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.
supported in part by the Deutsche Forschungsgemeinschaft, ME 872/1–1 and WE 1066/1–2, and the Leibniz Center for Research in Computer Science
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References
M.Ben Or: Lower bounds for algebraic computation trees, Proc. 15th ACM STOC, 80–86, 1983.
L.Babai, B.Just, F.Meyer auf der Heide: On the limits of computations with the floor functions, accepted for Information and Computation.
J.W.S. Cassels: An introduction to the geometry of numbers, Springer, Berlin 1959, second printing 1971.
D. Dobkin, R. Lipton: A lower bound of 1/2 n2 on linear search programs for the knapsack problem, J.C.S.S. 16, 417–421, 1975.
J.Hastad, B.Just, J.Lagarias, C.P.Schnorr: Polynomial time algorithms for finding integer relations among real numbers, Proc. STACS, 105–118, 1986.
P. Klein, F. Meyer auf der Heide: A lower bound for the knapsack problem on random access machines, Act. Inf. 19, 385–395, 1983.
H.W. Lenstra Jr.: Integer programming with a fixed number of variables, Report 81-03, Mathematisch Instituut, Amsterdam, 1983.
J.C.Lagarias, H.W.Lenstra Jr., C.P.Schnorr: K arkine-Zolotareff Bases and successive minima of a lattice and its reciprocal lattice, preprint 86.
F. Meyer auf der Heide: Lower bounds for solving linear diophantine equations on random access machines, J.ACM 32(4), 929–937, 1985.
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© 1988 Springer-Verlag Berlin Heidelberg
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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. https://doi.org/10.1007/BFb0035829
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DOI: https://doi.org/10.1007/BFb0035829
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