Algebraic topological methods for the synthesis of switching systems. I

i. The problem of synthesis of switching systems is one of ever increasing importance to modern technology; it arises in the design of digital computers, telephone switching systems and control mechanisms of all sorts. Behind, in fact, every pushbutton lies a switching circuit. In spite of its considerable importance there is as yet no well developed theory for the rational design of such systems. Professor H. H. Aiken A states that a lack of adequate mathematical methods for the investigation of the functional behavior of electronic control circuits represented the largest single obstacle to the rapid development of the subject. Even efforts to understand the human neural system have encountered this problem (cf. the provocative work of McCulloch and Pitts M-P). Previous attempts toward building a theory have considered the problem from the point of view of symbolic logic; this paper considers the problem as being inherently of a combinatorial topological character. This point of view yields a mechanization and visualization of the problen not evident from the logical approach. ii. The synthesis problem may be loosely described as follows. Let A he a binary variable, assuming values 0 and 1. A switch with variable A is a device having two states closed if A = 1 and open if A =0 (see illustration). _/_