Reasoning About Local Variables with Operationally-Based Logical Relations

  • Andrew M. Pitts
Part of the Progress in Theoretical Computer Science book series (PTCS)


A parametric logical relation between the phrases of an Algol-like language is presented. Its definition involves the structural operational semantics of the language, but was inspired by denotationally-based work of O’Heam and Reynolds on translating Algol into a predicatively polymorphic linear lambda calculus. The logical relation yields an applicative characterisation of contextual equivalence for the language and provides a useful (and complete) method for proving equivalences. Its utility is illustrated by giving simple and direct proofs of some contextual equivalences, including an interesting equivalence due to O’Hearn which hinges upon the undefinability of ‘snapback’ operations (and which goes beyond the standard suite of ‘Meyer-Sieber’ examples). Whilst some of the mathematical intricacies of denotational semantics are avoided, the hard work in this operational approach lies in establishing the ‘fundamental property’ for the logical relation—the proof of which makes use of a compactness property of fixpoint recursion with respect to evaluation of phrases. But once this property has been established, the logical relation provides a verification method with an attractively low mathematical overhead.


Local Variable Global Variable Logical Relation Operational Extensionality Denotational Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [HMST95]
    F. Honsell, I. A. Mason, S. F. Smith, and C. L. Talcott. A variable typed logic of effects. Information and Computation, 119 (1): 55–90, May 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [How89]
    D. J. Howe. Equality in lazy computation systems. In 4th Annual Symposium on Logic in Computer Science, pages 198–203. IEFF Computer Society Press, Washington, 1989.Google Scholar
  3. [How96]
    D. J. Howe. Proving congruence of bisimulation in functional programming languages. Information and Computation, 124 (2): 103–112, February 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Mi177]
    R. Milner. Fully abstract models of typed lambda-calculi. Theoretical Computer Science, 4: 1–22, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [Mit90]
    J. C. Mitchell. Type systems for programming languages. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B, pages 365458. North-Holland, Amsterdam, 1990.Google Scholar
  6. [MS88]
    A. Meyer and K. Sieber. Towards fully abstract semantics for local variables. In Proc. 15th Symp. on Principles of Programming Languages, San Diego,pages 191–203. ACM, 1988. See Chapter 7.Google Scholar
  7. [MST]
    I. A. Mason, S. F. Smith, and C. L. Talcott. From operational semantics to domain theory. Information and Computation (to appear). Revised and extended version of [Smi92].Google Scholar
  8. [01e85]
    F. J. 01es. Types algebras, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, chapter 15, pages 543–574. Cambridge University Press, 1985. See Chapter 11.Google Scholar
  9. [OR95a]
    P. W. O’Heam and U. S. Reddy. Objects, Interference and the Yoneda Embedding. In S. Brookes, M. Main, A. Melton, and M. Mislove, editors, Mathematical Foundations of Programming Semantics, Eleventh Annual Conference,volume 1 of Electronic Notes in Theoretical Computer Science,Tulane University, New Orleans, Louisiana, March 29-April 1 1995. Elsevier Science.Google Scholar
  10. [OR95b]
    P. W. O’Hearn and J. C. Reynolds. From ALGOL to polymorphic linear lambda calculus. Lectures at the Isaac Newton Institute for Mathematical Sciences, Cambridge UK, August 1995.Google Scholar
  11. [OT95]
    P. W. O’Hearn and R. D. Tennent. Parametricity and local variables. Journalof the ACM,42(3):658–709, 1995. See Chapter 16.Google Scholar
  12. [Pit96]
    A. M. Pitts. Relational properties of domains. Information and Computation, 1996. To appear. A preliminary version appeared as Cambridge Univ. Computer Laboratory Technical Report Number 321, December, 1993.Google Scholar
  13. [Pit]
    A. M. Pitts. Operationallybased theories of program equivalence. In P. Dybjer and A. M. Pitts, editors, Semantics and Logics of Computation. Cambridge University Press (to appear).Google Scholar
  14. [PS]
    A. M. Pitts and I. D. B. Stark. Operational reasoning for functions with local state. In A. D. Gordon and A. M. Pitts, editors, Higher Order Operational Techniques in Semantics. In preparation.Google Scholar
  15. [PS93]
    A. M. Pitts and I. D. B. Stark. Observable properties of higher order functions that dynamically create local names, or: What’s new? In Mathematical Foundations of Computer Science, Proc. 18th Int. Symp., Gdansk, 1993, volume 711 of Lecture Notes in Computer Science, pages 122–141. Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
  16. [Red95]
    U. S. Reddy. Global state considered unnecessary: Introduction to objectbased semantics. Lisp and Symbolic Computation,9(1):7–76, 1996. See Chapter 19.Google Scholar
  17. [Rey81]
    J. C. Reynolds. The essence of ALGOL. In J. W. de Bakker and J. C. van Vliet, editors, Algorithmic Languages. Proceedings of the International Symposium on Algorithmic Languages, pages 345–372. North-Holland, Amsterdam, 1981. See Chapter 3.Google Scholar
  18. [RP95]
    E. Ritter and A. M. Pitts. A fully abstract translation between a A-calculus with reference types and Standard ML. In 2nd Int. Con f. on Typed Lambda Calculus and Applications, Edinburgh, 1995, volume 902 of Lecture Notes in Computer Science, pages 397–413. Springer-Verlag, Berlin, 1995.Google Scholar
  19. [Sco93]
    D. S. Scott. A typetheoretical alternative to Cum, ISWIM, OwHY. Theoretical Computer Science, 121: 411–440, 1993.MathSciNetCrossRefGoogle Scholar
  20. [Sie95a]
    K. Sieber. Full abstraction for the second order subset of an ALGOL-like language. Technical Report A 04/95, Fach. Informatik, Univ. des Saarlandes, Saarbrücken, Germany, April 1995. To appear in Information and Computation. See Chapter 15.Google Scholar
  21. [Sie95b]
    K. Sieber. Full abstraction via logical relations. Habilitationsschrift, FB 14 Informatik, Universität des Saarlandes, July 1995.Google Scholar
  22. [Smi92]
    S. F. Smith. From operational to denotational semantics. In S. Brookes et al,editor, 7th International Conference on Mathematical Foundations of Programming Semantics, Pittsburgh PA, volume 598 of Lecture notes in Computer Science, pages 54–76. Springer-Verlag, Berlin, 1992.Google Scholar
  23. [Sta94]
    I. D. B. Stark. Names and Higher-Order Functions. PhD thesis, University of Cambridge, 1994. Also published as Technical Report 363, University of Cambridge Computer Laboratory, April 1995.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Andrew M. Pitts

There are no affiliations available

Personalised recommendations