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Solving Modal Logic Problems by Translation to Higher-Order Logic

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Logic and Argumentation (CLAR 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14156))

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Abstract

This paper 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 higher-order logic in the TPTP language using an embedding approach, and solved using higher-order logic ATP systems. Additionally, the results from native modal logic ATP systems are considered, and compared with those from the embedding approach. The findings are that the embedding process is reliable and successful, the choice of backend ATP system can significantly impact the performance of the embedding approach, native modal logic ATP systems outperform the embedding approach, and the embedding approach can cope with a wider range modal logics than the native modal systems considered.

T. Scholl–Independent researcher.

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Notes

  1. 1.

    All the problems and results are available from the QMLTP directory of the TPTP World’s non-classical logic Github repository at

    https://github.com/TPTPWorld/NonClassicalLogic.

  2. 2.

    http://www.iltp.de/qmltp.

  3. 3.

    The syntax format of QMLTP is not introduced here. For uniformity, the TPTP syntax standards and the extensions to modal logic are introduced in Sect. 3.

  4. 4.

    The development of TPTP World standards for writing ATP solutions beyond common derivations and models is still necessary; see, e.g., [37].

  5. 5.

    http://www.tptp.org/TPTP/SyntaxBNF.html.

  6. 6.

    This slightly unusual form was chosen to reflect the first-order functional style, but by making the application explicit the formulae can be parsed in Prolog – a long standing principle of the TPTP languages [51].

  7. 7.

    The property names presented in this work supersede those used in earlier works. The $designation used to be called $constants, while the $domains used to be called $quantification.

  8. 8.

    https://github.com/leoprover/logic-embedding.

  9. 9.

    http://www.iltp.de/qmltp/problems.html.

  10. 10.

    The automation pipeline presented here does support multi-modal logic reasoning. However, during experimentation, it was revealed that the expected results documented in the QMLTP library for the multi-modal MML problem domain seem to be erroneous, and they have been excluded from the evaluation. These issues will be assessed in more detail in future work.

  11. 11.

    As noted in Sect. 5.1, there are no expected results for decreasing domains (not documented in the QMLTP), and for some QMLTP problems it is unknown whether or not they are theorems with a given combination of properties.

  12. 12.

    The strength of a logic refers to the set of theorems of the particular logic, i.e., a logic \(L_1\) is stronger than a logic \(L_2\) if \(\textrm{theorems}(L_2) \subseteq \textrm{theorems}(L_1)\). This is not a complete nor linear order but rather a partial order relation, e.g., as visualized by the modal logic cube [22]. For example, S5 is stronger than K, assuming the other logic parameters (domain semantics, etc.) remain the same.

  13. 13.

    The conditions stated in [40] for “presenting results of modal ATP systems based on the QMLTP library” say that “no part of the problems may be modified”. As such the results presented in this paper for the corrected versions of the QMLTP problems cannot be called “results for problems from the QMLTP”. But pragmatically, the results on the set of (corrected) problems are comparable with the results on the original QMLTP problems.

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Acknowledgements

The authors thank the reviewers for their constructive feedback.

Author information

Authors and Affiliations

  1. University of Greifswald, Greifswald, Germany

    Alexander Steen

  2. University of Miami, Miami, USA

    Geoff Sutcliffe

  3. Berlin, Germany

    Tobias Scholl

  4. University of Bamberg and FU Berlin, Berlin, Germany

    Christoph Benzmüller

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  1. Alexander Steen
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  2. Geoff Sutcliffe
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  3. Tobias Scholl
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  4. Christoph Benzmüller
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Correspondence to Alexander Steen .

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  1. IRIT CNRS, Toulouse, France

    Andreas Herzig

  2. Zhejiang University, Hangzhou, China

    Jieting Luo

  3. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Pere Pardo

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Steen, A., Sutcliffe, G., Scholl, T., Benzmüller, C. (2023). Solving Modal Logic Problems by Translation to Higher-Order Logic. In: Herzig, A., Luo, J., Pardo, P. (eds) Logic and Argumentation. CLAR 2023. Lecture Notes in Computer Science(), vol 14156. Springer, Cham. https://doi.org/10.1007/978-3-031-40875-5_3

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