Journal of Automated Reasoning

, Volume 45, Issue 2, pp 157–188 | Cite as

A Framework for Proof Systems



Linear logic can be used as a meta-logic to specify a range of object-level proof systems. In particular, we show that by providing different polarizations within a focused proof system for linear logic, one can account for natural deduction (normal and non-normal), sequent proofs (with and without cut), and tableaux proofs. Armed with just a few, simple variations to the linear logic encodings, more proof systems can be accommodated, including proof system using generalized elimination and generalized introduction rules. In general, most of these proof systems are developed for both classical and intuitionistic logics. By using simple results about linear logic, we can also give simple and modular proofs of the soundness and relative completeness of all the proof systems we consider.


Linear logic Focusing Meta-logic Logical framework 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrusci, V.M., Ruet, P.: Non-commutative logic I: the multiplicative fragment. Ann. Pure Appl. Logic 101(1), 29–64 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. Log. Comput. 2(3), 297–347 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Artif. Intell. 4, 225–248 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chaudhuri, K., Pfenning, F., Price, G.: A logical characterization of forward and backward chaining in the inverse method. J. Autom. Reason. 40(2–3), 133–177 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5, 56–68 (1940)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ciabattoni, A., Galatos, N., Terui, K.: From axioms to analytic rules in nonclassical logics. In: 23th Symp. on Logic in Computer Science, pp. 229–240. IEEE Computer Society Press (2008)Google Scholar
  7. 7.
    D’Agostino, M., Mondadori, M.: The taming of the cut. Classical refutations with analytic cut. J. Log. Comput. 4(3), 285–319 (1994)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Danos, V., Joinet, J.-B., Schellinx, H.: LKT and LKQ: sequent calculi for second order logic based upon dual linear decompositions of classical implication. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic. London Mathematical Society Lecture Note Series, no. 222, pp. 211–224. Cambridge University Press (1995)Google Scholar
  9. 9.
    Dyckhoff, R., Lengrand, S.: LJQ: a strongly focused calculus for intuitionistic logic. In: Beckmann, A., et al. (eds.) Computability in Europe 2006. LNCS, vol. 3988, pp. 173–185. Springer (2006)Google Scholar
  10. 10.
    Felty, A., Miller, D.: Specifying theorem provers in a higher-order logic programming language. In: Ninth International Conference on Automated Deduction, pp. 61–80. Argonne, IL, Springer (1988)Google Scholar
  11. 11.
    Gentzen, G.: Investigations into logical deductions. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)Google Scholar
  12. 12.
    Girard, J.-Y.: Le Point Aveugle: Cours de Logique: Tome 1, Vers la Perfection. Hermann (2006)Google Scholar
  13. 13.
    Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. J. ACM 40(1), 143–184 (1993)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Henriksen, A.S.: Using LJF as a Framework for Proof Systems. Technical Report, U. of Copenhagen (2009). Available from
  16. 16.
    Hodas, J., Miller, D.: Logic programming in a fragment of intuitionistic linear logic. Inf. Comput. 110(2), 327–365 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hughes, D.: A minimal classical sequent calculus free of structural rule. Ann. Pure Appl. Logic 161(10), 1244–1253 (2010)CrossRefGoogle Scholar
  18. 18.
    Lambek, J.: The mathematics of sentence structure. Am. Math. Mon. 65, 154–169 (1958)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theor. Comput. Sci. 410(46), 4747–4768 (2009)MATHCrossRefGoogle Scholar
  20. 20.
    Maehara, S.: Eine darstellung der intuitionistischen logik in der klassischen. Nagoya Math. J. 7, 45–64 (1954)MATHMathSciNetGoogle Scholar
  21. 21.
    Miller, D.: Forum: a multiple-conclusion specification logic. Theor. Comput. Sci. 165(1), 201–232 (1996)MATHCrossRefGoogle Scholar
  22. 22.
    Miller, D., Nigam, V.: Incorporating tables into proofs. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007: Computer Science Logic. LNCS, vol. 4646, pp. 466–480. Springer (2007)Google Scholar
  23. 23.
    Miller, D., Pimentel, E.: Using linear logic to reason about sequent systems. In: Egly, U., Fermüller, C.G. (eds.) International Conference on Automated Reasoning with Analytic Tableaux and Related Methods. LNCS, vol. 2381, pp. 2–23. Springer (2002)Google Scholar
  24. 24.
    Miller, D., Pimentel, E.: Linear logic as a framework for specifying sequent calculus. In: van Eijck, J., van Oostrom, V., Visser, A. (eds.) Logic Colloquium ’99: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic. Lecture Notes in Logic, pp. 111–135. A K Peters Ltd (2004)Google Scholar
  25. 25.
    Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Logic 51, 125–157 (1991)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Negri, S., von Plato, J.: Structural Proof Theory. Cambridge University Press (2001)Google Scholar
  27. 27.
    Nigam, V.: Exploiting Non-canonicity in the Sequent Calculus. PhD thesis, Ecole Polytechnique (2009)Google Scholar
  28. 28.
    Nigam, V., Miller, D.: Focusing in linear meta-logic. In: Proceedings of IJCAR: International Joint Conference on Automated Reasoning. LNAI, vol. 5195, pp. 507–522. Springer (2008)Google Scholar
  29. 29.
    Nigam, V., Miller, D.: Focusing in Linear Meta-logic: Extended Report. Available at: (2008). Accessed 24 May 2008
  30. 30.
    Nigam, V., Miller, D.: Algorithmic specifications in linear logic with subexponentials. In: ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP), pp. 129–140 (2009)Google Scholar
  31. 31.
    Parigot, M.: Free deduction: an analysis of “computations” in classical logic. In: Proceedings of the First Russian Conference on Logic Programming, pp. 361–380. London, UK, Springer (1992)Google Scholar
  32. 32.
    Paulson, L.C.: The foundation of a generic theorem prover. J. Autom. Reason. 5, 363–397 (1989)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Pfenning, F.: Elf: a language for logic definition and verified metaprogramming. In: Logic in Computer Science, pp. 313–321. Monterey, CA (1989)Google Scholar
  34. 34.
    Pimentel, E., Miller, D.: On the specification of sequent systems. In: LPAR 2005: 12th International Conference on Logic for Programming, Artificial Intelligence and Reasoning. LNAI, no. 3835, pp. 352–366 (2005)Google Scholar
  35. 35.
    Pimentel, E.G.: Lógica Linear e a Especificação de Sistemas Computacionais. PhD thesis, Universidade Federal de Minas Gerais, Belo Horizonte, M.G., Brasil (2001). Written in EnglishGoogle Scholar
  36. 36.
    Prawitz, D.: Natural Deduction. Almqvist & Wiksell, Uppsala (1965)MATHGoogle Scholar
  37. 37.
    Schroeder-Heister, P.: A natural extension of natural deduction. J. Symb. Log. 49(4), 1284–1300 (1984)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Sieg, W., Byrnes, J.: Normal natural deduction proofs (in classical logic). Stud. Log. 60(1), 67–106 (1998)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Smullyan, R.M.: First-Order Logic. Springer, New York (1968)MATHGoogle Scholar
  40. 40.
    Smullyan, R.M.: Analytic cut. J. Symb. Log. 33(4), 560–564 (1968)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    von Plato, J.: Natural deduction with general elimination rules. Arch. Math. Log. 40(7), 541–567 (2001)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.INRIA Saclay & LIX/École PolytechniquePalaiseauFrance

Personalised recommendations