Skip to main content
SpringerLink
Log in

Some General Results About Proof Normalization

  • Published:
Logica Universalis Aims and scope Submit manuscript

Abstract

In this paper, we provide a general setting under which results of normalization of proof trees such as, for instance, the logicality result in equational reasoning and the cut-elimination property in sequent or natural deduction calculi, can be unified and generalized. This is achieved by giving simple conditions which are sufficient to ensure that such normalization results hold, and which can be automatically checked since they are syntactical. These conditions are based on basic properties of elementary combinations of inference rules which ensure that the induced global proof tree transformation processes do terminate.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aiguier, M., Arnould, A., Boin, C., Le Gall, P., Marre, B.: Testing from algebraic specifications: test data set selection by unfolding axioms. In: Grieskamp, W., Weise, C. (eds.) Formal Approaches to Testing of Software (FATES), Lecture Notes in Computer Science, vol. 3997, pp. 304–317. Springer (2005)

  2. Aiguier, M., Arnould, A., Le Gall, P., Longuet, D.: Test selection criteria for quantifier-free first-order specifications. In: Arbab, F., Sirjani, M. (eds.) International Symposium on Fundamentals of Software ENgineering (FSEN), Lecture Notes in Computer Science, vol. 4767, pp. 144–160. Springer (2007)

  3. Aiguier M., Bahrami D.: Structures for abstract rewriting. J. Automat. Reason. 38(4), 303–351 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aiguier, M., Bahrami, D., Dubois, C.: On a generalised logicality theorem. In: Artificial Intelligence, Automated Reasoning, and Symbolic Computation, Lecture Notes in Artificial Intelligence, vol. 2385, pp. 51–64. Springer (2002)

  5. Aiguier M., Barbier F.: An institution-independent proof of the Beth definability theorem. Studi. Logica 85(3), 333–359 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Aiguier M., Diaconescu R.: Stratified institutions and elementary homomorphisms. Inform. Process. Lett. 103(1), 5–13 (2007)

    Article  MathSciNet  Google Scholar 

  7. Aiguier, M., Longuet, D.: Test selection criteria for modal specifications of reactive systems. In: Joint IEEE/IFIP Symposium on Theoretical Aspects of Software Engineering (TASE), pp. 159–170. IEEE Computer Society (2007)

  8. Baader, F., Nipkow, T.: Term Rewriting and All That. C.U. Press (1998)

  9. Belnap N.D.: Display logic. J. Philos. Logic 11, 375–417 (1982)

    MATH  MathSciNet  Google Scholar 

  10. Berger U., Eberl M., Schwichtenberg H.: Term rewriting for normalization by evaluation. Inform. Comput. 183, 19–42 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Birkhoff G.: On the structure of abstract algebras. Proc. Camb. Philos. Soc. 31, 433–454 (1935)

    Article  Google Scholar 

  12. Bittar, E.T.: Strong normalization proofs for cut-elimination in Gentzen’s sequent calculi. In: Proceedings of the Symposium on Algebra and Computer Science, Helena Rasiowa in Memoriam, vol. 46, pp. 179–225. Banach Center Publication (1999)

  13. Borzyszkowski T.: Logical systems for structured specifications. Theor. Comput. Sci. 286, 197–245 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cichon, E.-A., Rusinowitch, M., Selhab, S.: Cut elimination and rewriting: termination proofs. Technical report, INRIA-Lorraine, Nancy, France (1996)

  15. Cornes, C., Courant, J., Fillâtre, J.-C., Huet, G., Manoury, P., Munoz, C., Murthy, C., Parent, C., Paulin-Mohring, Ch., Sabi, A., Werner, B.: The Coq proof assistant reference manual, version 5.10. Technical Report 177, INRIA, July (1995)

  16. Dawson, J., Gore, R.: A new machine-checked proof of strong normalisation for display logic. In: Harland, J. (ed.) Electronic Notes in Theoretical Computer Science, vol. 78. Elsevier (2003)

  17. Dawson, J.-E., Goré, R.: A general theorem on termination of rewriting. In: Computer Science Logic (CSL’04), vol. 3210 of Lecture Notes in Computer Science, pp. 100–114. Springer (2004)

  18. Dawson, J.-E., Goré, R.: Termination of abstract reduction systems computing. In: The Australian Theory Symposium, 2007 (CATS 2007), Conferences in Research and Practice in Information Technology (CRPIT), vol. 65, pp. 35–43. Australian Computer Society (2007)

  19. Dershowitz N.: A note on simplification orderings. Inform. Process. Lett. 9(5), 212–215 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  20. Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Formal Models and Semantics, vol. B: pp. 243–320. Elsevier (1990)

  21. Dershowitz, N., Kirchner, C.: Abstract saturation-based inference. In: Kolaitis, P. (ed.) Proceedings of the Annual Symposium on Logic in Computer Science. IEEE Publiser Society (2003)

  22. Dershowitz N., Kirchner C.: Abstract canonical presentations. Theor. Comput. Sci. 357(1–3), 53–69 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Diaconescu, R.: Institution-independent Model Theory. Studies in Universal Logic. Birkhuser (2008)

  24. Diaconescu, R., Goguen, J., Stefaneas, P.: Logical support for modularization. In: Huet, G., Plotkin, G. (eds.) Logical Environments, pp. 83–130. Proceedings of Workshop on Logical Frameworks (1991)

  25. Dowek, G.: Confluence as a cut-elimination property. In: Nieuwenhuis, R. (ed.) Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 2706, pp. 2–14. Springer (2003)

  26. Dowek G., Hardin Th., Kirchner C.: Theorem proving modulo. J. Automat. Reason. 31, 33–72 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Dowek, G., Werner, B.: An inconsistent theory modulo defined by a confluent and terminating rewrite system. Available at www.lix.polytechnique.fr/~dowek/

  28. Dowek, G., Werner, B.: Proof normalization modulo. In: International Workshop on Types for Proofs and Programs, pp. 62–77. Springer, London, UK (1999)

  29. Dowek G., Werner B.: Proof normalization modulo. J. Symb. Logic 68(4), 1289–1316 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Dragalin, A.-G.: Mathematical intuitionism, introduction to proof theory. Transl. Math. Monogr. 67, 185–199. Translation of the russian original from 1979 (1988)

  31. Fiadeiro, J., Sernadas, A.: Structuring theories on consequence. In: Recent Trends in Algebraic Development Techniques, Lecture Notes in Computer Science, vol. 332, pp. 44–72. Springer (1988)

  32. Gallier, J.-H.: Logic for Computer Science, Volume Foundations of Automatic Theorem Proving. Harper & Row (1986)

  33. Gentzen, G.: Untersuchungen ber das logische schlieen. Mathematische Zeitschrift 39(176–210): 405–431 (1935). English translation In: Szabo, M.-E. (ed.) The collected Papers of Gerhard Gentzen, pp. 68–131, North-Holland (1969)

  34. Girard J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. Girard J.-Y., Lafont Y., Taylor P.: Proofs and Types. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  36. Goguen J.A., Burstall R.-M.: Institutions: Abstract model theory for specification and programming. J. ACM 39(1), 95–146 (1987)

    Article  MathSciNet  Google Scholar 

  37. Herbelin, H.: A λ-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds.) Proceedings of the 8th International Workshop on Computer Science Logic, Lecture Notes in Computer Science, vol. 933, pp. 61–75. Springer (1995)

  38. Hermant, O.: A model-based cut elimination proof. In: 2nd St-Petersburg Days of Logic and Computability (2003)

  39. Hermant, O.: Semantic cut elmination in the intuitionistic sequent calculus. In: Urzyczyn, P. (ed.) International Conference on Typed Lambda Calculi and Applications (TLCA), Lecture Notes in Computer Science, vol. 3461, pp. 221–233. Springer (2007)

  40. Kaplan S.: Conditional rewrite rules. Theor. Comput. Sci. 33, 175–193 (1984)

    Article  MATH  Google Scholar 

  41. Kleene S.-C.: Introduction to Meta-mathematics. North-Holland, Amsterdam (1952)

    Google Scholar 

  42. Levy J., Agustí J.: Bi-rewrite systems. J. Symb. Comput. 22, 279–314 (1996)

    Article  MATH  Google Scholar 

  43. Longuet, D., Aiguier, M.: Specification-based testing for CoCasl’s modal specifications. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) Conference on Algebra and Coalgebra in Computer Science (CALCO), Lecture Notes in Computer Science, vol. 4624, pp. 356–371. Springer (2007)

  44. Longuet, D., Aiguier, M.: Integration testing from structured specifications via deduction modulo. In: International Colloquium on Theoretical Aspects of Computing (2009, To appear).

  45. Longuet, D., Aiguier, M., Le Gall, P.: Proof-guided test selection from first-order specifications with equality. J. Automat. Reason. (2009) To appear

  46. Meseguer, J.: General logics. In: Logic Colloquium’87, pp. 275–329. Holland (1989)

  47. Nipkow, T., Paulson, L., Wenzel, M.: Isabelle’s logics: HOL. http://isabelle.in.tum.de/doc/logics-HOL.pdf

  48. Prawitz D.: Natural Deduction: A Proof-Theoretical Study. Almqvist and Wiksell, Stockholm (1965)

    MATH  Google Scholar 

  49. Salibra, A., Scollo, G.: Compactness and Löwenheim-Skolem properties in categories of pre-institutions. In: Rauszer, C. (ed.) Algebraic Methods in Logic and in Computer Science, Banach Center Publ. vol. 28, pp. 67–94 (1993)

  50. Salibra A., Scollo G.: Interpolation and compactness in categories of pre-institutions. Math. Struct. Comput. Sci. 6, 261–286 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  51. Schorlemmer, W.M.: Term rewriting in a logic of special relations. In: 7th International Conference on Algebraic Methodology and Software Technology, Lecture Notes in Computer Science, vol. 1548, pp. 178–195. Springer (1998)

  52. Struth, G.: Non-symmetric rewriting. Technical report, MPI für Informatik (1996)

  53. Tait, W.-W.: Normal derivability in classical logic. In: Barwise, J. (ed.) The Syntax and Semantics of Infinitary Languages, pp. 204–236. Springer (1989)

  54. Tarlecki A.: On the existence of free models in abstract algebraic institutions. Theor. Comput. Sci. 37, 269–304 (1986)

    Article  MathSciNet  Google Scholar 

  55. Tarlecki A.: Quasi-varieties in abstract algebraic institutions. J. Comput. Syst. Sci. 33(3), 269–304 (1986)

    MathSciNet  Google Scholar 

  56. Tarlecki, A.: Algebraic Foundations of Systems Specification, Chapter Institutions: An Abstract Framework for Formal Specifications, pp. 105–131. IFIP State-of-the-Art Reports. Springer (1999)

  57. Urban C., Bierman G.-M.: Strong normalisation of cut-elimination in classical logic. Acta Informaticae 45(1–2), 123–155 (2001)

    MATH  MathSciNet  Google Scholar 

  58. van Oostrom, V.: Sub-Birkhoff. In: 7th International Symposium on Functional and Logic Programming, Lecture Notes in Computer Science, vol. 2998, pp. 180–195. Springer (2004)

  59. Wallen, L.-A.: Automated Deduction for Non Classical Logics. The MIT Press (1990)

  60. Yamada T., Avenhaus J., Loria-Saenz C., Middeldorp A.: Logicality of conditional rewrite systems. Theor. Comput. Sci. 236(1,2), 209–232 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MAS, École Centrale Paris (ECP), Châtenay-Malabry, France

    Marc Aiguier

  2. Univ. Paris-Sud, LRI UMR8623, 91405, Orsay, France

    Delphine Longuet

Authors
  1. Marc Aiguier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Delphine Longuet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Aiguier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aiguier, M., Longuet, D. Some General Results About Proof Normalization. Log. Univers. 4, 1–29 (2010). https://doi.org/10.1007/s11787-010-0011-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-010-0011-4

Mathematics Subject Classification (2000)

Keywords

Search

Navigation