A theory of computer instructions

Just as there is a theory of groups, or rings, or fields, or topological spaces, so there is presented here a theory of computer instructions. These are functions from image to image, where image is the set of states of a computer. Here image is a set of functions from image to image, where image is the memory (the set of variables) and image is a set of values; or alternatively, image is a cartesian product, over image as an index set, of the sets of values of all variables in image. For each instruction image there are defined the input region IRimage and the output region ORimage; these are subsets of image. An instruction takes data from its input region and places data in its output region. Here ORimage may be decomposed further into regions affected by subsets of image, which may be defined in either of two alternative ways. The theory presented here includes theorems concerning composition of instructions, decomposition of instructions, and the existence of instructions with specified regions. Many examples are given.