The Synthesis of Two-Terminal Switching Circuits 



By CLAUDE. E. SHANNON 

 PART I: GENERAL THEORY 



1. Introduction 



THE theory of switching circuits may be divided into two major divi- 

 sions, analysis and synthesis. The problem of analysis, determining 

 the manner of operation of a given switching circuit, is comparatively 

 simple. The inverse problem of finding a circuit satisfying certain given 

 operating conditions, and in particular the best circuit is, in general, more 

 difficult and more important from the practical standpoint. A basic part 

 of the general synthesis problem is the design of a two-terminal network 

 with given operating characteristics, and we shall consider some aspects of 

 this problem. 



Switching circuits can be studied by means of Boolean Algebra.'- This 

 is a branch of mathematics that was first investigated by George Boole in 

 connection with the study of logic, and has since been applied in various 

 other fields, such as an axiomatic formulation of Biology,^ the study of neural 

 networks in the nervous system,^ the analysis of insurance policies,^ prob- 

 ability and set theory, etc. 



Perhaps the simplest interpretation of Boolean Algebra and the one 

 closest to the application to switching circuits is in terms of propositions. 

 A letter A^, say, in the algebra corresponds to a logical proposition. The 

 sum of two letters A' + Y represents the proposition ".Y or I'" and the 

 product XY represents the proposition "A" and F". The symbol A'' is used 

 to represent the negation of proposition X, i.e. the proposition "not A"". 

 The constants 1 and represent truth and falsity respectively. Thus 

 A -\- Y = 1 means .Y or Y is true, while -Y + YZ' = means A' or (!' and 

 the contradiction of Z) is false. 



The interpretationof Boolean Algebra in terms of switching circuits*^-^''-'"' 

 is very similar. The symbol A' in the algebra is interpreted to mean a make; 

 (front) contact on a relay or switch. The negation of A', written .Y', 

 represents a break (back) contact on the relay or switch. The constants 

 and 1 represent closed and open circuits respectively and the combining 

 operations of addition and multiplication correspond to series and parallel 

 connections of the switching elements involved. These conventions are 

 shown in Fig. L With this identification it is possible to write an algebraic 



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