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International Conference on Automated Reasoning with Analytic Tableaux and Related Methods

TABLEAUX 2019: Automated Reasoning with Analytic Tableaux and Related Methods pp 355–373Cite as

Herbrand Constructivization for Automated Intuitionistic Theorem Proving

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

We describe a new method to constructivize proofs based on Herbrand disjunctions by giving a practically effective algorithm that converts (some) classical first-order proofs into intuitionistic proofs. Together with an automated classical first-order theorem prover such a method yields an (incomplete) automated theorem prover for intuitionistic logic. Our implementation of this prover approach, Slakje, performs competitively on the ILTP benchmark suite for intuitionistic provers: it solves 1674 out of 2670 problems (1290 proofs and 384 claims of non-provability) with Vampire as a backend, including 800 previously unsolved problems.

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Notes

  1. 1.

    Considering of course the law of excluded middle for arbitrary types, not just mere propositions.

  2. 2.

    Internally, Vampire, and in general most classical provers then refute \(\lnot \varphi \).

  3. 3.

    We use the convention that the quantifiers \(\forall , \exists \) bind stronger than \(\rightarrow , \wedge , \vee \). That is, \(\forall x\, P(x) \rightarrow Q\) is the same formula as \((\forall x\, P(x)) \rightarrow Q\). Furthermore, \(\rightarrow \) is right-associative, that is, \(P \rightarrow Q \rightarrow R\) is the same formula as \(P \rightarrow (Q \rightarrow R)\).

  4. 4.

    Proof systems (for propositional logic) are typically required to be polynomial-time checkable, as certificates to the coNP-complete validity problem. Expansion proofs are coNP-checkable certificates for the undecidable first-order validity problem.

  5. 5.

    Open source, and freely available at https://logic.at/gapt.

  6. 6.

    available at http://iltp.de/download/ILTP-v1.1.2-firstorder.tar.gz.

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Acknowledgements

The author would like to thank the anonymous reviewers for their suggestions which have led to a considerable improvement of this paper. This work has been supported by the Vienna Science and Technology Fund (WWTF) project VRG12-004.

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  1. TU Wien, Vienna, Austria

    Gabriel Ebner

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  1. Gabriel Ebner
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  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

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Ebner, G. (2019). Herbrand Constructivization for Automated Intuitionistic Theorem Proving. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_20

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