On Normalisation of Infinitary Combinatory Reduction Systems

  • Jeroen Ketema
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5117)

Abstract

For fully-extended, orthogonal infinitary Combinatory Reduction Systems, we prove that terms with perpetual reductions starting from them do not have (head) normal forms. Using this, we show that
  1. 1

    needed reduction strategies are normalising for fully-extended, orthogonal infinitary Combinatory Reduction Systems, and that

     
  2. 1

    weak and strong normalisation coincide for such systems as a whole and, in case reductions are non-erasing, also for terms.

     

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References

  1. 1.
    Kennaway, R., Klop, J.W., Sleep, R., de Vries, F.J.: Transfinite reductions in orthogonal term rewriting systems. I&C 119(1), 18–38 (1995)MATHGoogle Scholar
  2. 2.
    Kennaway, R., de Vries, F.J.: Infinitary rewriting. In: [19], ch.12Google Scholar
  3. 3.
    Kennaway, J.R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Infinitary lambda calculus. TCS 175(1), 93–125 (1997)MATHCrossRefGoogle Scholar
  4. 4.
    Klop, J.W., van Oostrom, V., van Raamsdonk, F.: Combinatory reduction systems: introduction and survey. TCS 121(1&2), 279–308 (1993)MATHCrossRefGoogle Scholar
  5. 5.
    Ketema, J., Simonsen, J.G.: Infinitary combinatory reduction systems. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 438–452. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Ketema, J., Simonsen, J.G.: On confluence of infinitary combinatory reduction systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 199–214. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Ketema, J., Simonsen, J.G.: Infinitary combinatory reduction systems. Technical Report D-558, Department of Computer Science, University of Copenhagen (2006)Google Scholar
  8. 8.
    van Oostrom, V.: Normalisation in weakly orthogonal rewriting. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 60–74. Springer, Heidelberg (1999)Google Scholar
  9. 9.
    Klop, J.W., de Vrijer, R.: Infinitary normalization. In: We Will Show Them: Essays in Honour of Dov Gabbay, vol. 2, pp. 169–192. College Publications (2005)Google Scholar
  10. 10.
    Huet, G., Lévy, J.J.: Computations in orthogonal rewriting systems. In: Computational Logic: Essays in honor of Alan Robinson, pp. 395–443. MIT Press, Cambridge (1991)Google Scholar
  11. 11.
    Glauert, J., Khasidashvili, Z.: Relative normalization in orthogonal expression reduction systems. In: Lindenstrauss, N., Dershowitz, N. (eds.) CTRS 1994. LNCS, vol. 968, pp. 144–165. Springer, Heidelberg (1995)Google Scholar
  12. 12.
    Khasidashvili, Z., Ogawa, M.: Perpetuality and uniform normalization. In: Hanus, M., Heering, J., Meinke, K. (eds.) ALP 1997 and HOA 1997. LNCS, vol. 1298, pp. 240–255. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. 13.
    Klop, J.W.: Term rewriting systems. In: Handbook of Logic in Computer Science, vol. 2, pp. 1–116. Oxford University Press, Oxford (1992)Google Scholar
  14. 14.
    Kennaway, R., van Oostrom, V., de Vries, F.J.: Meaningless terms in rewriting. The Journal of Functional and Logic Programming 1 (1999)Google Scholar
  15. 15.
    Arnold, A., Nivat, M.: The metric space of infinite trees. Algebraic and topological properties. Fundamenta Informaticae 3(4), 445–476 (1980)MathSciNetGoogle Scholar
  16. 16.
    Klop, J.W., van Oostrom, V., de Vrijer, R.: Orthogonality. In: [19], ch. 4Google Scholar
  17. 17.
    Middeldorp, A.: Call by need computations to root-stable form. In: POPL 1997, pp. 94–105 (1997)Google Scholar
  18. 18.
    van Raamsdonk, F.: Confluence and Normalisation for Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam (1996)Google Scholar
  19. 19.
    Terese (ed.): Term Rewriting Systems. Cambridge Tracts in Theorectical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jeroen Ketema
    • 1
  1. 1.Research Institute of Electrical CommunicationTohoku UniversitySendaiJapan

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