The two major parts of this report cover the two branches of our investigation of the theory of adjustable switching networks: Threshold Logic and Reliability of switching networks. Part A (Threshold Logic) presents the initial stages of bur work on the test-synthesis problem: For given switching functions, to determine whether or not they are threshold functions (test), and if they are, to obtain realizations (synthesis). Included here is a careful discussion of the preliminaries of test-synthesis: checking of unateness and positivization; checking of 2-monotonieity, ordering of arguments, and writing of the canonical form; and derivation of the complement and the reduced representations. Also included is a discussion of the basic system of linear equalities for test-synthesis and their reduction. One of two dual solution theorems, upon which our proposed test-synthesis procedure is based, is stated and proved. Finally, more recent work on' the test synthesis problem is summarized. We plan to write a comprehensive paper on test-synthesis which includes the material presented here (as its first part) and, in addition, a complete discussion of the results obtained more recently.
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