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The geometry of orthogonal reduction spaces

  • Zurab Khasidashvili
  • John Glauert
Session 16: Rewriting
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)

Abstract

We investigate mutual dependencies of subexpressions of a computable expression, in orthogonal rewrite systems, and identify conditions for their concurrent independent computation. To this end, we introduce concepts familiar from ordinary Euclidean Geometry (such as basis, projection, distance, etc.) for reduction spaces. We show how a basis for an expression can be constructed so that any reduction starting from that expression can be decomposed as the sum of its projections on the axes of the basis. To make the concepts more relevant computationally, we relativize them w.r.t. stable sets of results, and show that an optimal concurrent computation of an expression w.r.t. S consists of optimal computations of its S-independent subexpressions. All these results are obtained for Stable Deterministic Residual Structures, Abstract Reduction Systems with an axiomatized residual relation, which model all orthogonal rewrite systems.

Keywords

Optimal Computation Reduction Space Normalization Theorem Residual Relation Term Rewrite System 
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.

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References

  1. [AL94]
    Asperti A., Laneve C. Interaction Systems I: The theory of optimal reductions. MSCS 11:1–48, Cambridge University Press, 1993.Google Scholar
  2. [Bar84]
    Barendregt H. P. The Lambda Calculus, its Syntax and Semantics. North-Holland, 1984.Google Scholar
  3. [GK96]
    Glauert J.R.W., Khasidashvili Z. Relative normalization in deterministic residual structures. CAAP'96, Springer LNCS, vol. 1059, H. Kirchner, ed. 1996, p. 180–195.Google Scholar
  4. [GLM92]
    Gonthier G., Lévy J.-J., Melliès P.-A. An abstract Standardisation theorem. In: Proc. of LICS 1992, p. 72–81.Google Scholar
  5. [Hin69]
    Hindley R.J. An abstract form of the Church-Rosser theorem I. JSL, 34(4):545–560, 1969.Google Scholar
  6. [HL91]
    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
  7. [KKSV94]
    Kennaway J. R., Klop J. W., Sleep M. R., de Vries F.-J. On the adequacy of Graph Rewriting for simulating Term Rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493–523, 1994.Google Scholar
  8. [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
  9. [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
  10. [KG96]
    Khasidashvili Z., Glauert J. R. W. Discrete normalization and Standardization in Deterministic Residual Structures. In proc. of ALP'96, Springer LNCS, vol. 1139, M. Hanus, M. Rodríguez-Artalejo, eds. 1996, p.135–149.Google Scholar
  11. [KG97]
    Khasidashvili Z., Glauert J.R.W. Zig-zag, extraction and separable families in non-duplicating stable deterministic residual structures. Technical Report IR-420, Free University, February 1997.Google Scholar
  12. [KG97a]
    Khasidashvili Z., Glauert J. R. W. Relating conflict-free transition and event models. Submitted.Google Scholar
  13. [Klo80]
    Klop J. W. Combinatory Reduction Systems. Mathematical Centre Tracts n. 127, Amsterdam, 1980.Google Scholar
  14. [Klo92]
    Klop J. W. Term Rewriting Systems. In: S. Abramsky, D. Gabbay, and T. Maibaum eds. Handbook of Logic in Computer Science, vol. II, Oxford U. Press, 1992, p. 1–116.Google Scholar
  15. [Lév80]
    Lévy J.-J. Optimal reductions in the λ-calculus. In: To H. B. Curry: Essays on Combinatory Logic, Lambda-calculus and Formalizm, Hindley J. R., Seldin J. P. eds, Academic Press, 1980, p. 159–192.Google Scholar
  16. [Mar92]
    Maranget L. La stratégie paresseuse. Thèse de l'Université de Paris VII, 1992.Google Scholar
  17. [Mil92]
    Milner R. Functions as processes. MSCS 2(2):119–141, 1992.Google Scholar
  18. [Mel96]
    Melliès P.-A. Description Abstraite des Systèmes de Réécriture. Thèse de l'Université Paris 7, 1996.Google Scholar
  19. [Oos94]
    Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph.D. Thesis, Free University, Amsterdam, 1994.Google Scholar
  20. [Oos96]
    Van Oostrom V. Higher order families. In: Proc. of RTA'96, Springer LNCS, vol. 1103, Ganzinger, H., ed., 1996, p. 392–407.Google Scholar
  21. [Raa96]
    van Raamsdonk F. Confluence and normalisation for higher-order rewriting. Ph.D. Thesis, Free University, Amsterdam, 1996.Google Scholar
  22. [Sta89]
    Stark E. W. Concurrent transition systems. J. TCS, 64(3):221–270, 1989.Google Scholar
  23. [Win89]
    Winskel G. An introduction to Event Structures. Springer LNCS, vol. 354, 1989, p. 364–397.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Zurab Khasidashvili
    • 1
  • John Glauert
    • 2
  1. 1.NTT Basic Research LaboratoriesKanagawaJapan
  2. 2.School of Information SystemsUEANorwichEngland

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