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Synthesis of Modality Definitions and a Theorem Prover for Epistemic Intuitionistic Logic

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Logic-Based Program Synthesis and Transformation (LOPSTR 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12561))

Abstract

We propose a mechanism for automating discovery of definitions, that, when added to a logic system for which we have a theorem prover, extends it to support an embedding of a new logic system into it. As a result, the synthesized definitions, when added to the prover, implement a prover for the new logic.

As an instance of the proposed mechanism, we derive a Prolog theorem prover for an interesting but unconventional epistemic Logic by starting from the sequent calculus G4IP that we extend with operator definitions to obtain an embedding in intuitionistic propositional logic (IPC). With help of a candidate definition formula generator, we discover epistemic operators for which axioms and theorems of Artemov and Protopopescu’s Intuitionistic Epistemic Logic (IEL) hold and formulas expected to be non-theorems fail.

We compare the embedding of IEL in IPC with a similarly discovered successful embedding of Dosen’s double negation modality, judged inadequate as an epistemic operator. Finally, we discuss the failure of the necessitation rule for an otherwise successful S4 embedding and share our thoughts about the intuitions explaining these differences between epistemic and alethic modalities in the context of the Brouwer-Heyting-Kolmogorov semantics of intuitionistic reasoning and knowledge acquisition.

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Notes

  1. 1.

    Logics stronger than intuitionistic but weaker than classical.

  2. 2.

    Actually infinitely many, as there’s an infinite lattice of intermediate logics between classical and intuitionistic logic.

  3. 3.

    Originally called the LJT calculus in [8]. Restricted here to its key implicational fragment.

  4. 4.

    https://github.com/ptarau/TypesAndProofs/blob/master/third_party/dyckhoff_orig.pro.

  5. 5.

    At http://iltp.de.

  6. 6.

    https://github.com/ptarau/TypesAndProofs/blob/master/tester.pro.

  7. 7.

    In fact, our prover is faster than both fCube and Dyckhoff’s prover on the set of formulas of small size on which our definition induction algorithm will run.

  8. 8.

    Not totally accidentally named, given the way Archimedes expressed his sudden awareness about the volume of water displaced by his immersed body.

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Acknowledgement

We thank the participants to the EELP’2019 workshop (A forum with no formal proceedings but insightful presentations and lively discussions on epistemic extensions of logic programming systems) and the anonymous reviewers of LOPSTR’2020 for their constructive comments and suggestions.

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  1. Department of Computer Science and Engineering, University of North Texas, Denton, USA

    Paul Tarau

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  1. Paul Tarau
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Correspondence to Paul Tarau .

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  1. King’s College London, London, UK

    Maribel Fernández

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Tarau, P. (2021). Synthesis of Modality Definitions and a Theorem Prover for Epistemic Intuitionistic Logic. In: Fernández, M. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2020. Lecture Notes in Computer Science(), vol 12561. Springer, Cham. https://doi.org/10.1007/978-3-030-68446-4_17

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  • DOI: https://doi.org/10.1007/978-3-030-68446-4_17

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