A Theory of Computer Instructions

A computer is a set <italic>M</italic> (the memory), a set <italic>B</italic>, a class of maps <italic>S</italic>: <italic>M</italic> → <italic>B</italic>, known as states, and a class @@@@ of maps <italic>T</italic>: @@@@ → @@@@, known as instructions. Each instruction <italic>I</italic> has an input region <italic>IR</italic>(<italic>I</italic>), an output region <italic>OR</italic>(<italic>I</italic>), and affected regions <italic>AR</italic>(<italic>M′</italic>, <italic>I</italic>), for <italic>M′</italic> ⊆ <italic>IR</italic>(<italic>I</italic>). For example, let <italic>I</italic> be the instruction (<italic>CLA Y</italic>) on the IBM 7094. If <italic>L</italic> is the location counter and <italic>AC</italic> is the accumulator, then <italic>IR</italic>(<italic>I</italic>) = <italic>Y</italic> ∪ <italic>L</italic> and <italic>OR</italic>(<italic>I</italic>) = <italic>AC</italic> ∪ <italic>L</italic>; if <italic>M′</italic> is the address portion of <italic>Y</italic>, then <italic>AR</italic>(<italic>M′</italic>, <italic>I</italic>) is the address portion of <italic>AC</italic>. The fundamental properties of all these notions are derived, and computers are related to other models, such as sequential machines. The existence problem (how arbitrarily the input, output and affected regions of an instruction can be specified) is fully settled for countable memory <italic>M</italic>.