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Constructive Decision via Redundancy-Free Proof-Search

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Automated Reasoning (IJCAR 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10900))

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Abstract

We give a constructive account of Kripke-Curry’s method which was used to establish the decidability of Implicational Relevance Logic (\(\mathbf R_{{\rightarrow }}\)). To sustain our approach, we mechanize this method in axiom-free Coq, abstracting away from the specific features of \(\mathbf R_{{\rightarrow }}\) to keep only the essential ingredients of the technique. In particular we show how to replace Kripke/Dickson’s lemma by a constructive form of Ramsey’s theorem based on the notion of almost full relation. We also explain how to replace König’s lemma with an inductive form of Brouwer’s Fan theorem. We instantiate our abstract proof to get a constructive decision procedure for \(\mathbf R_{{\rightarrow }}\) and discuss potential applications to other logical decidability problems.

Work partially supported by the TICAMORE project (ANR grant 16-CE91-0002).

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Notes

  1. 1.

    As for coverability in BVASS, it seems that the arguments developed in [7] cannot easily be converted to constructive ones (private communication with S. Demri).

  2. 2.

    Dickson’s lemma states that under pointwise order, \(\mathbb N^k\) is a WQO for any \(k\in \mathbb N\).

  3. 3.

    Unrestricted contraction would generate infinitely branching proof-search.

  4. 4.

    This result is known as Dickson’s lemma when restricted to \(\mathbb N^k\) with the point-wise product order. The inclusion relation between multisets built from the finite set \(\mathcal S\) is a particular case of the product order \(\mathbb N^k\) where k is the cardinal of \(\mathcal S\).

  5. 5.

    The braces around specify an implicit argument.

  6. 6.

    and _ _ are shorthand notations for the two list constructors.

  7. 7.

    The notation is a shortcut for , the (finitary) membership predicate.

  8. 8.

    Typically, systems which include a cut-rule do not satisfy the property which is why cut-elimination is viewed as a critical requisite to design sequent-based decision procedures. The same remark holds for the modus-ponens rule of Hilbert systems, usually making them unsuited for decision procedures.

  9. 9.

    For this, we need a notion of sub-statement that is reflexive, transitive and such that valid rules instances possess the sub-statement property.

  10. 10.

    Branches are read from the root to leaves, hence the use of to reverse lists.

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Authors and Affiliations

  1. Université de Lorraine, CNRS, LORIA, Nancy, France

    Dominique Larchey-Wendling

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  1. Dominique Larchey-Wendling
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Correspondence to Dominique Larchey-Wendling .

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Editors and Affiliations

  1. Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Didier Galmiche

  2. Baden-Wuerttemberg Cooperative State University, Stuttgart, Germany

    Stephan Schulz

  3. University of Trento, Trento, Italy

    Roberto Sebastiani

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Larchey-Wendling, D. (2018). Constructive Decision via Redundancy-Free Proof-Search. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_28

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  • DOI: https://doi.org/10.1007/978-3-319-94205-6_28

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