ABSTRACT
This chapter consists of two independent parts. The first deals with programs involving indexed data sets such as dense arrays and indexed computations such as loops. Our position is that high-level mathematical equations are the most natural way to express a large class of such computations, and furthermore, such equations are amenable to powerful static analyses that would enable a compiler to derive very efficient code, possibly significantly better than what a human would write. We illustrate this by describing a simple equational language and its semantic foundations and by illustrating the analyses we can perform, including one that allows the compiler to reduce the degree of the polynomial complexity of the algorithm embodied in the program.
