Skip to main content
SpringerLink
Log in

Typed Applicative Structures and Normalization by Evaluation for System Fω

  • Conference paper
Computer Science Logic (CSL 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5771))

Included in the following conference series:

Abstract

We present a normalization-by-evaluation (NbE) algorithm for System Fω with βη-equality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the algorithm. The algorithm is verified through a general model construction using typed applicative structures, called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem.

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

Access this chapter

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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abel, A.: Weak βη-normalization and normalization by evaluation for System F. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 497–511. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Abel, A.: Typed applicative structures and normalization by evaluation for System Fω (full version) (2009), http://www.tcs.ifi.lmu.de/~abel/fomegaNbe.pdf

  3. Abel, A., Coquand, T., Dybjer, P.: Normalization by evaluation for Martin-Löf Type Theory with typed equality judgements. In: LICS 2007, pp. 3–12. IEEE CS Press, Los Alamitos (2007)

    Google Scholar 

  4. Abel, A., Coquand, T., Dybjer, P.: Verifying a semantic βη-conversion test for Martin-Löf type theory. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 29–56. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  5. Altenkirch, T., Dybjer, P., Hofmann, M., Scott, P.J.: Normalization by evaluation for typed lambda calculus with coproducts. In: LICS 2001, pp. 303–310. IEEE CS Press, Los Alamitos (2001)

    Google Scholar 

  6. Altenkirch, T., Hofmann, M., Streicher, T.: Reduction-free normalisation for a polymorphic system. In: LICS 1996, pp. 98–106. IEEE CS Press, Los Alamitos (1996)

    Google Scholar 

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

    MATH  Google Scholar 

  8. Barral, F.: Decidability for non-standard conversions in lambda-calculus. PhD thesis, Ludwig-Maximilians-University, Munich (2008)

    Google Scholar 

  9. Balat, V., Di Cosmo, R., Fiore, M.P.: Extensional normalisation and type-directed partial evaluation for typed lambda calculus with sums. In: POPL 2004, pp. 64–76. ACM, New York (2004)

    Google Scholar 

  10. Bruce, K.B., Meyer, A.R.: The semantics of second order polymorphic lambda calculus. In: Plotkin, G., MacQueen, D.B., Kahn, G. (eds.) Semantics of Data Types 1984. LNCS, vol. 173, pp. 131–144. Springer, Heidelberg (1984)

    Chapter  Google Scholar 

  11. Berger, U., Schwichtenberg, H.: An inverse to the evaluation functional for typed λ-calculus. In: LICS 1991, pp. 203–211. IEEE CS Press, Los Alamitos (1991)

    Google Scholar 

  12. Chapman, J., Altenkirch, T., McBride, C.: Epigram reloaded: a standalone typechecker for ETT. In: TFP 2005. Trends in Functional Programming, vol. 6, pp. 79–94. Intellect (2007)

    Google Scholar 

  13. Coquand, T., Dybjer, P.: Intuitionistic model constructions and normalization proofs. MSCS 7(1), 75–94 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Coquand, T.: An algorithm for type-checking dependent types. In: MPC 1995. SCP, vol. 26, pp. 167–177. Elsevier, Amsterdam (1995)

    Google Scholar 

  15. Danvy, O.: Type-directed partial evaluation. In: Hatcliff, J., Thiemann, P. (eds.) DIKU 1998. LNCS, vol. 1706, pp. 367–411. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  16. Dybjer, P.: A general formulation of simultaneous inductive-recursive definitions in type theory. JSL 65(2), 525–549 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: ICFP 2002. SIGPLAN Notices, vol. 37, pp. 235–246. ACM, New York (2002)

    Google Scholar 

  18. Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in TCS, vol. 7. CUP (1989)

    Google Scholar 

  19. INRIA. The Coq Proof Assistant, Version 8.1. INRIA (2007), http://coq.inria.fr/

  20. Norell, U.: Towards a practical programming language based on dependent type theory. PhD thesis, Chalmers, Göteborg, Sweden (2007)

    Google Scholar 

  21. Pollack, R.: Closure under alpha-conversion. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 313–332. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  22. Seely, R.A.G.: Categorical semantics for higher order polymorphic lambda calculus. JSL 52(4), 969–989 (1987)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Ludwig-Maximilians-University Munich, Germany

    Andreas Abel

Authors
  1. Andreas Abel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematische Grundlagen der Informatik, RWTH Aachen, 52056, Aachen, Germany

    Erich Grädel

  2. CENTRIA and Departamento de Matemática, Universidade Nova de Lisboa, 2829-516, Caparica, Portugal

    Reinhard Kahle

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Abel, A. (2009). Typed Applicative Structures and Normalization by Evaluation for System Fω . In: Grädel, E., Kahle, R. (eds) Computer Science Logic. CSL 2009. Lecture Notes in Computer Science, vol 5771. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04027-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04027-6_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04026-9

  • Online ISBN: 978-3-642-04027-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation