Skip to main content
SpringerLink
Log in

Formalizing Adequacy: A Case Study for Higher-order Abstract Syntax

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

Adequacy is an important criterion for judging whether a formalization is suitable for reasoning about the actual object of study. The issue is particularly subtle in the expansive case of approaches to languages with name-binding. In prior work, adequacy has been formalized only with respect to specific representation techniques. In this article, we give a general formal definition based on model-theoretic isomorphisms or interpretations. We investigate and formalize an adequate interpretation of untyped lambda-calculus within a higher-order metalanguage in Isabelle/HOL using the Nominal Datatype Package. Formalization elucidates some subtle issues that have been neglected in informal arguments concerning adequacy.

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. Abel, A.: A third-order representation of the λμ-calculus. In: Ambler, S., Crole, R., Momigliano, A. (eds.) Electronic Notes in Theoretical Computer Science, vol. 58. Elsevier Science Publishers (2001)

  2. Cheney, J.: A simple nominal type theory. In: Logical Frameworks and Meta-Languages: Theory and Practice, pp. 90–104 (2008)

  3. Cheney, J., Urban, C.: Nominal logic programming. ACM Trans. Program. Lang. Syst. 30(5), 26 (2008)

    Article  Google Scholar 

  4. Coquand, T.: An algorithm for testing conversion in type theory. In: Logical Frameworks, pp. 255–279. Cambridge University Press, New York (1991)

    Chapter  Google Scholar 

  5. Crary, K.: Logical relations and a case study in equivalence checking. In: Pierce, B.C. (ed.) Advanced Topics in Types and Programming Languages, pp. 223–244. MIT Press (2005)

  6. Crary, K., Harper, R.: Higher-order abstract syntax: setting the record straight. SIGACT News 37(3), 93–96 (2006)

    Article  Google Scholar 

  7. Crole, R.: The representational adequacy of hybrid. Math. Struct. Comput. Sci. (2011, to appear)

  8. Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Form. Asp. Comput. 13, 341–363 (2002)

    Article  MATH  Google Scholar 

  9. Gardner, P.: Equivalences between logics and their representing type theories. Math. Struct. Comput. Sci. 5(3), 323–349 (1995)

    Article  MATH  Google Scholar 

  10. Harper, R., Honsell, F., Plotkin, G.D.: A framework for defining logics. J. ACM 40(1), 143–184 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harper, R., Licata, D.R.: Mechanizing metatheory in a logical framework. J. Funct. Program. 17(4–5), 613–673 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harper, R., Pfenning, F.: On equivalence and canonical forms in the LF type theory. ACM Trans. Comput. Log. 6(1), 61–101 (2005)

    Article  MathSciNet  Google Scholar 

  13. Hodges, W.: A Shorter Model Theory. Cambridge University Press (1997)

  14. Miller, D., Nadathur, G.: A logic programming approach to manipulating formulas and programs. In: Haridi, S. (ed.) IEEE Symposium on Logic Programming, pp. 379–388. San Francisco (1987)

  15. Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Logic 51, 125–157 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Narboux, J., Urban, C.: Formalising in nominal Isabelle Crary’s completeness proof for equivalence checking. In: LFMTP, vol. 196 of ENTCS (2007)

  17. Norrish, M., Vestergaard, R.: Proof pearl: De Bruijn terms really do work. In: Schneider, K., Brandt, J. (eds.) TPHOLs. Lecture Notes in Computer Science, vol. 4732, pp. 207–222. Springer (2007)

  18. Pfenning, F.: Logical frameworks. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. II, chapter 17, pp. 1063–1147. Elsevier Science (2001)

  19. Pitts, A.: Nominal system T. In: POPL, pp. 159–170 (2010)

  20. Pitts, A.M.: Alpha-structural recursion and induction. J. ACM 53(3), 459–506 (2006)

    Article  MathSciNet  Google Scholar 

  21. Urban, C.: Nominal techniques in Isabelle/HOL. J. Autom. Reason. 40(4), 327–356 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Urban, C., Berghofer, S., Norrish, M.: Barendregt’s variable convention in rule inductions. In: CADE. LNAI, vol. 4603, pp. 35–50 (2007)

  23. Urban, C., Cheney, J., Berghofer, S.: Mechanizing the metatheory of LF. ACM Trans. Comput. Log. (2010, in press)

  24. Watkins, K., Cervesato, I., Pfenning, F., Walker, D.: A concurrent logical framework I: judgments and properties. Technical Report CMU-CS-02-101, Carnegie Mellon University (2003)

Download references

Author information

Authors and Affiliations

  1. Laboratory for Foundations of Computer Science, University of Edinburgh, Edinburgh, Scotland, UK

    James Cheney

  2. Canberra Research Lab, NICTA, Canberra, Australia

    Michael Norrish

  3. Australian National University, Canberra, Australia

    Michael Norrish

  4. Research Center for Integrated Science, JAIST, Ishikawa, Japan

    René Vestergaard

Authors
  1. James Cheney
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Norrish
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. René Vestergaard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Cheney.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheney, J., Norrish, M. & Vestergaard, R. Formalizing Adequacy: A Case Study for Higher-order Abstract Syntax. J Autom Reasoning 49, 209–239 (2012). https://doi.org/10.1007/s10817-011-9221-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-011-9221-6

Keywords

Search

Navigation