Chapter IV: Part 4

At first glance, however, binary numbers seem a hopeless conglomeration of ones and zeros. This is so only because we have become conditioned to the decimal system, which was even more hopeless to us as youngsters. We may have forgotten, with the contempt of familiarity, that our number system is built on the idea of powers. In grade school we learned that starting at the right we had units, tens, hundreds, thousands, and so on. In the decimal number 111, for example, we mean 1 times 10^2, plus 1 times 10^1, plus 1. We have handled so many numbers so many times we have usually forgotten just what we are doing, and how.

The binary system uses only two numbers: 1 and 0. So it is five times as simple as the decimal system. It uses powers of two rather than ten, again far simpler. Let’s take the binary number 111 and break it down just as we do a decimal number. Starting at the left, we have 1 times 2^2, plus 1 times 2^1, plus 1. This adds up to 7, and there is our answer.

The decimal system is positional; this is what made it so much more effective in the simple expression of large numbers than the Roman numeral system. Binary is positional too, and for larger numbers we continue moving toward the left, increasing our power of two each time. Thus 1111 is 2^3 plus 2^2 plus 2^1 plus 1.

_System Development Corp._

A computer teaching machine answering a question about the binary
system.
]

We are familiar with decimal numbers like 101. This means 1 hundred, no tens, and 1 unit. Likewise in binary notation 101 means one 4, no 2’s, and one 1. For all its seeming complexity, then, the binary system is actually simpler than the “easy” decimal one we are more familiar with. But despite its simplicity, the binary system is far from being inferior to the decimal system. You can prove this by doing some counting on your fingers.

Normally we count, or tally, by bending down a finger for each new unit we want to record. With both hands, then, we can add up only ten units, a quite limited range. We can add a bit of sophistication, and assign a different number to each finger; thus 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Now, believe it or not, we can tally up to 55 with our hands! As each unit is counted, we raise and lower the correct finger in turn. On reaching 10, we leave that finger—thumb, actually—depressed, and start over with 1. On reaching 9, we leave it depressed, and so on. We have increased the capacity of our counting machine by 5-1/2 times without even taking off our shoes. The mathematician, by the way, would say we have a capability of not 55 but 56 numbers, since all fingers up would signify 0, which can be called a number. Thus our two hands represent to the mathematician a modulo-56 counter.

This would seem to vanquish the lowly binary system for good, but let’s do a bit more counting. This time we will assign each finger a number corresponding to the powers of 2 we use in reading our binary numbers. Thus we assign the numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. How many units can we count now? Not 10, or 55, but a good bit better than that. Using binary notation, our ten digits can now record a total of 1,023 units. True, it will take a bit of dexterity, but by bending and straightening fingers to make the proper sums, when you finally have all fingers down you will have counted 1,023, or 1,024 if you are a mathematical purist.

Once convinced that the binary method does have its merits, it may be a little easier to pursue a mastery of representing numbers in binary notation, difficult as it may seem at the outset. The usual way to convert is to remember, or list, the powers of 2, and start at the left side with the largest power that can be divided into the decimal number we want to convert. Suppose we want to change the number 500 into binary. First we make a chart of the positions:

Power of 2 8 7 6 5 4 3 2 1 0
─────────────────────────────────────────────────────────────
Decimal Number 256 128 64 32 16 8 4 2 1
─────────────────────────────────────────────────────────────
Binary Number 1 1 1 1 1 0 1 0 0

Since 256 is the largest number that will go into 500, we start there, knowing that there will be nine binary digits, or “bits” in our answer. We place a 1 in that space to indicate that there is indeed an eighth power of 2 included in 500. Since 128 will go into the remainder, we put a 1 in that space also. Continuing in this manner, we find that we need 1’s until we reach the “8” space which we must skip since our remainder does not contain an 8. We mark a 1 in the 4 space, but skip the 2 and the 1. Our answer, then, in binary notation is 111110100. This number is called “pure binary.” It can also lead to pure torture for human programmers whose eyes begin to bug with this “bit chasing,” as it has come to be called. Everything is of course relative, and the ancient Roman might gladly have changed DCCCLXXXVIII to binary 1101111000, which is two digits shorter.

There is a simpler way of converting that might be interesting to try out. We’ll start with our same 500. Since it is an even number, we put a 0 beneath it. Moving to the left, we divide by two and get 250. This also is an even number, so we mark down a 0 in our binary equivalent. The next division gives 125, an odd number, so we put down a 1. We continue to divide successively, marking a zero for each even remainder, and a 1 for the odd. Although it may not be obvious right away, we are merely arriving at powers of two by a process called mediation, or halving.

Decimal 1 3 7 15 31 62 125 250 500
─────────────────────────────────────────────────────
Binary 1 1 1 1 1 0 1 0 0

Obviously we can reverse this procedure to convert binary numbers to their decimal equivalents.

There is an interesting extension of this process called duplication by which multiplication can be done quite simply. Let us multiply 95 times 36. We will halve our 95 as we did in the earlier example, while doubling the 36. This time when we have an even number in the left column, we will simply cancel out the corresponding number in the right column.

95 36
47 72
23 144
11 288
5 576
2 ****
1 2304
——
3420

This clever bit of mathematics is called Russian peasant multiplication, although it was also known to early Egyptians and many others. It permits unschooled people, with only the ability to add and divide, to do fairly complex multiplication problems. Actually it is based on our old stand-by, the binary system. What we have done is to represent the 95 “dyadically,” or by twos, and to multiply 36 successively by each of these powers as applicable. We will not digress further, but leave this as an example of the tricks possible with the seemingly simple binary system.

Even after we have learned to convert from the decimal numbers we are familiar with into binary notation almost by inspection, the results are admittedly unwieldy for human handling. An employee who is used to getting $105 a week would be understandably confused if the computer printed out a check for him reading $1101001. For this reason the computer programmer has reached a compromise with the machine. He speaks decimal, it speaks binary; they meet each other halfway with something called binary-coded decimal. Here’s the way it works.

A little thought will show that the decimal numbers from 0 through 9 can be presented in binary using four bits. Thus:

_Decimal_ _Binary_
0 0
1 1
2 110
3 111
4 1100
5 1101
6 1110
7 10111
8 11000
9 11001

In the interest of uniformity we fill in the blanks with 0’s, so that each decimal number is represented by a four-digit block, or word, of binary code. Now when the computer programmer wants to feed the number 560 into the computer in binary he breaks it into separate words of 5, 6, and 0; or 0101, 0110, and 0000. In effect, we have changed $5 words into four-bit words! The computer couldn’t care less, since it handles binary digits at the rate of millions a second; and the human is better able to keep his marbles while he works with the computer. Of course, there are some computers that are classed as pure binary machines. These work on mathematical problems, with none of the restrictions imposed by human frailty. For the computer the pure binary system is more efficient than the binary decimal compromise.

The four-digit words can be made to represent not only numbers, but letters as well. When this is done it is called an alpha-numeric or alphameric code. Incidentally, it is conceivable that language could be made up of only 1’s and 0’s, or perhaps _a_’s and _b_’s would be better. All it would take would be the stringing together of enough letters to cover all the words there are. The result would be rather dull, with words like _aabbababaabbaaba_, _bbaabbaabababaaabab_, and _aaaaaaaaabaaa_; it is doubtful that the computer will make much headway with a binary alphabet for its human masters.

In the early days of binary computer work, the direct conversion to binary code we have discussed was satisfactory, but soon the designers of newer machines and calculating methods began to juggle the digits around for various reasons. For one thing, a decimal 0 was represented by four binary 0’s. Electrically, this represents no signal at all in the computer’s inner workings. If trouble happened, say a loose connection, or a power failure for a split second, the word 0000 might be printed out and accepted as a valid zero when it actually meant a malfunction. So the designers got busy trying other codes than the basic binary.

One clever result is the “excess-3” code. In this variation 3 is added to each decimal number before conversion. A decimal 30 is then represented by the word 0011 instead of 0000. There is, in fact, no such computer word as 0000 in excess-3 code. This eliminates the possibility of an error being taken for a 0. Excess-3 does something else too. If each digit is changed, that is, if 1’s become 0’s and 0’s become 1’s, the new word is the “9’s complement” of the original. For example, the binary code for 4 in excess-3 is 0111. Changing all the digits, we get 1000, which is decimal 5. This is not just an interesting curiosity, but the 9’s complement of 4 (9 minus 4). Anyone familiar with an adding machine is used to performing subtraction by using complements of numbers. The computer cannot do anything but add; by using the excess-3 code it can subtract by adding. Thus, while the computer cannot subtract 0110 from 1000, it can quite handily add 1001 to 1000 to get the same result.

There are many other reasons for codes, among them being the important one of checking for errors. “Casting out nines” is a well-known technique of the bookkeeper for locating mistakes in work. Certain binary codes, containing what is called a “parity bit,” have the property of self-checking, in a manner similar to casting out nines. A story is told of some pioneer computer designers who hit on the idea of another means of error checking not as effective as the code method.

The idea was clever enough, it being that identical computers would do each problem and compare answers, much like the pairs of abacus-wielders in Japan’s banks. In case both computers did not come up with the same answer, a correction would be made. With high hopes, the designers fed a problem into the machines and sat back to watch. Soon enough a warning light blinked on one machine as it caught an error. But simultaneously a light blinked on the other. After that, chaos reigned until the power plugs were finally pulled. Although made of metal and wires, the computers demonstrated a remarkably human trait; each thought the other was wrong and was doing its best to change its partner’s answer! The solution, of course, was to add a third computer.

Binary decimal, as we have pointed out, is a wasteful code. The decimal number 100 in binary decimal coding is 0001 0000 0000, or 12 digits. Pure binary is 1100100, or only 7 digits. By going to a binary-octal code, using eight numbers instead of ten, the words can be 3-bit instead of 4-bit. This is called an “economy” code, and finds some application. There are also “Gray” codes, reflected binary codes, and many more, each serving a particular purpose. Fortunately for the designer, he can be prodigal with his use of codes. With 4-bit words, 29 _billion_ codes are available, so a number of them are still unused.

Having translated our decimal numbers into code intelligible to our computer, we still have the mathematical operations to perform on it. With a little practice we can add, subtract, multiply, and divide our binary numbers quite easily, as in the examples that follow.

Addition: 1100 (12)
0111 ( 7)
—— ——
10011 (19)

Subtraction: 1010 (10)
- 0010 ( 2)
——— ——
1000 (8)

Multiplication: 0110 (6)

× 0011 (3)

——— –——

0110

0110

0000

0000

———

10010 (18)

TN1 Division: 1010 ÷ 10 = 0101 (10 ÷ 2 =
5)

The rules should be obvious from these examples. Just as we add 5 and 5 to get 0 with 1 to carry, we add 1 and 1 and get 0 with 1 to carry in binary. Adding 1 and 0 gives 1, 0 and 0 gives 0. Multiplying 1 times 1 gives 1, 1 times 0 gives 0, and 0 times 0 gives 0. One divides into 1 once, and into 0 no times. Thus we can manipulate in just the manner we are accustomed to.

The computer does not even need to know this much. All it is concerned with is addition: 1 plus 1 gives 0 and 1 to carry; 1 plus 0 gives 1; and 0 plus 0 gives 0. This is all it knows, and all it needs to know. We have described how it subtracts by adding complements. It can multiply by repetitive additions, or more simply, by shifting the binary number to the left. Thus, 0001 becomes 0010 in one shift, and 0100 in two shifts, doubling each time. This is of course just the way we do it in the decimal system. Shifting to the right divides by two in the binary system.

The simplest computer circuitry performs additions in a serial manner, that is, one operation at a time. This is obviously a slow way to do business, and by adding components so that there are enough to handle the digits in each row simultaneously the arithmetic operation is greatly speeded. This is called parallel addition. Both operations are done by parts understandably called adders, which are further broken down into half-adders.

There are refinements to basic binary computation, of course. By using a decimal point, or perhaps a binary point, fractions can be expressed in binary code. If the position to the left of the point is taken as 2 to the zero power, then the position just to the right of the point is logically 2 to the minus one, which if you remember your mathematics you’ll recognize as one-half. Two to the minus two is then one-fourth, and so on. While we are on the subject of the decimal point, sophisticated computers do what is called “floating-point arithmetic,” in which the point can be moved back and forth at will for much more rapid arithmetical operations.

No matter how many adders we put together and how big the computer eventually gets, it is still operating in what seems an awkward fashion. It is counting its fingers, of which it has two. The trick is in the speed of this counting, so fast that one million additions a second is now a commonplace. Try that for size in your own decimally trained head and you will appreciate the computer a little more.

_The Logical Algebra_

We come now to another most important reason for the effectiveness of the digital computer; the reason that makes it the “logical” choice for not only mathematics but thinking as well. For the digital computer and logic go hand in hand.

Logic, says Webster, is “the science that deals with canons and criteria of validity in thought and demonstration.” He admits to the ironic perversion of this basic definition; for example, “artillery has been called the ‘_logic_ of kings,’” a kind of logic to make “argument useless.” Omar Khayyám had a similar thought in mind when he wrote in _The Rubáiyát_,

The grape that can with logic absolute,
The Two-and-Seventy Sects confute.

Other poets and writers have had much to say on the subject of logic through the years, words of tribute and words of warning. Some, like Lord Dunsany, counsel moderation even in our logic. “Logic, like whiskey,” he says, “loses its beneficial effect when taken in too large quantities.” And Oliver Wendell Holmes asks,

Have you heard of the wonderful one-hoss shay
That was built in such a logical way
It ran a hundred years to the day?

The words logic and logical are much used and abused in our language, and there are all sorts of logic, including that of women, which seems to be a special case. For our purposes here it is best to stick to the primary definition in the dictionary, that of validity in thought and demonstration.

Symbolic logic, a term that still has an esoteric and almost mystical connotation, is perhaps mysterious because of the strange symbology used. We are used to reasoning in words and phrases, and the notion that truth can be spelled out in algebraic or other notation is hard to accept unless we are mathematicians to begin with.

We must go far back in history for the beginnings of logic. Aristotelian logic is well known and of importance even though the old syllogisms have been found not as powerful as their inventors thought. Modern logicians have reduced the 256 possible permutations to a valid 15 and these are not as useful as the newer kind of logic that has since come into being.

Leibniz is conceded to be the father of modern symbolic logic, though he probably neither recognized what he had done nor used it effectively. He did come up with the idea of two-valued logic, and the cosmological notion of 1 and 0, or substance and nothingness. In his _Characteristica Universalis_ he was groping for a universal language for science; a second work, _Calculus Ratiocinator_, was an attempt to implement this language. Incidentally, Leibnitz was not yet twenty years old when he formulated his logic system.

Unfortunately it was two centuries later before the importance of his findings was recognized and an explanation of their potential begun. In England, Sir William Hamilton began to refine the old syllogisms, and is known for his “quantification of the predicate.” Augustus De Morgan, also an Englishman, moved from the quantification of the predicate to the formation of thirty-two rules or propositions that result. The stage was set now for the man who has come to be known as the father of symbolic logic. His name was George Boole, inventor of Boolean algebra.

In 1854, Boole published “An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities.” In an earlier pamphlet, Boole had said, “The few who think that there is that in analysis which renders it deserving of attention for its own sake, may find it worth while to study it under a form in which every equation can be solved and every solution interpreted.” He was a mild, quiet man, though nonconformist religiously and socially, and his “Investigation” might as well have been dropped down a well for all the immediate splash it made in the scientific world. It was considered only academically interesting, and copies of it gathered dust for more than fifty years.

Only in 1910 was the true importance given to Boole’s logical calculus, or “algebra” as it came to be known. Then Alfred North Whitehead and Bertrand Russell made the belated acknowledgment in their _Principia Mathematica_, and Russell has said, “Pure mathematics was discovered by Boole, in a work he called ‘The Laws of Thought.’” While his praise is undoubtedly exaggerated, it is interesting to note the way in which mathematics and thought are considered inseparable. In 1928, the first text on the new algebra was published. The work of Hilbert and Ackermann, _Mathematical Logic_, was printed first in German and then in English.

What was the nature of this new tool for better thinking that Boole had created? Its purpose was to make possible not merely precise, but _exact_ analytical thought. Historically we think in words, and these words have become fraught with semantic ditches, walls, and traps. Boole was thinking of thought and not mathematics or science principally when he developed his logic algebra, and it is indicative that symbolic logic today is often taught by the philosophy department in the university.

Russell had hinted at the direction in which symbolic logic would go, and it was not long before the scientist as well as the mathematician and logician did begin to make use of the new tool. One pioneer was Shannon, mentioned in the chapter on history. In 1938, Claude Shannon was a student at M.I.T. He would later make scientific history with his treatise on and establishment of a new field called information theory; his early work was titled “A Symbolic Analysis of Relay and Switching Circuits.” In it he showed that electrical and electronic circuitry could best be described by means of Boolean logic. Shannon’s work led to great strides in improving telephone switching circuits and it also was of much importance to the designer of digital computers. To see why this is so, we must now look into Boolean algebra itself. As we might guess, it is based on a two-valued logic, a true-false system that exactly parallels the on-off computer switches we are familiar with.

The Biblical promise “Ye shall know the truth, and the truth shall make you free” applies to our present situation. The best way to get our feet wet in the Boolean stream is to learn its so-called “truth tables.”

_Conjunctive Boolean Operation_

A _and_ B equal C A B C
(A · B = C) ———
0 0 0
1 0 0
0 1 0
1 1 1

_Disjunctive Boolean Operation_

A _or_ B equals C A B C
(Ā ∨ B = C) ———
0 0 0
1 0 1
0 1 1
1 1 1

In the truth tables, 1 symbolizes true, 0 is false. In the conjunctive AND operation, we see that only if both A and B are true is C true. In the disjunctive OR operation, if _either_ A _or_ B is true, then C is also true. From this seemingly naïve and obvious base, the entire Boolean system is built, and digital computers can perform not only complex mathematical operations, but logical ones as well, including the making of decisions on a purely logical basis.

Before going on to the few additional conditions and combinations that complete the algebra, let’s study some analogies that will make clear the AND/OR principles of operation. We can think of AND as two bridges in sequence over two rivers. We can reach our destination only if both bridges are working. However, suppose there are two parallel bridges and only one river. We can then cross if either or both of the bridges is working. A closer example is that of electrical switches. Current will flow through our AND circuit if—and only if—both switches are closed. When the switches are in parallel—an OR circuit—current will flow if either, or both, are closed.

The truth tables resemble the bridge or switch arrangements. We can proceed across the line of 1’s and 0’s in the first table only if both switches are closed. The symbol 1 means that the switch is closed, so we can cross only the bottom line. In the second table, we are told we can proceed across the line if either switch is closed. Thus we can cross lines 2, 3, and 4. We can use many symbols in our two-valued system.

_Symbol_

Bridge No
Bridge

Power No Power

1 0

True False

A little imagination suggests a logic computer of sorts with one switch, a battery, and a light bulb. Suppose we turn on the switch when we drive into our garage. A light in the hallway then indicates that the car is available. By using two switches we can indicate that a second car is also in the garage; or that either of them is, simply by choosing between AND logic and OR logic. Childish as this seems, it is the principle of even our most complex thinking processes. You will remember that the brain is considered a digital computer, since neurons can only be on or off. All it takes is 10 billion neuron switches!

_Remington Rand UNIVAC_

AND and OR gates in series. Switches 1 _and_ 2, plus 3 _or_ 4, are
needed to light the bulb.
]

In addition to the conjunctives AND and OR, Boolean algebra makes use of the principle of negation. This is graphically illustrated thus:

_Original_ _Negation_
A Ā
1 0
0 1

The negation device used in computer circuitry is called an inverter, since it changes its input from a 1 to a 0, or vice versa. The usefulness of such an element is obvious when we remember the computer trick of subtracting by adding complements. The inverter circuit used with a code like the excess-3 readily forms these complements.

Further sophistication of the basic Boolean forms leads to units other than the AND and OR gates. Possible are NOT, NOR, and exclusive-OR forms. In the latter, there is an output if one and only one input is present. The NOR circuit is interesting in that it was made possible with the introduction of the transistor; the vacuum tube does not permit this configuration.

_Computer Control Co._

The functions of two binary variables.
]

Present-day symbolic logic is not the pure Boolean as presented back in 1854. Boole’s OR was the exclusive, one and only one, type. Today the logician generally assumes the either-or connotation. The logic has also been amplified, using the commutative, associative, and distributive laws much like those of conventional algebra. We are indebted to De Morgan for most of this work, showing that A and B equals B and A; A and (A and B) equals (A and B) and A; and so on. While these seem intuitively true, the implications are nonetheless of great importance both in pure logic and its practical use in circuitry.

A graphic representation of the metamorphosis from symbolic to actual implementation of Boolean equations follows: The implication of importance is that logic applies equally well whether we are making a qualifying statement such as “A man must have strength _and_ courage to win a barehanded fight with a lion,” or wiring a defensive missile so that it will fire only if a target is within range _and_ is unfriendly.

In the early period of computer design the engineer was faced with the problem of building his own switches and gates. Today many companies offer complete “packaged” components—AND gates, OR gates, and the other configurations. This is the modular approach to building a computer and the advantages are obvious. The designer can treat the components simply as “black boxes” that will respond in a prescribed way to certain input conditions. If he wants, the engineer can go a step further and buy a ready-built logic panel consisting of many components of different types. All he need do to form various logic circuits is to interconnect the proper components with plug-in leads. This brings us to the point of learning what we can do with these clever gates and switches now that we have them available and know something about the way they work.

We talked about the computer adder circuit earlier in this chapter. It is made up of two half-adders, remember, with perhaps an additional OR gate, flip-flop, etc. Each half-adder is composed of two AND gates and an OR gate. So we have put together several basically simple parts and the result is a piece of equipment that will perform addition at a rate to make our heads swim.

There are other things we can do with Boolean logic besides arithmetic. A few gates will actuate a warning signal in a factory in case either of two ventilators is closed and the temperature goes up beyond a safe point; or in case both vents are closed at the same time. We can build a logic computer that will tell us when three of four assembly lines are shut down at the same time, and also which three they are.

_General Electric Co., Computer Dept._

Electronic computers are built up of many “building blocks” like this
one.
]

Logic problems abound in puzzle books, and many of us spend sleepless nights trying to solve them in our heads. An example is the “Farnsworth Car Pool” problem. Rita Farnsworth asks her husband if someone in his car pool can drive for him tomorrow so that she may use the car. Joe Farnsworth replies, “Well, when I asked Pete if he would take my turn he said he was flying to Kansas City today, but he’d be glad to drive tomorrow if he didn’t have to stay over and that his wife has been staying home lately and he will drive her car if she doesn’t go to work. Oscar said that since his own car is due back from the garage tomorrow he can drive it even if his wife does use hers, provided the garage gets his back to him. But if this cold of mine gets any worse I’m going to stay home even if those fellows have to walk to work, so you can certainly have the car if I don’t go to work.” This dialogue of Joe’s confuses Rita and most of us are in the same state.

_Autonetics Division, North American Aviation, Inc._

Testing an assembled digital computer.
]

The instruction manual for BRAINIAC, a do-it-yourself computer that sells for a few dollars, gives a simple wiring diagram for solving Rita’s dilemma. Electrically the problem breaks down into three OR gates and one AND gate. All Mrs. Farnsworth has to do is set in the conditions and watch the indicator light. If it glows, she gets the car!

These are of course simple tasks and it might pay to hire a man to operate the vents, and ride to work on the bus when the car pool got complicated. But even with relatively few variables, decision-making can quickly become a task requiring a digital computer operating with Boolean logic principles.

_Science Materials Center_

Problem in logic reduced to electrical circuits.
]

The Smith-Jones-Robinson type of problem in which we must find who does what and lives where is tougher than the car pool—tough enough that it is sometimes used in aptitude tests. Lewis Carroll carried this form of logical puzzler to complicated extremes involving not just three variables but a dozen. To show how difficult such a problem is, an IBM 704 required four minutes to solve a Carroll puzzle as to whether any magistrates indulge in snuff-taking. The computer did it the easy way, without printing out a complete “truth table” for the problem—the method a man would have to use to investigate all the combinations of variables. This job would have taken 13 hours! While the question of the use of snuff is perhaps important only to tobacconists and puzzle-makers, our technical world today does encounter similar problems which are not practical of solution without a high-speed computer. A recent hypothetical case discussed in an electronics journal illustrates this well.

A missile system engineer has the problem of modifying a Nike-Ajax launching site so that it can be used by the new Nike-Hercules missile. He must put in switching equipment so that a remote control center can choose either an Ajax system, or one of six Hercules systems. To complicate things, the newer Hercules can be equipped with any of three different warheads and fly either of two different missions. When someone at the control center pushes a button, the computer must know immediately which if any of the missiles are in acceptable condition to be fired.

This doesn’t sound like too big a problem. However, since there are twelve on-off signals to be considered, and since each has two possible states, there are 4,096 possible missile combinations. Not all these are probable, of course, but there is still sufficient variation to make it humanly impossible to check all of them and close a firing switch in the split second the control center can allow.

The answer lies in putting Boolean algebra on the job, with a system of gates and inverters capable of juggling the multiplicity of combinations. Then when the word comes requesting a missile launch, the computer handles the job in microseconds without straining itself unduly.

Just as Shannon pointed out twenty-five years ago, switching philosophy can be explained best by Boolean logic, and the method can be used not only to implement a particular circuit, but also to actually design the circuit in the first place. A simple example of this can be shown with the easy-to-understand AND and OR gates. A technician experimenting with an AND gate finds that if he simply reverses the direction of current, he changes the gate into an OR gate. This might come as a surprise to him if he is unfamiliar with Boolean logic, but a logician with no understanding of electrical circuits could predict the result simply by studying the truth tables for AND and OR.

Reversing the polarity is equivalent to changing a 1 to a 0 and vice versa. If we do this in the AND gate table, we should not be surprised to find that the result looks exactly like the OR table! It acts like it too, as the technician found out.

Boolean logic techniques can be applied to existing circuits to improve and/or simplify them. Problems as simple as wiring a light so that it can be turned on and off from two or more locations, and those as complex as automating a factory, yield readily to the simple rules George Boole laid down more than a hundred years ago.

Watching a high-speed electronic digital computer solve mathematical problems, or operate an industrial control system with speed and accuracy impossible for human monitors, it is difficult to believe that the whole thing hinges on something as simple as switches that must be either open or closed. If Leibnitz were alive, he could well take this as proof of his contention that there was cosmological significance in the concept of 1 and 0. Maybe there is, after all!

_Industrial Electronic Engineering & Maintenance_

“Luckily I brought along a ‘loaner’ for you to use while I repair your
computer.”
]

------------------------------------------------------------------------

“_Whatever that be which thinks, understands, wills, and
acts, it is something celestial and divine._”

—Cicero

6: The Electronic Brain

The idea of a man-made “brain” is far from being new. Back in 1851, Dr. Alfred Smee of England proposed a machine made up of logic circuits and memory devices which would be able to answer any questions it was asked. Doctor Smee was a surgeon, keenly interested in the processes of the mind. Another Britisher, H. G. Wells, wrote a book called _Giant Brain_ in 1938 which proposed much the same thing: a machine with all knowledge pumped into it, and capable of feeding back answers to all problems.

If it was logical to credit “human” characteristics to the machines man contrived, the next step then was to endow the machine with the worst of these attributes. In works including Butler’s _Erewhon_, the diabolical aspects of an intelligent machine are discussed. The Lionel Britton play, _Brain_, produced in 1930, shows the machine gradually becoming the master of the race. A more physical danger from the artificial brain is the natural result of giving it a body as well. We have already mentioned Čapek’s _R.U.R._ and the Ambrose Bierce story about a chess-playing robot without a built-in sense of humor, who strangles the human being who beats him at a game. With these stories as models, other writers have turned out huge quantities of work involving mechanical brains capable of all sorts of mischief. Most of these authors were not as well-grounded scientifically as the pioneering Dr. Smee who admitted sadly that his “brain” would indeed be a giant, covering an area about the size of London!

The idea of the giant brain was given new lease by the early electronic computers that began appearing in the 1940’s. These vacuum-tube and mechanical-relay machines with their rows of cabinets and countless winking lights were seized on gleefully by contemporary writers, and the “brain” stories multiplied gaudily.

Many of the acts of these fictional machines were monstrous, and most of the stories were calculated to make scientists ill. Many of these gentlemen said the only correct part of the name “giant brain” was the adjective; that actually the machine was an _idiot savant_, a sort of high-speed moron. This opinion notwithstanding, the name stuck. One scholar says that while it is regrettable that such a vulgar term has become so popular, it is hardly worth while campaigning against its use.

An amusing contemporary fiction story describes an angry crowd storming a laboratory housing a “giant brain,” only to be placated by a calm, sensibly arguing scientist. The mob dispersed, he goes back inside and reports his success to the machine. The “brain” is pleased, and issues him his next order.

“Nonsense!” scoff most computer people. A recent text on operation of the digital computer says, “Where performance comparable with that of the human brain is concerned, man need have little fear that he will ever be replaced by this machine. It cannot think in any way comparable to a human being.” Note the cautious use of “little,” however.

Another authority admits that the logic machines of the monk Ramón Lull were very clever in their proof of God’s existence, but points out that the monk who invented them was far cleverer since no computer has ever invented a monk who could prove anything at all!

The first wave of ridiculous predictions has run its course and been followed by loud refutations. Now there is a third period of calmer and more sensible approach. A growing proportion of scientists take a middle-of-the-stream attitude, weighing both sides of the case for the computer, yet some read like science fiction.

Cyberneticist Norbert Wiener, more scientist than fictioneer, professes to foresee computerized robots taking over from their masters, much as a Greek slave once did. Mathematician John Williams of the Rand Corporation thinks that computers can, and possibly will, become more intelligent than men.

Equally reputable scientists take the opposite view. Neuro-physiologist Gerhard Werner of Cornell Medical College doubts that computers can ever match the creativity of man. He seems to share the majority view today, though many who agree will add, tongue in cheek, that perhaps we’d _better_ keep one hand on the wall plug just in case.

_Thinking Defined_

The first step in deciding whether or not the computer thinks is to define thinking. Far from being a simple task, this definition turns out to be a slippery thing. In fact, if the computer has done no more than demand this sort of reappraisal of the human brain’s working, it has justified its existence. Webster lists meanings for “think” under two headings, for the transitive and intransitive forms of the verb. These meanings, respectively, start out with “To form in the mind,” and “To exercise the powers of judgment ... to reflect for the purpose of reaching a conclusion.”

Even a fairly simple computer would seem to qualify as a thinker by these yardsticks. The storing of data in a computer memory may be analogous to forming in the mind, and manipulating numbers to find a square root certainly calls for some sort of judgment. Learning is a part of thinking, and computers are proving that they _can_ learn—or at least be taught. Recall of this learning from the memory to solve problems is also a part of the thinking process, and again the computer demonstrates this capability.

One early psychological approach to the man-versus-machine debate was that of classifying living and nonliving things. In _Outline of Psychology_, the Englishman William McDougall lists seven attributes of life. Six of these describe “goal-seeking” qualities; the seventh refers to the ability to learn. In general, psychologist McDougall felt that purposive behavior was the key to the living organism. Thus any computer that is purposive—and any commercial model had better be!—is alive, in McDougall’s view. A restating of the division between man and machine is obviously in order.

Dr. W. Ross Ashby, a British scientist now working at the University of Illinois, defines intelligence as “appropriate selection” and goal-seeking as the intelligent process _par excellence_, whether the selecting is done by a human being or by a machine. Ashby does split off the “non goal-seeking” processes occurring in the human brain as a distinct class: “natural” processes neither good nor bad in themselves and resulting from man’s environment and his evolution.

Intelligence, to Ashby, who long ago demonstrated a mechanical “homeostat” which showed purposive behavior, is the utilization of information by highly efficient processing to achieve a high intensity of appropriate selection. Intelligent is as intelligent does, no distinction being made as to man or machine. _Humanoid_ and _artificial_ would thus be meaningless words for describing a computer. Ashby makes another important point in that the intelligence of a brain or a machine cannot exceed what has been put into it, unless we admit the workings of magic. Ashby’s beliefs are echoed in a way by scientist Oliver Selfridge of Lincoln Laboratory. Asked if a machine can think, Selfridge says, “Certainly; although the machine’s intelligence has an elusive, _unnatural_ quality.”

“Think, Hell, COMPUTE!” reads the sign on the wall of a computer laboratory. But much of our thinking, perhaps some of the “natural” processes of our brains, doesn’t seem to fit into computational patterns. That part of our thinking, the part that includes looking at pretty girls, for example, will probably remain peculiar to the human brain.

_The Human Brain_

Mundy Peale, president of Republic Aviation Corporation, addressing a committee studying the future of manned aircraft, had this to say:

Until someone builds, for $100 or less with unskilled labor, a computer no larger than a grapefruit, requiring only a tenth of a volt of electricity, yet capable of digesting and transmitting incoming data in a fraction of a second and storing 10,000 times as much data as today’s largest computers, the pilots of today have nothing to worry about.

The human brain is obviously a thing of amazing complexity and fantastic ability. Packed into the volume Mr. Peale described are some 10 _billion_ neurons, the nerve cells that seem to be the key to the operation of our minds. Hooked up like some ultra-complicated switchboard, the network of interconnections stores an estimated 200,000,000,000,000,000,000 bits of information during a lifetime! By comparison, today’s most advanced computers do seem pathetically unimpressive.

We have discussed both analog and digital computers in preceding chapters. It is interesting to find that the human brain is basically a digital type, though it does have analog overtones as well. Each of the neurons is actually a switch operated by an electric current on a go/no-go, all-or-nothing basis. Thus a neuron is not partly on or partly off. If the electrical impulse exceeds a certain “threshold” value, the switch operates.

Tied to the neurons are axons, the long “wires” that carry the input and output. The axons bring messages from the body’s sensors to the neurons, and the output to other neurons or to the muscles and other control functions. This grapefruit-size collection of electrochemical components thus stores our memories and effects the operation we call thinking.

Since brain impulses are electrical in nature, we speak of them in electrical terms. The impulses have an associated potential of 50 millivolts, that is, fifty thousandths of a volt. The entire brain dissipates about 10 watts, so that each individual neuron requires only a billionth of a watt of power. This amount is far less than that of analogous computer parts.

A neuron may take a ten-thousandth of a second to respond to a stimulus. This seemingly rapid operation time turns out to be far slower than present-day computer switches, but the brain makes up for this by being a “parallel operation” system. This means that many different connections are being made simultaneously in different branches, rather than being sequential, or a series of separate actions.

Packaging 10 billion parts in a volume the size of a grapefruit is a capability the computer designer admires wistfully. Since the brain has a volume of about 1,000 cubic centimeters, 10 million neurons fit into a space of one cubic centimeter! A trillion would fit in one cubic foot, and man-made machines with even a million components per cubic foot are news today.

Even when we are resting, with our eyes closed, a kind of stand-by current known as the alpha rhythm is measurable in our brains. This current, which has a frequency of about 10 cycles per second, changes when we see or feel something, or when we exercise the power of recall. It disappears when we sleep soundly, and is analogous to the operating current in a computer. Also, there is “power” available locally at the neurons to “amplify” weak signals sufficiently to trigger off following branches of neurons.

Philosophers have proposed two general concepts of the human brain and how it functions. The _a priori_ theory presupposes a certain amount of “wired-in” knowledge: instincts, ideals, and so on. The other theory, that of the _tabula rasa_ or clean slate new brain, argues that each of us organizes an essentially random net of nerves into ordered intelligence. Both theories are being investigated with computers, and as a result light is beginning to be shed on the workings of our brains.

_The Upjohn Company, Ezra Stoller Associates Photo_

“A moment at a concert” is diagrammed by brain model, showing eyes,
ears, nerves, and structures analogous to brain. Picture at top
represents perception.
]

There is another division of philosophical thought in the mechanistic versus _elan vital_ argument. In other words, is the entire mind to be found in its constituent parts, or is there an intangible extra something that really breathes life into us? Whatever the correct concept, the brain does record impressions it can later recall. No one yet knows just how this is done, but several theories have been advanced. One of these describes a “chain circuit” set up in a neuron network by messages from the body’s sensors. This circuit, once started, continues to circle through the brain and is on tap whenever that particular experience needs to be recalled. The term “reverberate” is used in connection with this kind of memory, seeming to be a good scientific basis for the poetic “echoes of the past.” Reverberation circuits also provide the memory for some computers.

Among other explanations of memory is that of conditioning the neurons to operate more “easily,” so that certain paths are readily traversed by brain impulses. This could be effected by chemical changes locally, and such a technique too is used in computers.

However the brain accomplishes its job, it is certain that it evolved in its present form as a result of the environment its cells have had to function in for billions of years. Its prime purpose has been one of survival, and for this reason some argue that it is not particularly well adapted to abstract reasoning. Although the brain can do a wide variety of things from dreaming to picking out one single voice amid the hubbub of noise at a social gathering—a phenomenon scientists have given the descriptive name of “cocktail party effect”—men like Ashby consider it a very inflexible piece of equipment not well suited to pure logic. As a test of your brain as a logical device, consider the following problem from the Litton Industries “Problematical Recreations.”

If Sara shouldn’t, then Wanda would. It is impossible that the statements: “Sara should” and “Camille couldn’t” can both be true at the same time. If Wanda could, then Sara should and Camille could. Therefore Camille could. Is this conclusion valid?

If your head starts to swim, you are not alone. Very few humans solve such problems easily. Interestingly, those who do, make good computer programmers.

_The Computer’s Brain_

Just as we have an anthropomorphic God, many people have done their best to endow the computer with human characteristics. Not only in fiction but also in real life, the electronic brains have been described as neurotic and frustrated on occasion, and also as being afraid and even having morning sickness! A salesman for a line of computers was asked to explain in understandable terms the difference between two computers whose specifications confused a customer. “Let’s put it this way,” the salesman said, “The 740 thinks the 690 is a moron!”

We can begin to investigate the question of computer intelligence by again looking up a definition. The word “compute” means literally to think, or reckon, with. Early computers such as counting sticks, the abacus, and the adding machine are obviously something man thinks with. Even though we may know the multiplication tables, we find it easier and _safer_ to use a mechanical device to remember and even to perform operations for us.

These homely devices do not possess sufficient “intelligence” to raise any fears in our minds. The abacus, for example, displays only what we might charitably call the property of memory. It has a certain number of rows, each row with a fixed number of beads. While it is not fallible, as is the human who uses it, it is far more limited in scope. All it can ever do is help us to add or subtract, and if we are clever, to multiply, divide, do square roots, and so on. If we are looking for purposive behavior in computing machines, it is only when we get to the adding machine that a glimmer appears. When a problem is set in and the proper button pushed, this device is compelled to go through the gear-whirring or whatever required to return it to a state of equilibrium with its problem solved.

So far we might facetiously describe the difference in the goal-seeking characteristics of man and machine by recalling that man seeks lofty goals like climbing mountains simply because they are there, while the computer seeks its goal much like the steel ball in the pinball machine, impelled by gravity and the built-in springs and chutes of the device. When we come to a more advanced computer, however, we begin to have difficulty in assessing characteristics. For the JOHNNIAC, built by Rand and named for John von Neumann, can prove the propositions in the _Principia Mathematica_ of Whitehead and Russell. It can also “learn” to play a mediocre game of chess.

If we investigate the workings of a digital computer, we find much to remind us of the human brain. First is the obvious similarity of on-off, yes-no operation. This implies a power source, usually electrical, and a number of two-position switches. The over-all configuration of the classic computer resembles, in principal if not physical appearance, that of the human brain and its accessories.

As we have learned, the electronic computer has an input section, a control, an arithmetic (or logic) section, a memory, and an output. Looking into the arithmetic and memory sections, we find a number of comparisons with the brain. The computer uses power, far more than the brain. A single transistor, which forms only part of a neuron, may use a tenth of a watt; the brain is ahead on this score by a factor of millions to one.

Electronic switches have an advantage over the neuron in that they are much faster acting. So fast have they become that engineers have had to coin new terms like nanosecond and picosecond, for a billionth and a trillionth of a second. Thus, the computer’s individual elements are perhaps 100,000 times faster than those of the brain.