296 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1960 



the next instruction ; if it is negative, clear the accumulator and go to 

 the storage location specified for the next instruction," is an example 

 used in the compound interest problem. Most digital computei-s can 

 use any of several decision operations. 



The binary notation incidentally is convenient for logical opera- 

 tions. Tlie 1 and can represent words — true and false^ yes and no — 

 as well as they can represent numbers. Furthermore, we noted that 

 electronic computers use logical elements such as and, or, and not 

 to do arithmetic, and since logical problems are also generally stated 

 in these terms, obviously the same devices used for arithmetic opera- 

 tions can be used for strictly logical operations. 



As an example of a logical problem, let us consider the "logic" of 

 a two-way switch. Assume two switches, A (the upstairs switch) 

 and B (the downstairs switch) where in each case represents the 

 switch in the down position and 1 represents the switch in the up 

 position. Assume further that the hall light is represented by S 

 where is the light being off and 1 is the light being on. Further, 

 we know that "the hall light is on when the upstairs switch is up and 

 the downstairs switch is down or the upstairs switch is down mid the 

 downstairs switch is up, but not when both switches are up or when 

 both switches are down." How can this be represented in terms which 

 have already been considered ? 



Tlie answer is the A, B, and Sum columns of table 3. There the 

 A and B represented binary numbers being added, but the logic is 

 the same. When the condition of the hall light being on is restated 

 as "the upstairs switch is up or the downstairs switch is up and both 

 the upstairs switch and the downstairs switch are not up" then it is 

 directly analogous to the word statement previously given for the 

 sum digit in binary addition. 



Of course, practical logical problems are much more complicated 

 than indicated by this example, the number of different possibilities 

 being enormous. To illustrate, let us consider how computers have 

 been used for a process well recognized as a model of logic, that is, 

 the proving of Euclidean plane geometry theorems. 



Plane geometry theorem pi'oving. — The use of a digital computer 

 for proving theorems of plane geometry is illustrated by the example 

 in figure 2, in which is given the machine proof that a certain con- 

 struction involving the midpoints of two sides and two diagonals of 

 a quadrilateral results in a parallelogram. 



Tlie general procedure used here for theorem proving is to work 

 backward. Given as its goal to prove that a quadrilateral EFGII is 

 a parallelogram, the computer first selects subgoals which would 

 allow EFGH to meet the definition of a parallelogram. Each sub- 

 goal causes further subgoals to be generated, and so on. There may 



