Least Upper Bounds on the Size of Church-Rosser Diagrams in Term Rewriting and λ-Calculus

  • Jeroen Ketema
  • Jakob Grue Simonsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6009)


We study the Church-Rosser property—which is also known as confluence—in term rewriting and λ-calculus. Given a system R and a peak t ←* s →* t′ in R, we are interested in the length of the reductions in the smallest corresponding valley t →* s′ ←* t′ as a function vsR(m, n) of the size m of s and the maximum length n of the reductions in the peak. For confluent term rewriting systems (TRSs), we prove the (expected) result that vsR(m, n) is a computable function. Conversely, for every total computable function ϕ(n) there is a TRS with a single term s such that vsR( ∣ s ∣ , n) ≥ ϕ(n) for all n. In contrast, for orthogonal term rewriting systems R we prove that there is a constant k such that vsR(m, n) is bounded from above by a function exponential in k and independent of the size of s. For λ-calculus, we show that vsR(m,n) is bounded from above by a function contained in the fourth level of the Grzegorczyk hierarchy.


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  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    Barendregt, H.P.: The Lambda Calculus: Its Syntax and Semantics. In: Studies in Logic and the Foundations of Mathematics, rev. edn., vol. 103. North-Holland, Amsterdam (1985)Google Scholar
  3. 3.
    de Vrijer, R.: A direct proof of the finite developments theorem. Journal of Symbolic Logic 50(2), 339–343 (1985)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Endrullis, J., Geuvers, H., Zantema, H.: Degrees of undecidability in term rewriting. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 255–270. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Fernandez, M.: Models of Computation: An Introduction to Computability Theory. Undergraduate topics in computer science. Springer, Heidelberg (2009)MATHGoogle Scholar
  6. 6.
    Grzegorczyk, A.: Some classes of recursive functions. Rozpr. Mat. 4, 1–45 (1953)MathSciNetGoogle Scholar
  7. 7.
    Jones, N.D.: Computability and Complexity from a Programming Perspective. MIT Press, Cambridge (1997)MATHGoogle Scholar
  8. 8.
    Khasidashvili, Z.: The longest perpetual reductions in orthogonal expression reduction systems. In: Matiyasevich, Y.V., Nerode, A. (eds.) LFCS 1994. LNCS, vol. 813, pp. 191–203. Springer, Heidelberg (1994)Google Scholar
  9. 9.
    Klop, J.W.: Term rewriting systems. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 1–116. Oxford University Press, Oxford (1992)Google Scholar
  10. 10.
    Odifreddi, P.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics, vol. II, 143. North-Holland, Amsterdam (1999)Google Scholar
  11. 11.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  12. 12.
    Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge (1987)Google Scholar
  13. 13.
    Sipser, M.: Introduction to the Theory of Computation, 2nd edn. Thomson Course Technology (2006)Google Scholar
  14. 14.
    Statman, R.: The typed lambda calculus is not elementary recursive. Theoretical Computer Science 9, 73–81 (1979)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Terese: Term Rewriting Systems. In: Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar
  16. 16.
    Xi, H.: Upper bounds for standardizations and an application. Journal of Symbolic Logic 64(1), 291–303 (1999)CrossRefMathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jeroen Ketema
    • 1
  • Jakob Grue Simonsen
    • 2
  1. 1.Faculty EEMCSUniversity of TwenteEnschedeThe Netherlands
  2. 2.Department of Computer ScienceUniversity of Copenhagen (DIKU)Copenhagen ØDenmark

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