Advertisement

Discrete Normalization and Standardization in Deterministic Residual Structures

  • Zurab Khasidashvili
  • John Glauert
Term Rewriting
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1139)

Abstract

We prove a version of the Standardization Theorem and the Discrete Normalization Theorem in stable Deterministic Residual Structures, Abstract Reduction Systems with axiomatized notions of residual, which model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions Lévy-equivalent (or permutation-equivalent) to a given, finite or infinite, regular (or semi-linear) reduction, based on the neededness concept of Huet and Lévy. This and other results of this paper add to the understanding of Lévy-equivalence of reductions, and consequently, Lévy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner.

Keywords

Normal Form Relative Normalization Lambda Calculus Discrete Normalization Residual Relation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AbOn93]
    Abramsky S., Ong C.-H. L. Full abstraction in the lazy lambda calculus. Inf.& Comp., 105:159–267, 1993.Google Scholar
  2. [AEH94]
    Antoy S., Echahed R., Hanus M. A needed narrowing strategy. In: Proc. of POPL'94, Portland, Oregon, 1994.Google Scholar
  3. [AnMi94]
    Antoy S. and Middeldorp A. A Sequential Reduction Strategy. In Proc. of ALP'94, Springer LNCS, vol. 850, p. 168–185, 1994.Google Scholar
  4. [AsLa96]
    Asperti A., Laneve C. Interaction Systems II: the practice of optimal reductions. TCS, 159(2):191–244, 1996.Google Scholar
  5. [Bar84]
    Barendregt H. P. The Lambda Calculus, its Syntax and Semantics. North-Holland, 1984.Google Scholar
  6. [BKKS87]
    Barendregt H. P., Kennaway J. R., Klop J. W., Sleep M. R. Needed Reduction and spine strategies for the lambda calculus. Inf.& Comp., 75(3):191–231, 1987.Google Scholar
  7. [BeLé79]
    Berry G., Lévy J.-J. Minimal and optimal computations of recursive programs. JACM 26, 1979, p. 148–175.Google Scholar
  8. [Bou85]
    Boudol G. Computational semantics of term rewriting systems. In: Algebraic methods in semantics. Nivat M., Reynolds J.C., eds. Cambr. Univ. Press, 1985, p. 169–236.Google Scholar
  9. [CuFe58]
    Curry H. B., Feys R. Combinatory Logic. vol. 1, North-Holland, 1958.Google Scholar
  10. [Gar94]
    Gardner P. Discovering needed reductions using type theory. In: Proc. of TACS'94, Springer LNCS, v. 789, M. Hagiya, J. C. Mitchell, eds. Sendai, 1994, p. 555–574.Google Scholar
  11. [GlKh94]
    Glauert J.R.W., Khasidashvili Z. Relative Normalization in Orthogonal Expression Reduction Systems. In: Proc. of CTRS'94, Springer LNCS, vol. 968, N. Dershowitz, N. Lindenstrauss, eds. Jerusalem, 1994, p. 144–165.Google Scholar
  12. [GlKh94a]
    Glauert J.R.W., Khasidashvili Z. Minimal and optimal relative normalization in Expression Reduction Systems. Report SYS-C94-06, UEA, Norwich, 1994.Google Scholar
  13. [GlKh96]
    Glauert J.R.W., Khasidashvili Z. Relative normalization in deterministic residual structures. In [CAAP'96], p. 180–195.Google Scholar
  14. [GlKh96a]
    Glauert J.R.W., Khasidashvili Z. Relative normalization in stable deterministic residual structures. Report SYS-C96-01, UEA, Norwich, 1996.Google Scholar
  15. [GLM92]
    Gonthier G., Lévy J.-J., Melliès P.-A. An abstract Standardisation theorem. In: Proc. of LICS'92, IEEE Computer Society Press, 1992, p. 72–81.Google Scholar
  16. [Hin64]
    Hindley R.J. An abstract form of the Church-Rosser theorem I. JSL, 34(4):545–560, 1969.Google Scholar
  17. [HuLé91]
    Huet G., Lévy J.-J. Computations in Orthogonal Rewriting Systems. In: Computational Logic, Essays in Honor of Alan Robinson, J.-L. Lassez and G. Plotkin, eds. MIT Press, 1991, p. 394–443.Google Scholar
  18. [Kat90]
    Kathail V. Optimal Interpreters for Lambda-calculus based functional languages, PhD thesis, MIT, 1990.Google Scholar
  19. [Ken89]
    Kennaway J.R. Sequential evaluation strategy for parallel-or and related reduction systems. Annals of Pure and Applied Logic, 43:31–56, 1989.Google Scholar
  20. [KeSl89]
    Kennaway J. R., Sleep M. R. Neededness is hypernormalizing in regular combinatory reduction systems. Preprint, University of East Anglia, Norwich, 1989.Google Scholar
  21. [KKSV95]
    Kennaway J. R., Klop J. W., Sleep M. R., de Vries F.-J. Transfinite reductions in orthogonal term rewriting. Information and Computation, 119(1):18–38, 1995.Google Scholar
  22. [KOV96]
    Kennaway J.R., van Oostrom V., de Vries F.J. Meaningless terms in rewriting. In: Proc. of the 5th International Conference on Algebraic and Logic Programming, ALP'96, Aachen, 1996. To Appear.Google Scholar
  23. [Kha88]
    Khasidashvili Z. β-reductions and β-developments of λ-terms with the least number of steps. In: Proc. of the Int. Conference on Computer Logic COLOG'88, Tallinn 1988, Springer LNCS, v. 417, P. Martin-Löf and G. Mints, eds. 1990, p. 105–111.Google Scholar
  24. [Kha92]
    Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.Google Scholar
  25. [Kha93]
    Khasidashvili Z. Optimal normalization in orthogonal term rewriting systems. In: Proc. of RTA'93, Springer LNCS, vol. 690, C. Kirchner, ed. Montreal, 1993, p. 243–258.Google Scholar
  26. [Kha94]
    Khasidashvili Z. On higher order recursive program schemes. In: Proc. of CAAP'94, Springer LNCS, vol. 787, S. Tison, ed. Edinburgh, 1994, p. 172–186.Google Scholar
  27. [KhGl96]
    Khasidashvili Z., Glauert J.R.W. Discrete normalization and standardization in stable deterministic residual structures. Report SYS-C96-02, UEA, Norwich, 1996.Google Scholar
  28. [KhGl96a]
    Khasidashvili Z., Glauert J.R.W. Zig-zag and extraction families in non-duplicating stable Deterministic Residual Structures. Submitted for publication. (Available at ftp://ftp.sys.uea.ac.uk/pub/zurab/)Google Scholar
  29. [CAAP'96]
    Kirchner H., editor. Proc. of the 19th International Colloquium on Trees in Algebra and Programming, CAAP'96, Springer LNCS, vol. 1059, 1996.Google Scholar
  30. [Klo80]
    Klop J. W. Combinatory Reduction Systems. Mathematical Centre Tracts n. 127, Amsterdam, 1980.Google Scholar
  31. [Klo92]
    Klop J. W. Term Rewriting Systems. In: S. Abramsky, D. Gabbay, and T. Maibaum eds. Handbook of Logic in Comp. Science, vol. II, Oxford Univ. Press, 1992, p. 1–116.Google Scholar
  32. [KlMi90]
    Klop J.W., Middeldorp A. Sequentiality in Orthogonal Term Rewriting Systems. Journal of Symbolic Computation 12:161–195, 1991.Google Scholar
  33. [Lam90]
    Lamping J. An algorithm for optimal lambda calculus reduction. In: Proc. of POPL'90, 1990, p.16–30.Google Scholar
  34. [Lév76]
    Lévy J.-J. An algebraic interpretation of the λβK-calculus; and an application of a labelled λ-calculus. TCS 2(1):97–114, 1976.Google Scholar
  35. [Lév80]
    Lévy J.-J. Optimal reductions in the Lambda-calculus. In: To H. B. Curry: Essays on Combinatory Logic, Lambda-calculus and Formalism, Hindley J. R., Seldin J. P. eds, Academic Press, 1980, p. 159–192.Google Scholar
  36. [Lis96]
    Lisper B. Computing in unpredictable environments: semantics, reduction strategies, and program transformations. In [CAAP'96], p. 165–179.Google Scholar
  37. [Lon83]
    Longo G. Set theoretic models of lambda calculus: theories, expansions and isomorphisms. Annals of pure and applied logic, 24:153–188, 1983.Google Scholar
  38. [Mar92]
    Maranget L. La stratégie paresseuse. Thèse de l'Université de Paris VII, 1992.Google Scholar
  39. [Mel96]
    Melliès P.-A. Description Abstraite des Systèmes de Réécriture. Thèse de l'Université Paris 7, 1996.Google Scholar
  40. [Nip93]
    Nipkow T. Orthogonal higher-order rewrite systems are confluent. In: Proc. of TLCA'93, Springer LNCS, vol. 664, Bazem M., Groote J.F., eds. 1993, p. 306–317.Google Scholar
  41. [Niv75]
    Nivat M. On the interpretation of recursive polyadic program schemes. Symposia Mathematica, 15:255–281, 1975.Google Scholar
  42. [Nök94]
    Nöcker E. Efficient Functional Programming. Compilation and Programming Techniques. Ph.D. Thesis, Katholic University of Nijmegen, 1994.Google Scholar
  43. [Oos94]
    Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph.D. Thesis, Free University of Amsterdam, 1994.Google Scholar
  44. [Oos96]
    Van Oostrom V. Higher order families. In: Proc. of RTA'96. To appear.Google Scholar
  45. [Raa96]
    Van Raamsdonk F. Confluence and normalisation for higher-order rewriting. Ph.D. Thesis, CWI Amsterdam, 1996.Google Scholar
  46. [SeRa93]
    Sekar R.C., Ramakrishnan I.V. Programming in Equational Logic: Beyond Strong Sequentiality. Information and Computation, 104(1):78–109, 1993.Google Scholar
  47. [SPE93]
    Sleep M. R., Plasmeijer M. J., van Eekelen M. C. J. D., eds. Term Graph Rewriting: Theory and Practice. John Wiley, 1993, p.141–156.Google Scholar
  48. [Sta89]
    Stark E. W. Concurrent transition systems. TCS, 64(3):221–270, 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Zurab Khasidashvili
    • 1
  • John Glauert
    • 1
  1. 1.School of Information SystemsUEANorwichEngland

Personalised recommendations