Lower Bounds for Computations with the Floor Operation

A general lower bound technique is developed for computation trees with operations $\{ + , - , * ,/,\lfloor \cdot \rfloor , < \} $ and constants $\{ 0,1\} $, for functions that have as their input a single n-bit integer. The technique applies to many natural functions, such as perfect square root (deciding if the square root of the input is integral or not), computing the parity of $\lfloor {\log x} \rfloor $ , etc. The arguments are then extended to obtain the same lower bounds on the time complexity of any RAM program with operations $\{ + , - , * ,/,\lfloor \cdot \rfloor , < \} $ that solves the problem. Another related result is described in a companion paper [Proc. 29th IEEE Symposium on Foundations of Computer Science, 1988] and [J. Assoc. Comput. Mach., 1991, to appear].