Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Using Linear Logic to Reason about Sequent Systems

  • Conference paper
  • First Online:
Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2381))

Abstract

Linear logic can be used as a meta-logic for the specification of some sequent calculus proof systems. We explore in this paper properties of such linear logic specifications. We show that derivability of one proof system from another has a simple decision procedure that is implemented simply via bounded logic programming search. We also provide conditions to ensure that an encoded proof system has the cut-elimination property and show that this can be decided again by simple, bounded proof search algorithms.

Miller has been supported in part by NSF grants CCR-9912387, CCR-9803971, INT-9815645, and INT-9815731. Both authors wish to thank L’Institut de Mathématiques de Luminy, University Aix-Marseille 2 for the support to attend the Logic and Interaction Weeks in February 2002, during which much of this paper was written.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Arnon Avron and Iddo Lev. Canonical propositional gentzen-type systems. In R. Goré, A. Leitsch, and T. Nipkow, editors, IJCAR 2001, volume 2083 of LNAI, pages 529–544. Springer-Verlag, 2001.

    Google Scholar 

  2. Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297–347, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  3. V. Michele Abrusci and Paul Ruet. Non-commutative logic I: The multiplicative fragment. Annals of Pure and Applied Logic, 101(1):29–64, 1999.

    Article  MathSciNet  Google Scholar 

  4. Jawahar Chirimar. Proof Theoretic Approach to Specification Languages. PhD thesis, University of Pennsylvania, February 1995.

    Google Scholar 

  5. Alonzo Church. A formulation of the simple theory of types. Journal of Symbolic Logic, 5:56–68, 1940.

    Article  MATH  MathSciNet  Google Scholar 

  6. Vincent Danos, Jean-Baptiste Joinet, and Harold Schellinx. LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication. In Girard, Lafont, and Regnier, editors, Workshop on Linear Logic, pages 211–224. London Mathematical Society Lecture Notes 222, Cambridge University Press, 1995.

    Google Scholar 

  7. Nachum Dershowitz and Zohar Manna. Proving termination with multiset orderings. Communications of the ACM, 22(8):465–476, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  8. Giorgio Delzanno and Maurizio Martelli. Objects in Forum. In Proceedings of the International Logic Programming Symposium, 1995.

    Google Scholar 

  9. Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3):795–807, September 1992.

    Google Scholar 

  10. Amy Felty and Dale Miller. Specifying theorem provers in a higher-order logic programming language. In Ninth International Conference on Automated Deduction, pages 61–80, Argonne, IL, May 1988. Springer-Verlag.

    Google Scholar 

  11. Gerhard Gentzen. Investigations into logical deductions. In M. E. Szabo, editor, The Collected Papers of Gerhard Gentzen, pages 68–131. North-Holland Publishing Co., Amsterdam, 1969.

    Google Scholar 

  12. Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201–217, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  13. Alessio Guglielmi and Lutz Straßburger. Non-commutativity and MELL in the calculus of structures. In L. Fribourg, editor, CSL 2001, volume 2142 of LNCS, pages 54–68, 2001.

    Google Scholar 

  14. Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. Journal of the ACM, 40(1):143–184, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  15. Patrick Lincoln, Andre Scedrov, and Natarajan Shankar. Linearizing intuitionistic implication. In Annals of Pure and Applied Logic, pages 151–177, 1993.

    Google Scholar 

  16. Dale Miller. The π-calculus as a theory in linear logic: Preliminary results. In E. Lamma and P. Mello, editors, Proceedings of the 1992 Workshop on Extensions to Logic Programming, number 660 in LNCS, pages 242–265. Springer-Verlag, 1993.

    Google Scholar 

  17. Dale Miller. Forum: A multiple-conclusion specification language. Theoretical Computer Science, 165(1):201–232, September 1996.

    Google Scholar 

  18. Dale Miller. Abstract syntax for variable binders: An overview. In John Lloyd and et. al., editors, Computational Logic-CL 2000, number 1861 in LNAI, pages 239–253. Springer, 2000.

    Google Scholar 

  19. Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91–119, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  20. Dale Miller, Gopalan Nadathur, Frank Pfenning, and Andre Scedrov. Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic, 51:125–157, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  21. Dale Miller and Elaine Pimentel. Linear logic as a framework for specifying sequent calculus. To appear in the Proceedings of Logic Colloquium 1999.

    Google Scholar 

  22. Gopalan Nadathur and Dale Miller. An Overview of λProlog. In Fifth International Logic Programming Conference, pages 810–827, Seattle, August 1988. MIT Press.

    Google Scholar 

  23. Lawrence C. Paulson. The foundation of a generic theorem prover. Journal of Automated Reasoning, 5:363–397, September 1989.

    Google Scholar 

  24. Elaine Gouvêa Pimentel. Lógica linear e a especificação de sistemas computacionais. PhD thesis, Universidade Federal de Minas Gerais, Belo Horizonte, M.G., Brasil, December 2001. (written in English).

    Google Scholar 

  25. Christian Retoré. Pomset logic: a non-commutative extension of classical linear logic. In Proceedings of TLCA, volume 1210, pages 300–318, 1997.

    Google Scholar 

  26. Giorgia Ricci. On the expressive powers of a Logic Programming presentation of Linear Logic (FORUM). PhD thesis, Department of Mathematics, Siena University, December 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science and Engineering Department, 220 Pond Lab, Pennsylvania State University, University Park, PA, 16802-6106, USA

    Dale Miller

  2. Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, M.G., Brasil

    Elaine Pimentel

Authors
  1. Dale Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elaine Pimentel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Abteilung Wisssenbasierte Systeme 184/3, Technische Univesität Wien, Favoritenstr. 9-11, 1040, Vienna, Austria

    Uwe Egly

  2. Institut für Computersprachen AG Theoretische Informatik und Logik 185.2, Technische Universität Wien, Favoritenstr. 9-11, 1040, Vienna, Austria

    Chritian G. Fermüller

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Miller, D., Pimentel, E. (2002). Using Linear Logic to Reason about Sequent Systems. In: Egly, U., Fermüller, C.G. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2002. Lecture Notes in Computer Science(), vol 2381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45616-3_2

Download citation

  • DOI: https://doi.org/10.1007/3-540-45616-3_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43929-5

  • Online ISBN: 978-3-540-45616-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation