Studia Logica

, Volume 77, Issue 2, pp 209–240 | Cite as

Algebraic Aspects of Cut Elimination

  • Francesco Belardinelli
  • Peter Jipsen
  • Hiroakira Ono
Article

Abstract

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].

Algebraic Gentzen systems cut elimination substructural logics residuated lattices finite model property 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Avigad, J., \lsAlgebraic proofs of cut elimination\rs, Journal of Logic and Algorithmic Programming 49 (2001), 15–30.CrossRefGoogle Scholar
  2. [2]
    Belardinelli, F., Aspetti semantici delle logiche sottostrutturali, graduate thesis, University of Pisa, 2002.Google Scholar
  3. [3]
    Blok, W. J., and C. J. van Alten, \lsThe finite embeddability property for residuated lattices, pocrims and BCK-algebras\rs, Algebra Universalis 48 (2002), 253–271.CrossRefGoogle Scholar
  4. [4]
    Fitting, M., \lsModel existence theorems for modal and intuitionistic logics\rs, Journal of Symbolic Logic 38 (1973), 613–627.Google Scholar
  5. [5]
    Gentzen, G., \lsUntersuchungen \:uber das logische Schliessen\rs, Mathematische Zeitschrift 39 (1934), 176–210, 405–413.Google Scholar
  6. [6]
    Girard, J.-Y., \lsLinear logic\rs, Theoretical Computer Science 50 (1987), 1–102.CrossRefGoogle Scholar
  7. [7]
    Girard, J.-Y., Proof Theory and Logical Complexity, vol. I, Studies in Proof Theory, Bibliopolis, 1987.Google Scholar
  8. [8]
    Grishin, V. N., \lsPredicate and set-theoretic calculi based on logic without contraction\rs, Math. USSR Izvestiya 18 (1982), 41–59.Google Scholar
  9. [9]
    Jipsen, P., and C. Tsinakis, \lsA survey of residuated lattices\rs, in J. Martinez, (ed.), Ordered Algebraic Structures, Kluwer Academic Publishers, 2002, pp. 19–56.Google Scholar
  10. [10]
    Komori, Y., \lsPredicate logics without the structural rules\rs, Studia Logica 45 (1986), 393–404.Google Scholar
  11. [11]
    Kowalski, T., and H. Ono, Residuated Lattices: An algebraic glimpse at logics without contraction, monograph, March, 2001.Google Scholar
  12. [12]
    Lafont, Y., \lsThe finite model property for various fragments of linear logic\rs, Journal of Symbolic Logic 62 (1997), 1202–1208.Google Scholar
  13. [13]
    Maehara, S., \lsLattice-valued representation of the cut-elimination theorem, Tsukuba Journal of Mathematics 15 (1991), 509–521.Google Scholar
  14. [14]
    Meyer, R. K., Topics in modal and many-valued logic, Doctoral dissertation, University of Pittsburgh, 1966.Google Scholar
  15. [15]
    R. K. Meyer, and H. Ono, \lsThe finite model property for BCK and BCIW\rs, Studia Logica 53 (1994), 107–118.Google Scholar
  16. [16]
    Okada, M., \lsPhase semantics for higher order completeness, cut-elimination and normalization proofs (extended abstract)\rs, Electronic Notes in Theoretical Computer Science 3 (1996).Google Scholar
  17. [17]
    Okada, M., and K. Terui, \lsThe finite model property for various fragments of intuitionistic linear logic\rs, Journal of Symbolic Logic 64 (1999), 790–802.Google Scholar
  18. [18]
    Ono, H., \lsSemantics for substructural logics\rs, in: K. Do\<sen and P. Schroeder-Heister, (eds.), Substructural Logics, Oxford University Press, 1993, pp. 259–291.Google Scholar
  19. [19]
    Ono, H., \lsDecidability and the finite model property of substructural logics\rs, in J. Ginzburg et. al., (eds.), Tbilisi Symposium on Logic, Language and Computation: Selected Papers (Studies in Logic, Language and Information), CSLI, 1998, pp. 263–274.Google Scholar
  20. [20]
    Ono, H., \lsProof-theoretic methods for nonclassical logic \3-an introduction\rs, in M. Takahashi, M. Okada and M. Dezani-Ciancaglini, (eds.), Theories of Types and Proofs (MSJ Memoirs 2), Mathematical Society of Japan, 1998, pp. 207–254.Google Scholar
  21. [21]
    Ono, H., \lsClosure operators and complete embeddings of residuated lattices\rs, Studia Logica 74 (2003), 427–440.CrossRefGoogle Scholar
  22. [22]
    Ono, H., \lsCompletions of algebras and completeness of modal and substructural logics\rs, in P. Balbiani et al, (eds.), Advances in Modal Logic 4, King's College Publications, 2003, pp. 335–353.Google Scholar
  23. [23]
    Ono, H. and Y. Komori, \lsLogics without the contraction rule\rs, Journal of Symbolic Logic 50 (1985), 169–201.Google Scholar
  24. [24]
    Sch\:utte, K., \lsSyntactical and semantical properties of simple type theory\rs, Journal of Symbolic Logic 25 (1960), 305–325.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Francesco Belardinelli
    • 1
  • Peter Jipsen
    • 2
  • Hiroakira Ono
    • 3
  1. 1.Classe di Lettere e Filosofia, Scuola Normale SuperioreP.zza dei CavalieriPisaItaly
  2. 2.Department of Mathematics, CS, PhysicsChapman UniversityOrangeUSA
  3. 3.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan

Personalised recommendations