On computations with integer division

extended abstract
  • Bettina Just
  • Fb Mathematik
  • Friedhelm Meyer auf der Heide
  • Fb Informatik
  • Avi Wigderson
Contributed Papers Algorithms

DOI: 10.1007/BFb0035829

Part of the Lecture Notes in Computer Science book series (LNCS, volume 294)
Cite this paper as:
Just B., Mathematik F., Meyer auf der Heide F., Informatik F., Wigderson A. (1988) On computations with integer division. In: Cori R., Wirsing M. (eds) STACS 88. STACS 1988. Lecture Notes in Computer Science, vol 294. Springer, Berlin, Heidelberg

Abstract

We consider computation trees (CT's) with operations S ⊂ {+, −, *, DIV, DIVC}, where DIV denotes integer division and DIVC integer division by constants. We characterize the families of languages L ⊂ ℕ that can be recognized over {+, −, DIVC}, {+, −, DIV}, and {+, −, *, DIV}, resp. and show that they are identical. Furthermore we prove lower bounds for CT's with operations {+, −, DIVC} for languages L ⊂ ℕ which only contain short arithmetic progressions. We cannot apply the classical component counting arguments as for operation sets S ⊂ {+, −, *,./.} because of the DIVC - operation. Such bounds are even no longer true. Instead we apply results from the Geometry of Numbers about arithmetic progressions on integer points in high-dimensional convex sets for our lower bounds.

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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Bettina Just
    • 1
  • Fb Mathematik
    • 1
  • Friedhelm Meyer auf der Heide
    • 2
  • Fb Informatik
    • 2
  • Avi Wigderson
    • 3
  1. 1.Johann Wolfgang Goethe UniversitätFrankfurt a.M.Fed. Rep. of Germany
  2. 2.Universität DortmundDortmundFed. Rep. of Germany
  3. 3.Computer Science DepartmentHebrew UniversityJerusalemIsrael

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