Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Science Logic

CSL 2007: Computer Science Logic pp 238–252Cite as

The Theory of Calculi with Explicit Substitutions Revisited

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4646))

Abstract

Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with meta-level substitutions) they were implementing.

In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proof-nets.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arbiser, A., Bonelli, E., Ríos, A.: Perpetuality in a lambda calculus with explicit substitutions and composition. In: WAIT (2000)

    Google Scholar 

  2. Abadi, M., Cardelli, L., Curien, P.L., Lévy, J.-J.: Explicit substitutions. JFP 4(1), 375–416 (1991)

    Google Scholar 

  3. Arbiser, A.: Explicit Substitution Systems and Subsystems. PhD thesis, Universidad Buenos Aires (2006)

    Google Scholar 

  4. Barendregt, H.: The Lambda Calculus: Its Syntax and Semantics. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  5. Barendregt, H.: Lambda calculus with types. Handbook of Logic in Computer Science 2 (1992)

    Google Scholar 

  6. Benaissa, Z.-E.-A., Briaud, D., Lescanne, P., Rouyer-Degli, J.: λυ, a calculus of explicit substitutions which preserves strong normalisation. JFP  (1996)

    Google Scholar 

  7. Bloo, R., Geuvers, H.: Explicit substitution: on the edge of strong normalization. TCS 6(5), 699–722 (1999)

    Google Scholar 

  8. Bloo, R.: Preservation of Termination for Explicit Substitution. PhD thesis, Eindhoven University of Technology (1997)

    Google Scholar 

  9. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  10. Bloo, R., Rose, K.: Preservation of strong normalization in named lambda calculi with explicit substitution and garbage collection. In: Computer Science in the Netherlands (1995)

    Google Scholar 

  11. de Bruijn, N.: Lambda-calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem. Indag. Mat. 5(35), 381–392 (1972)

    Google Scholar 

  12. de Bruijn, N.: Lambda-calculus notation with namefree formulas involving symbols that represent reference transforming mappings. Indag. Mat. 40, 356–384 (1978)

    Google Scholar 

  13. Di Cosmo, R., Kesner, D., Polonovski, E.: Proof nets and explicit substitutions. In: Tiuryn, J. (ed.) ETAPS 2000 and FOSSACS 2000. LNCS, vol. 1784, Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  14. David, R., Guillaume, B.: A λ-calculus with explicit weakening and explicit substitution. MSCS 11, 169–206 (2001)

    MATH  MathSciNet  Google Scholar 

  15. Dowek, G., Hardin, T., Kirchner, C.: Higher-order unification via explicit substitutions. I&C 157, 183–235 (2000)

    MATH  MathSciNet  Google Scholar 

  16. Dyckhoff, R., Urban, C.: Strong normalisation of Herbelin’s explicit substitution calculus with substitution propagation. In: WESTAPP 2001 (2001)

    Google Scholar 

  17. de Flavio Moura, M.A.-R., Kamareddine, F.: Higher order unification: A structural relation between Huet’s method and the one based on explicit substitution. Available from http://www.macs.hw.ac.uk/~fairouz/papers/

  18. Forest, J.: A weak calculus with explicit operators for pattern matching and substitution. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. Girard, J.-Y.: Linear logic. TCS 50(1), 1–101 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Goubault-Larrecq, J.: Conjunctive types and SKInT. In: Altenkirch, T., Naraschewski, W., Reus, B. (eds.) TYPES 1998. LNCS, vol. 1657, Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  21. Hardin, T.: Résultats de confluence pour les règles fortes de la logique combinatoire catégorique et liens avec les lambda-calculs. Thèse de doctorat, Université de Paris VII (1987)

    Google Scholar 

  22. Herbelin, H.: A ł-calculus structure isomorphic to sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  23. Hardin, T., Lévy, J.-J.: A confluent calculus of substitutions. In: France-Japan Artificial Intelligence and Computer Science Symposium (1989)

    Google Scholar 

  24. Hardin, T., Maranget, L., Pagano, B.: Functional back-ends within the lambda-sigma calculus. In: ICFP (1996)

    Google Scholar 

  25. Huet, G.: Résolution d’équations dans les langages d’ordre 1,2, ..., ω. Thèse de doctorat d’état, Université Paris VII (1976)

    Google Scholar 

  26. Khasidashvili, Z.: Expression reduction systems. In: Proceedings of IN Vekua Institute of Applied Mathematics, Tbilisi, vol. 36 (1990)

    Google Scholar 

  27. Kesner, D.: Confluence properties of extensional and non-extensional λ-calculi with explicit substitutions. In: Ganzinger, H. (ed.) Rewriting Techniques and Applications. LNCS, vol. 1103, Springer, Heidelberg (1996)

    Google Scholar 

  28. Kesner, D.: The theory of calculi with explicit substitutions revisited (2006), Available as http://hal.archives-ouvertes.fr/hal-00111285/

  29. Kesner, D., Lengrand, S.: Extending the explicit substitution paradigm. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, Springer, Heidelberg (2005)

    Google Scholar 

  30. Klop, J.-W.: Combinatory Reduction Systems. PhD thesis, Mathematical Centre Tracts 127, CWI, Amsterdam (1980)

    Google Scholar 

  31. Khasidashvili, Z., Ogawa, M., van Oostrom, V.: Uniform Normalization Beyond Orthogonality. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  32. Kamareddine, F.: A λ-calculus à la de Bruijn with explicit substitutions. In: Swierstra, S.D. (ed.) PLILP 1995. LNCS, vol. 982, Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  33. Lengrand, S., Dyckhoff, R., McKinna, J.: A sequent calculus for type theory. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  34. Lengrand, S.: Normalisation and Equivalence in Proof Theory and Type Theory. PhD thesis, University Paris 7 and University of St Andrews (2006)

    Google Scholar 

  35. Lescanne, P.: From λ σ to λ υ , a journey through calculi of explicit substitutions. In: POPL (1994)

    Google Scholar 

  36. Lins, R.: A new formula for the execution of categorical combinators. In: Siekmann, J.H. (ed.) 8th International Conference on Automated Deduction. LNCS, vol. 230, Springer, Heidelberg (1986)

    Google Scholar 

  37. Lins, R.: Partial categorical multi-combinators and Church Rosser theorems. Technical Report 7/92, Computing Laboratory, University of Kent at Canterbury (1992)

    Google Scholar 

  38. Lévy, J.-J., Maranget, L.: Explicit substitutions and programming languages. In: Pandu Rangan, C., Raman, V., Ramanujam, R. (eds.) Foundations of Software Technology and Theoretical Computer Science. LNCS, vol. 1738, Springer, Heidelberg (1999)

    Google Scholar 

  39. Lescanne, P., Rouyer-Degli, J.: Explicit substitutions with de Bruijn levels. In: Hsiang, J. (ed.) Rewriting Techniques and Applications. LNCS, vol. 914, Springer, Heidelberg (1995)

    Google Scholar 

  40. Melliès, P.-A.: Typed λ-calculi with explicit substitutions may not terminate. In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, Springer, Heidelberg (1995)

    Google Scholar 

  41. Muñoz, C.: Un calcul de substitutions pour la représentation de preuves partielles en théorie de types. PhD thesis, Université Paris 7 (1997)

    Google Scholar 

  42. Nederpelt, R.: Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Eindhoven University of Technology (1973)

    Google Scholar 

  43. Polonovski, E.: Substitutions explicites, logique et normalisation. Thèse de doctorat, Université Paris 7 (2004)

    Google Scholar 

  44. Rose, K.: Explicit cyclic substitutions. In: Rusinowitch, M., Remy, J.-L. (eds.) Conditional Term Rewriting Systems. LNCS, vol. 656, Springer, Heidelberg (1993)

    Google Scholar 

  45. Sakurai, T.: Strong normalizability of calculus of explicit substitutions with composition. Available on http://www.math.s.chiba-u.ac.jp/~sakurai/papers.html

  46. Sinot, F.-R., Fernández, M., Mackie, I.: Efficient reductions with director strings. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  47. Terese.: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. PPS, Université Paris 7 and CNRS (UMR 7126), France

    Delia Kesner

Authors
  1. Delia Kesner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jacques Duparc Thomas A. Henzinger

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kesner, D. (2007). The Theory of Calculi with Explicit Substitutions Revisited. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-74915-8_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74914-1

  • Online ISBN: 978-3-540-74915-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation