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Stable Computational Semantics of Conflict-Free Rewrite Systems (Partial Orders with Duplication)

  • Zurab Khasidashvili
  • John Glauert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2706)

Abstract

We study orderings ⊴S on reductions in the style of Lévy reflecting the growth of information w.r.t. (super)stable sets S of ‘values’ (such as head-normal forms or Böhm-trees). We show that sets of co-initial reductions ordered by ⊴S form finitary ω-algebraic complete lattices, and hence form computation and Scott domains. As a consequence, we obtain a relativized version of the computational semantics proposed by Boudol for term rewriting systems. Furthermore, we give a pure domain-theoretic characterization of the orderings ⊴S in the spirit of Kahn and Plotkin’s concrete domains. These constructions are carried out in the framework of Stable Deterministic Residual Structures, which are abstract reduction systems with an axiomatized residual relations on redexes, that model all orthogonal (or conflict-free) reduction systems as well as many other interesting computation structures.

Keywords

Complete Lattice Reduction Ordering Initial Reduction Reduction Space 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. [AGM]
    Abramsky S., Gabbay D., and Maibaum T., eds. Handbook of Logic in Computer Science, vol. 1–4, Oxford University Press, 1992–1995.Google Scholar
  2. [Bar84]
    Barendregt H. P. The Lambda Calculus, its Syntax and Semantics. North-Holland, 1984.Google Scholar
  3. [Ber96]
    Berarducci A. Infinite lambda-calculus and non-sensible models. Logic and Algebra, Lecture Notes in Pure and Applied Mathematics 180, Marcel Dekker Inc., 1996, p. 339–378.Google Scholar
  4. [Ber79]
    Berry G. Modéles complétement adéquats et stables des λ-calculs typés. Thèse de l’Université de Paris VII, 1979.Google Scholar
  5. [BL79]
    Berry G., Lévy J.-J. Minimal and optimal computations of recursive programs. JACM 26, 1979, p. 148–175.zbMATHCrossRefGoogle Scholar
  6. [BCL85]
    Berry G., Curien P.-L., Lévy J.-J. Full abstraction for sequential languages: the state of the art. In [NR85], p. 89–132.Google Scholar
  7. [Bou85]
    Boudol G. Computational semantics of term rewriting systems. In [NR85], p. 169–236.Google Scholar
  8. [CGW89]
    Coquand T., Gunter C.A., Winskel G. dI-domains as a model of polymorphism. In Proc. of MFPLS’87, Springer LNCS, vol. 298, 1987, p. 344–363.Google Scholar
  9. [CGM95]
    Corradini A., Gadduchi F., Montanari U. Relating two categorical models of term rewriting. In: Proc. of RTA’95, springer LNCS, Vol. 914, 1995, p. 25–240.Google Scholar
  10. [Cur86]
    Curien P.-L. Categorical combinators, Sequential algorithms and functional programming. John Wiley & Sons, 1986.Google Scholar
  11. [DP90]
    Davey, B.A., Priestley H.A., Introduction to Lattices and Order Cambridge University Press, 1990.Google Scholar
  12. [Gir86]
    Girard J.Y. The system F of variable types: fifteen years later. TCS 45:159–192, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Gir87]
    Girard J.Y. Linear logic. TCS 50:1–101, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [GKK00]
    Glauert J. R. W., Kennaway J. R., Khasidashvili Z. Stable results and relative normalization. J. Log. and Comp., vol. 10(3), OUP, 2000, p. 1–26.MathSciNetGoogle Scholar
  15. [GK96]
    Glauert J.R.W., Khasidashvili Z. Relative normalization in deterministic residual structures. In: Proc. of CAAP’96, Springer LNCS, vol. 1059, 1996, p. 180–195.Google Scholar
  16. [GK02]
    Glauert J. R. W, Khasidashvili Z. An abstract Böhm-normalization. in: Proc. WRS’02, Electronic Notes in Computer Science, Elsevier Science B.V., 2002.Google Scholar
  17. [GLM92]
    Gonthier G., Lévy J.-J., Melliès P.-A. An abstract Standardisation theorem. In: Proc. of LICS’92, 1992, p. 72–81.Google Scholar
  18. [Gun93]
    Gunter C.A. Semantics of programming languages: structures and techniques. MIT Press, 1993.Google Scholar
  19. [HL91]
    Huet G., Lévy J.-J. Computations in Orthogonal Rewriting Systems. In: Computational Logic, Essays in Honor of Alan Robinson, MIT Press, 1991.Google Scholar
  20. [KP93]
    Kahn G., Plotkin G.D. Concrete domains. TCS 121:186–277, 1993.CrossRefMathSciNetGoogle Scholar
  21. [KKSV93]
    Kennaway J. R., Klop J. W., Sleep M. R, de Vries F.-J. Event structures and orthogonal term graph rewriting. In: Term Graph Rewriting: Theory and Practice. John Wiley, 1993, p. 141–156.Google Scholar
  22. [KG97]
    Khasidashvili Z., Glauert J.R.W. Zig-zag, extraction and separable families in non-duplicating stable deterministic residual structures. Report IR-420, Free University, February 1997.Google Scholar
  23. [KG02]
    Khasidashvili Z., Glauert J.R.W. Relating conflict-free stable transition and event models. Theoretical Computer Science, vol. 286, 2002, p. 65–95.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [KG03]
    Khasidashvili Z., Glauert J.R.W. Stable Computational Semantics of Conflictfree Rewrite Systems, available from http://www.cmp.uea.ac.uk/~jrwg/papers/KhaGlaRTARep.pdf.
  25. [Lan94]
    Laneve C. Distributive evaluations of λ-calculus, Fundamenta Informaticae, 20(4):333–352, 1994.zbMATHMathSciNetGoogle Scholar
  26. [LW93]
    Larsen K., Winskel G. Using information systems to solve recursive domain equations effectively. In: Proc. of Int. Symposium on the Semantics of Data Types, Springer LNCS, vol. 173, 1984.Google Scholar
  27. [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.zbMATHCrossRefGoogle Scholar
  28. [Lév78]
    Lévy J.-J. Réductions correctes et optimales dans le lambda-calcul, Thèse de l’Université de Paris VII, 1978.Google Scholar
  29. [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
  30. [Lon83]
    Longo G. Set theoretic models of lambda calculus: theories, expansions and isomorphisms. Annals of Pure and Applied Logic, 24:153–188, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [Mel96]
    Melliès P.-A. Description Abstraite des Systèmes de Réécriture. Thèse de l’Universit é Paris 7, 1996.Google Scholar
  32. [Mel97]
    Melliès P.-A. A factorization theorem in rewriting theory. In; Proc. of CTCS’97, Springer LNCS vol. 1290.Google Scholar
  33. [Mel98]
    Melliès P.-A. A stability theorem in rewriting theory. In: Proc. LICS’98, 1998.Google Scholar
  34. [Mid97]
    Middeldorp A. Call by need computations to root-stable form, in: POPL’97, 1997, 94–105.Google Scholar
  35. [NPW81]
    Nielsen M., Plotkin G., Winskel G. Petri nets, event structures and domains. Part 1. TCS 13:85–108, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [NR85]
    Nivat M., Reynolds J.C., eds. Algebraic methods in semantics. CUP, 1985.Google Scholar
  37. [Ong95]
    Ong C.-H. Correspondence between operational and denotational semantics: the full abstraction problem for PCF. In: [AGM], vol. 4, 1995, p. 269–356.Google Scholar
  38. [Plo83]
    Plotkin G. Domains. University of Edinburgh, 1983 (Manuscript).Google Scholar
  39. [Sco82]
    Scott D. Domains for denotational semantics. In Proc. ICALP’82, Springer LNCS, vol. 140, pp. 577–613, 1982.Google Scholar
  40. [Sta89]
    Stark E. W. Concurrent transition systems. TCS 64(3):221–270, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [Ter03]
    Terese. Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, Volume 55, Cambridge University Press, 2003.Google Scholar
  42. [Win80]
    Winskel G. Events in Computation. Ph.D. Thesis, Univ. Edinburgh, 1980.Google Scholar
  43. [Win89]
    Winskel G. An introduction to Event Structures. Springer LNCS, vol. 354, 1989, p. 364–397. Extended version in Cam. Univ. Comp. Lab. Report 95, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Zurab Khasidashvili
    • 1
  • John Glauert
    • 2
  1. 1.Logic and Validation Technology IntelIDCHaifaIsrael
  2. 2.School of Computing SciencesUEANorwichUK

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