Abstract
It is well known how to use an intuitionistic meta-logic to specify natural deduction systems. It is also possible to use linear logic as a meta-logic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and non-normal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. of Logic and Computation 2(3), 297–347 (1992)
D’Agostino, M., Mondadori, M.: The taming of the cut. Classical refutations with analytic cut 4(3), 285–319 (1994)
Felty, A., Miller, D.: Specifying theorem provers in a higher-order logic programming language. In: Ninth International Conference on Automated Deduction, Argonne, IL, May 1988, pp. 61–80. Springer, Heidelberg (1988)
Gentzen, G.: Investigations into logical deductions. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)
Girard, J.-Y.: Le Point Aveugle: Cours de logique: Tome 1, Vers la perfection. Hermann (2006)
Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the ACM 40(1), 143–184 (1993)
Hodas, J., Miller, D.: Logic programming in a fragment of intuitionistic linear logic. Information and Computation 110(2), 327–365 (1994)
Liang, C., Miller, D.: Focusing and polarization in intuitionistic logic. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 451–465. Springer, Heidelberg (2007)
Miller, D.: Forum: A multiple-conclusion specification logic. Theoretical Computer Science 165(1), 201–232 (1996)
Miller, D., Nigam, V.: Incorporating tables into proofs. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 466–480. Springer, Heidelberg (2007)
Miller, D., Pimentel, E.: Using linear logic to reason about sequent systems. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 2–23. Springer, Heidelberg (2002)
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 1999: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, pp. 111–135. A. K. Peters Ltd (2004)
Nigam, V., Miller, D.: Focusing in linear meta-logic: Extended report, http://hal.inria.fr/inria-00281631
Negri, S., Von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)
Parigot, M.: Free deduction: An analysis of “computations” in classical logic. In: Proceedings of the First Russian Conference on Logic Programming, London, UK, pp. 361–380. Springer, Heidelberg (1992)
Pfenning, F.: Elf: A language for logic definition and verified metaprogramming. In: Fourth Annual Symposium on Logic in Computer Science, Monterey, CA, June 1989, pp. 313–321 (1989)
Pfenning, F.: Structural cut elimination I. intuitionistic and classical logic 157(1/2), 84–141 (2000)
Pimentel, E.G.: Lógica linear e a especificação de sistemas computacionais. PhD thesis, Universidade Federal de Minas Gerais, Belo Horizonte, M.G., Brasil, Written in English (December 2001)
Pimentel, E., Miller, D.: On the specification of sequent systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 352–366. Springer, Heidelberg (2005)
Prawitz, D.: Natural Deduction. Almqvist & Wiksell, Uppsala (1965)
Sieg, W., Byrnes, J.: Normal natural deduction proofs (in classical logic). Studia Logica 60(1), 67–106 (1998)
Schroeder-Heister, P.: A natural extension of natural deduction. Journal of Symbolic Logic 49(4), 1284–1300 (1984)
Smullyan, R.M.: Analytic cut. J. of Symbolic Logic 33(4), 560–564 (1968)
von Plato, J.: Natural deduction with general elimination rules. Archive for Mathematical Logic 40(7), 541–567 (2001)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nigam, V., Miller, D. (2008). Focusing in Linear Meta-logic. In: Armando, A., Baumgartner, P., Dowek, G. (eds) Automated Reasoning. IJCAR 2008. Lecture Notes in Computer Science(), vol 5195. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71070-7_42
Download citation
DOI: https://doi.org/10.1007/978-3-540-71070-7_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71069-1
Online ISBN: 978-3-540-71070-7
eBook Packages: Computer ScienceComputer Science (R0)