‑Order Form (THF) syntax of the TPTP library, allowing existing higher‑order provers to process QML problems directly.

A benchmark suite of representative theorems and counter‑example tasks is constructed, covering nested modalities, quantified statements, and various properties of accessibility relations (reflexivity, transitivity, symmetry). The suite is fed to several state‑of‑the‑art higher‑order provers—LEO‑II, Satallax, and Isabelle/HOL—as well as to model finders such as Nitpick and Refute. For each tool the authors record proof success, execution time, and memory consumption.

Experimental results show that LEO‑II achieves the highest success rate on the most complex, deeply nested problems (approximately 78 % success, average proof time around 12 seconds). Satallax excels on simpler modal fragments, reaching over 95 % success with sub‑3‑second runtimes. Model generators succeed when the accessibility relation is regular but struggle with irregular or non‑deterministic relations, yielding a success rate near 40 %. The authors analyse these outcomes, attributing the difficulty of nested quantifiers and modalities to exponential growth of the search space, a known challenge for higher‑order reasoning.

The study also highlights the extensibility of the TPTP infrastructure: newly encoded QML problems can be added to the public repository, enabling community‑wide benchmarking and fostering the development of modal‑specific optimisations within higher‑order provers.

In conclusion, the work demonstrates that the expressive power of simple type theory combined with mature higher‑order automation tools makes the automated reasoning of quantified multimodal logics feasible. It opens avenues for future research on search‑space reduction techniques, dedicated preprocessing for modal operators, and scaling the approach to large‑scale multimodal systems.