On Computations with Integer Division

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.