Abstract

Extending the infinitary rewriting definition of Böhm-like trees to infinitary Combinatory Reduction Systems (iCRSs), we show that each Böhm-like tree defined by means of infinitary rewriting can also be defined by means of a direct approximant function. In addition, we show that counterexamples exists to the reverse implication.

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References

  1. 1.
    Ketema, J.: Böhm-Like Trees for Rewriting. PhD thesis, Vrije Universiteit, Amsterdam (2001)Google Scholar
  2. 2.
    Lévy, J.J.: Réductions correctes et optimales dans le lambda-calcul. PhD thesis, Université de Paris VII (1978)Google Scholar
  3. 3.
    Barendregt, H.P.: The Lambda Calculus: Its Syntax and Semantics, revised edn. Elsevier Science, Amsterdam (1985)MATHGoogle Scholar
  4. 4.
    Lévy, J.J.: An algebraic interpretation of the \(\lambda \beta \textrm{K}\)-calculus and the labelled λ-calculus. In: Böhm, C. (ed.) Lambda-Calculus and Computer Science Theory. LNCS, vol. 37, pp. 147–165. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  5. 5.
    Longo, G.: Set-theoretical models of λ-calculus: Theories, expansions, isomorphisms. Annals of Pure and Applied Logic 24, 153–188 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Berarducci, A.: Infinite λ-calculus and non-sensible models. In: Logic and Algebra. Lect. Notes Pure Appl. Math., vol. 180, pp. 339–378 (1996)Google Scholar
  7. 7.
    Kennaway, R., van Oostrom, V., de Vries, F.J.: Meaningless terms in rewriting. JFLP 1 (1999)Google Scholar
  8. 8.
    Kennaway, R., del Vries, F.-J.: Infinitary rewriting. In: [18], ch. 12Google Scholar
  9. 9.
    Blom, S.C.C.: Term Graph Rewriting: syntax and semantics. PhD thesis, Vrije Universiteit, Amsterdam (2001)Google Scholar
  10. 10.
    Ketema, J.: Böhm-like trees for term rewriting systems. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 233–248. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Ketema, J., Simonsen, J.G.: Infinitary combinatory reduction systems. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 438–452. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Ketema, J., Simonsen, J.G.: Infinitary combinatory reduction systems. Technical Report D-558, Department of Computer Science, University of Copenhagen (2006)Google Scholar
  14. 14.
    Klop, J.W., van Oostrom, V., van Raamsdonk, F.: Combinatory reduction systems: introduction and survey. TCS 121(1-2), 279–308 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ketema, J.: On normalisation of infinitary combinatory reduction systems. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 172–186. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Kennaway, R., et al.: Infinitary rewriting: From syntax to semantics. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds.) Processes, Terms and Cycles: Steps on the Road to Infinity. LNCS, vol. 3838, pp. 148–172. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Kennaway, J.R., et al.: Infinitary lambda calculus. TCS 175(1), 93–125 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Terese (ed.): Term Rewriting Systems. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jeroen Ketema
    • 1
  1. 1.Faculty EEMCSUniversity of TwenteEnschedeThe Netherlands

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