Simon’s Mentality — Possible Range of Information
We can say that Simon has a mentality of 4. We mean not age 4 but just the simple fact that Simon knows only 4 numbers and can do only 4 operations with them. But Simon can keep on doing these operations in all sorts of routines as long as Simon has instructions. We decide that Simon will know just 4 numbers, 0, 1, 2, 3, in order to keep our model mechanical brain very simple. Then, for any register, we need only 2 relays; for any answer, we need only 2 lights.
Any calculating machine has a mentality, consisting of the whole collection of different ideas that the machine can ever actually express in one way or another. For example, a 10-place desk calculating machine can handle numbers up to 10 decimal digits without additional capacity. It cannot handle bigger numbers.
What are the 4 operations with numbers which Simon can carry out? Let us consider some simple operations that we can perform with just 4 numbers. Suppose that they stood for 4 directions in the order east, north, west, south (see Fig. 2). Or suppose that they stood for a turn counterclockwise through some right angles as follows:
0: Turn through 0°, or no right angles.
1: Turn through 90°, or 1 right angle.
2: Turn through 180°, or 2 right angles.
3: Turn through 270°, or 3 right angles.
Then we could have the operations of addition and negation, defined as follows:
| Addition | Negation | |||||||
|---|---|---|---|---|---|---|---|---|
| c = a + b | c = -a | |||||||
| b: | 0 | 1 | 2 | 3 | ||||
| a: | a | c | ||||||
| 0 | 0 | 1 | 2 | 3 | 0 | 0 | ||
| 1 | 1 | 2 | 3 | 0 | 1 | 3 | ||
| 2 | 2 | 3 | 0 | 1 | 2 | 2 | ||
| 3 | 3 | 0 | 1 | 2 | 3 | 1 | ||
For example, the first table says, “1 plus 3 equals 0.” This means that, if we turn 1 right angle and then turn in the same direction 3 more right angles, we face in exactly the same way as we did at the start. This statement is clearly true. For another example, the second table says, “2 is the negative of 2.” This means that, if we turn to the left 2 right angles, we face in exactly the same way as if we turn to the right 2 right angles, and this statement also is, of course, true.
With only these two operations in Simon, we should probably find him a little too dull to tell us much. Let us, therefore, put into Simon two more operations. Let us choose two operations involving both numbers and logic: in particular, (1) finding which of two numbers is greater and (2) selecting. In this way we shall make Simon a little cleverer.
It is easy to teach Simon how to find which of two numbers is the greater when all the numbers that Simon has to know are 0, 1, 2, 3. We put all possible cases of two numbers a and b into a table:
| b: | 0 | 1 | 2 | 3 | |
| a: | |||||
| 0 | |||||
| 1 | |||||
| 2 | |||||
| 3 | |||||
Then we tell Simon that we shall mark with 1 the cases where a is greater than b and mark with 0 the cases where a is not greater than b:
| Greater Than | |||||
|---|---|---|---|---|---|
| b: | 0 | 1 | 2 | 3 | |
| a: | |||||
| 0 | 0 | 0 | 0 | 0 | |
| 1 | 1 | 0 | 0 | 0 | |
| 2 | 1 | 1 | 0 | 0 | |
| 3 | 1 | 1 | 1 | 0 | |
For example, “2 is greater than 3” is false, so we put 0 in the table on the 2 line in the 3 column. We see that, for the 16 possible cases, a is greater than b in 6 cases and a is not greater than b in 10 cases.
There is a neat way of saying what we have just said, using the language of mathematical logic (see Chapter 9 and Supplement 2). Suppose that we consider the statement “a is greater than b” where a and b may be any of the numbers 0, 1, 2, 3. We can say that the truth value p of a statement P is 1 if the statement is true and that it is 0 if the statement is false:
p = 1 if P is true, 0 if P is false
The truth value of a statement P is conveniently denoted as T(P) (see Supplement 2):
p = T(P)
Now we can say that the table for the operation greater than shows the truth value of the statement “a is greater than b”:
p = T(a > b)
Let us turn now to the operation selection. By selecting we mean choosing one number a if some statement P is true and choosing another number b if that statement is not true. As before, let p be the truth value of that statement P, and let it be equal to 1 if P is true and to 0 if P is false. Then the operation of selection is fully expressed in the following table and logical formula (see Supplement 2):
| Selection | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| c = a·p + b·(1 - p) | |||||||||
| p: | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | |
| b: | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 | |
| a: | |||||||||
| 0 | 0 | 1 | 2 | 3 | 0 | 0 | 0 | 0 | |
| 1 | 0 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | |
| 2 | 0 | 1 | 2 | 3 | 2 | 2 | 2 | 2 | |
| 3 | 0 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | |
For example, suppose that a is 2 and b is 3 and the statement P is the statement “2 is greater than 0.” Since this statement is true, p is 1, and a·p + b·(1 - p) = 2(1) + 3(0) = 2
This result is the same as selecting 2 if 2 is greater than 0 and selecting 3 if 2 is not greater than 0.
Thus we have four operations for Simon that do not overstrain his mentality; that is, they do not require him to go to any numbers other than 0, 1, 2, and 3. These four operations are: addition, negation, greater than, selection. We label these operations also with the numbers 00 to 11 as follows: addition, 00; negation, 01; greater than, 10; selection, 11.