Subformula Linking as an Interaction Method

  • Kaustuv Chaudhuri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7998)


Current techniques for building formal proofs interactively involve one or several proof languages for instructing an interpreter of the languages to build or check the proof being described. These linguistic approaches have a drawback: the languages are not generally portable, even though the nature of logical reasoning is universal. We propose a somewhat speculative alternative method that lets the user directly manipulate the text of the theorem, using non-linguistic metaphors. It uses a proof formalism based on linking subformulas, which is a variant of deep inference (inference rules are allowed to apply in any formula context) where the relevant formulas in a rule are allowed to be arbitrarily distant. We substantiate the design with a prototype implementation of a linking-based interactive prover for first-order classical linear logic.


Inference Rule Formal Proof Linear Logic Proof Tree Positive Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. of Logic and Computation 2(3), 297–347 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baelde, D.: Least and greatest fixed points in linear logic. ACM Trans. on Computational Logic 13(1) (April 2012)Google Scholar
  3. 3.
    Bertot, Y.: The CtCoq system: Design and architecture. Formal Aspects of Computing 11(3), 225–243 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertot, Y., Kahn, G., Théry, L.: Proof by pointing. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 141–160. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  5. 5.
    Boespflug, M., Carbonneaux, Q., Hermant, O.: The λΠ-calculus modulo as a universal proof language. In: Pichardie, D., Weber, T. (eds.) Proceedings of PxTP 2012: Proof Exchange for Theorem Proving, pp. 28–43 (2012)Google Scholar
  6. 6.
    Chaudhuri, K.: Profound (2013),
  7. 7.
    Chaudhuri, K., Guenot, N., Straßburger, L.: The Focused Calculus of Structures. In: Computer Science Logic: 20th Annual Conference of the EACSL, Leibniz International Proceedings in Informatics (LIPIcs), pp. 159–173. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (September 2011)Google Scholar
  8. 8.
    Chaudhuri, K., Pfenning, F., Price, G.: A logical characterization of forward and backward chaining in the inverse method. J. of Automated Reasoning 40(2-3), 133–177 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gacek, A.: The Abella system and homepage (2009),
  10. 10.
    Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Thery, G.K.L., Bertot, Y.: Real theorem provers deserve real user-interfaces. In: Proceedings of the Fifth ACM SIGSOFT Symposium on Software Development Environments. Software Engineering Notes, vol. 17(5). ACM Press (1992)Google Scholar
  12. 12.
    Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410(46), 4747–4768 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    McLaughlin, S., Pfenning, F.: Imogen: Focusing the polarized focused inverse method for intuitionistic propositional logic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 174–181. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Snow, Z., Baelde, D., Nadathur, G.: A meta-programming approach to realizing dependently typed logic programming. In: ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP), pp. 187–198 (2010)Google Scholar
  15. 15.
    Straßburger, L.: Linear Logic and Noncommutativity in the Calculus of Structures. PhD thesis, Technische Universität Dresden (2003)Google Scholar
  16. 16.
    Straßburger, L.: Some observations on the proof theory of second order propositional multiplicative linear logic. In: Curien, P.-L. (ed.) TLCA 2009. LNCS, vol. 5608, pp. 309–324. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Stump, A.: Proof checking technology for satisfiability modulo theories. In: Abel, A., Urban, C. (eds.) Logical Frameworks and Meta-Languages: Theory and Practice (2008)Google Scholar
  18. 18.
    Tiu, A.: A Logical Framework for Reasoning about Logical Specifications. PhD thesis, Pennsylvania State University (May 2004)Google Scholar
  19. 19.
    Troelstra, A.S.: Lectures on Linear Logic. CSLI Lecture Notes 29, Center for the Study of Language and Information, Stanford, California (1992)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kaustuv Chaudhuri
    • 1
  1. 1.INRIAFrance

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