Some remarks on abstract machines

Introduction. In 1954 the mathematical entity called a (sequential) machine was found to be a valuable tool in designing sequential switching circuits [2; 8; 9]. Since then there has been considerable mathematical activity by mathematicians and nonmathematicians relating to the analysis and the synthesis of these machines. As was to be expected of a topic which arose because of an engineering need, most of these results have appeared in engineering and computing journals. Recently though, some of the papers have appeared in mathematical journals [3; 4; 5; 6; 10]. Also, much of the recent literature has dealt with questions almost exclusively of mathematical, as contrasted with engineering, interest [1; 3; 4; 5; 6; 10; 12]. The present paper is written in that spirit. The Moore-Mealy (complete, sequential) machine is defined [8; 9] as a nonempty set K (of "states"), a nonempty set D (of 'inputs"), a nonempty set F (of "outputs"), and two functions a (the "next state" function), and X (the "output" function), 5 mapping KXD into K and X mapping KXD into F. Then 5 and X are extended to sequences of inputs I1 ... Ik (written without commas) by