Part 7
A third perhaps not as important advantage the digital machine has is its compactness. We are speaking now of later computers, and not the pioneer electromechanical giants, of course. The transistor and other small semiconductor devices supplanted the larger tubes, and magnetic cores took the place of cruder storage components. Now even more exotic devices are quietly ousting these, as magnetic films and cryotrons begin to be used in computers.
[Illustration:
_Science Materials Center_
BRAINIAC, another do-it-yourself computer. This digital machine is here being programmed to solve a logic problem involving a will. ]
This drastic shrinking of size by thinking small on the part of computer designers increases the capacity of the digital computer at no sacrifice in accuracy or reliability. The analog, unfortunately, cannot make use of many of these solid-state devices. Again, the bugaboo of accuracy is the reason; let’s look further into the problem.
The most accurate and reliable analog computers are mechanical in nature. We can cut gears and turn shafts and wheels to great accuracy and operate them in controlled temperature and humidity. Paradoxically, this is because mechanical components are nearer to digital presentations than are electrical switches, magnets, and electronic components. A gear can have a finite number of teeth; when we deal with electrons flowing through a wire we leave the discrete and enter the continuous world. A tiny change in voltage or current, or magnetic flux, compounded several hundred times in a complex computer, can change the final result appreciably if the errors are cumulative, that is, if they are allowed to pile up. This is what happens in the analog computer using electrical and electronic components instead of precisely machined cams and gears.
The digital device, on the other hand, is not so penalized. Though it uses electronic switches, these can be so set that even an appreciable variation in current or voltage or resistance will not affect the proper operation of the switch. We can design a transistor switch, for example, to close when the current applied exceeds a certain threshold. We do not have to concern ourselves if this excess current is large or small; the switch will be on, no more and no less. Or it will be completely off. Just as there is no such thing as being a little bit dead, there is no such thing as a partly off digital switch. So our digital computer can make use of the more advanced electronic components to become more complex, or smaller, or both. The analog must sacrifice its already marginal accuracy if it uses more electronics. The argument here is simplified, of course; there are electronic analog machines in operation. However, the problem of the “drift” of electronic devices is inherent and a limiting factor on the performance of the analog.
These, then, are some of the advantages the digital computer has over its analog relative. It is more flexible in general—though there are _some_ digital machines that are more specialized than _some_ analog types; it is more accurate and apparently will remain so; and it is more amenable to miniaturization and further complexity because its designer can use less than perfect parts and produce a perfect result.
In the disadvantage department the digital machine’s only drawback seems to be its childish way of solving problems. About all it knows how to do is to add 1 and 1 and come up with 2. To multiply, it performs repetitive additions, and solving a difficult equation becomes a fantastically complex problem when compared with the instantaneous solution possible in the analog machine. The digital computer redeems itself by performing its multitudinous additions at fabulous speeds.
Because it must be fed digits in its input, the digital machine is not economically feasible in many applications that will probably be reserved for the analog. A digital clock or thermometer for household use would be an interesting gimmick, but hardly worth the extra trouble and expense necessary to produce. Even here, though, first glances may be wrong and in some cases it may prove worth while to convert analog inputs to digital with the reverse conversion at the output end. One example of this is the airborne digital computer which has taken over many jobs earlier done by analog devices.
There is another reason for the digital machines ubiquitousness, a reason it does not seem proper to list as merely a relative advantage over the analog. We have described the analog computer used as an aid to psychological testing procedures, and its ability to handle a multiplicity of problems at once. This perhaps tends to obscure the fact that the digital machine by its very on-off, yes-no nature is ideally suited to the solving of problems in logic. If it achieves superiority in mathematics in spite of its seemingly moronic handling of numbers, it succeeds in logic because of this very feature.
While it might seem more appropriate that music be composed by analogy, or that a chess-playing machine would likely be an analog computer, we find the digital machine in these roles. The reason may be explained by our own brains, composed of billions of neurons, each capable only of being on or off. While many philosophers build a strong case for the yes-no-maybe approach with its large areas of gray, the discipline of formal logic admits to only two states, those that can so conveniently be represented in the digital computer’s flip-flop or magnetic cores.
The digital computer, then, is not merely a counting machine, but a decision-maker as well. It can decide whether something should be added, subtracted, or ignored. Its logical manipulations can by clever circuitry be extended from AND to OR, NOT, and NOR. It thus can solve not only arithmetic, but also the problems of logic concerning foxes, goats, and cabbages, or cannibals and missionaries that give us human beings so much trouble when we encounter them.
The fact that the digital computer is just such a rigorously logical and unbending machine poses problems for it in certain of its dealings with its human masters. Language ideally should be logical in its structure. In general it probably is, but man is so perverse that he has warped and twisted his communications to the point that a computer sticking strictly to book logic will hit snags almost as soon as it starts to translate human talk into other human talk, or into a logical machine command or answer.
For instance, we have many words with multiple meanings which give rise to confusion unless we are schooled in subtleties. There are stories, some of them apocryphal but nonetheless pointing up the problem, of terms like “water goat” cropping up in an English-to-Russian translation. Investigation proved that the more meaningful term would have been “hydraulic ram.” In another interesting experiment, the expression, “the spirit is willing but the flesh is weak” was machine translated into Russian, and then that result in turn re-translated back into English much in the manner of the party game of “Telephone” in which an original message is whispered from one person to another and finally back to the originator. In this instance, the final version was, “The vodka is strong, but the meat is rotten.”
It is a fine distinction here as to who is wrong, the computer or man and his irrational languages. Chances are that in the long run true logic will prevail, and instead of us confusing the computer it will manage instead to organize our grammar into the more efficient tool it should be. With proper programming, the computer may even be able to retain sufficient humor and nuance to make talk interesting and colorful as well as utilitarian.
We can see that the digital machine with its flexibility, accuracy, and powerful logical capability is the fair-haired one of the computer family. Starting with _a_ for abacus, digital computer applications run through practically the entire alphabet. Its take-over in the banking field was practically overnight; it excels as a tool for design and engineering, including the design and engineering of other computers. Aviation relies heavily on digital computers already, from the sale of tickets to the control of air traffic.
Gaming theory is important not only to the Saturday night poker-player and the Las Vegas casino operator, but to military men and industrialists as well. Manufacturing plants rely more and more on digital techniques for controls. Language translation, mentioned lightly above, is a prime need at least until we all begin speaking Esperanto, Io, or Computerese. Taxation, always with us, may at least be more smoothly handled when the computers take over. Insurance, the arrangement of music, spaceflight guidance, and education are random fields already dependent more or less on the digital computer. We will not take the time here to go thoroughly into all the jobs for which the computer has applied for work and been hired; that will be taken up in later chapters. But from even a quick glance the scope of the digital machine already should be obvious. This is why it is usually a safe assumption that the word computer today refers to the digital type.
_Hybrid Computers_
We have talked of the analog and the digital; there remains a further classification that should be covered. It is the result of a marriage of our two basic types, a result naturally hybrid. The analog-digital computer is third in order of importance, but important nonetheless.
[Illustration:
_Minneapolis-Honeywell_
Nerve center of Philadelphia Electric Company’s digital computer-directed automatic economic dispatch system is this console from which power directors operate and supervise loading of generating units at minimum incremental cost. ]
Necessity, as always, mothered the invention of the analog-digital machine. We have talked of the relative merits of the two types; the analog is much faster on a complex problem such as solving simultaneous equations. The digital machine is far more accurate. As an example, the Psychological Matrix Rotator described earlier could solve its twelve equations practically instantaneously. A digital machine might take seconds—a terribly long time by computer standards. If we want an accurate high-speed differential analyzer, we must combine an analog with a digital computer.
Because the two are hardly of the same species, this breeding is not an easy thing. But by careful study, designers effected the desired mating. The hybrid is not actually a new type of computer, but two different types tied together and made compatible by suitable converters.
The composite consists of a high-speed general-purpose digital computer, an electronic analog computer, an analog-to-digital converter, a digital-to-analog and a suitable control for these two converters. The converters are called “transducers” and have the ability of changing the continuous analog signal into discrete pulses of energy, or vice versa.
Sometimes called digital differential analyzers, the hybrid computers feature the ease of programming of the analog, plus its speed, and the accuracy and much broader range of the digital machine. Bendix among others produced such machines several years ago. The National Bureau of Standards recently began development of what it calls an analog-digital differential analyzer which it expects to be from ten to a hundred times more accurate than earlier hybrid computers. The NBS analyzer will be useful in missile and aircraft design work.
Despite its apparent usefulness as a compromise and happy medium between the two types, the hybrid would seem to have as limited a future as any hybrid does. Pure digital techniques may be developed that will be more efficient than the stopgap combination, and the analog-digital will fall by the wayside along the computer trail.
_Summary_
Historically, the digital computer was first on the scene. The analog came along, and for a time was the more popular for a variety of reasons. One of these was the naïve, cumbersome mode of operation the digital computer is bound to; another its early lack of speed. Both these drawbacks have been largely eliminated by advances in electronics, and apparently this is only the beginning. In a few years the technology has progressed from standard-size vacuum tubes through miniature tubes and the shrinking of other components, to semiconductors and other tinier devices, and now we have something called integrated circuitry, with molecular electronics on the horizon. These new methods promise computer elements approaching the size of the neurons in our own brains, yet with far faster speed of operation.
Such advances help the digital computer more than the analog, barring some unexpected breakthrough in the accuracy problem of the latter. Digital building blocks become ever smaller, faster, cheaper, and more reliable. Computers that fit in the palm of the hand are on the market, and are already bulky by comparison with those in the laboratory. The analog-digital hybrid most likely will not be new life for the analog, but an assimilating of its better qualities by the digital.
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“‘_What’s one and one and one and one and one and one and one and one and one and one?_’
‘_I don’t know,’ said Alice. ‘I lost count._’
‘_She can’t do Addition,’ the Red Queen interrupted._”
—Lewis Carroll
5: The Binary Boolean Bit
In this world full of “bigness,” in which astronomical numbers apply not only to the speed of light and the distance to stars but to our national debt as well, it is refreshing to recall that some lucky tribes have a mathematical system that goes, “One—two—plenty!” Such an uncluttered life at times seems highly desirable, and we can only envy those who lump all numbers from three to billions as simply “plenty.”
Instead we are faced today with about as many different number systems as there are numbers, having come a long way from the dawn of counting when an even simpler method than “one—two—plenty” prevailed. Man being basically self-centered, he first thought in terms of “me,” or one. Two was not a concept, but two “ones”; likewise, three “ones” and so on. Pebbles were handy, and to represent the ten animals slain during the winter, a cave man could make ten scratches on the wall or string out that many stones.
It is said that the ancient cabbies in Rome had a taximeter that dropped pebbles one by one onto a plate as the wheels turned the requisite number of revolutions. This plate of stones was presented to the passenger at the end of his ride—perhaps where we get the word “fare”! Prices have risen so much that it would take quite a bag of pebbles in the taximeter today.
Using units in this manner to express a sum is called the unitary system. It is the concept that gives rise to the “if all the dollars spent in this country since such and such a time were laid end to end—” analogies. Put to practice, this might indeed have a salutary effect, but long ago man learned that it was not practical to stick to a one-for-one representation.
How long it was before we stumbled onto the fact that we had a “handy” counting system attached to our wrists is not positively known, but we eventually adopted the decimal system. In some places the jump from one to ten was not made completely. The Pueblo Indians, for instance, double up one fist each time a sum of five is reached. Thus the doubled fist and two fingers on the other hand signifies seven. In the mathematician’s language, this is a modulo-5 system. The decimal system is modulo-10; in other words we start over each time after reaching 10.
Besides the word digit in our vocabulary to tie fingers and numbers, the Roman numerals V and X are graphic representations of one hand with thumb widespread, and two hands crossed, respectively. A point worth remembering is that the decimal system was chosen arbitrarily because we happen to have ten digits. There is no divine arithmetical significance in the number 10; in fact mathematicians would prefer 12, since it can be divided more ways.
The ancient Mayans, feeling that if 10 were ten times as good as 1, then surely 20 would be twice the improvement of the decimal system. So they pulled off their boots and added toes to fingers for a modulo-20 number system. Their word for 20, then, is the same as that for “the whole man” for very good reason. Other races adopted even larger base systems, the base of 60 being an example.
If we look to natural reasons for the development of number systems, we might decide that the binary, or two-valued system, did not attain much prominence in naïve civilizations because there are so few one-legged, two-toed animals! Only when man built himself a machine uniquely suited to two-valued mathematics did the binary system come into its own.
Numbers are merely conventions, rigorous conventions to be sure with no semantic vagueness. God did not ordain that we use the decimal system, as evidenced in the large number of other systems that work just fine. Some abacuses use the biquinary system, and there are septal, octal, and sexagesimal systems. We can even express numbers in an ABC or XYZ notation. So a broad choice was available for the computer designer when he began to look about for the most efficient system for his new machine.
Considering only the question of a radix, or base, which will permit the fewest elements to represent the desired numbers, mathematicians can show us that a base of not 10, or 12, or any other whole number is most efficient, but the fraction 2.71828. The ideal model is not found in man, then, since man does not seem to have 2.71828 of anything. However, the strange-looking number does happen to be the base of the system of natural logarithms.
Now a system of mathematics based on 2.71828 might make the most efficient use of the components of the computer, but it would play hob with other factors, including the men who must work with such a weird set of numbers. As is often done, a compromise was made between ideal and practical choices. Since the computer with the most potential seems to be the electronic computer, and since its operation hinges on the opening and closing of simple or sophisticated switches, a two-valued mathematical system, the binary system, was chosen. It wasn’t far from the ideal 2.71828, and there was another even more powerful reason for the choice. Logic is based on a yes-no, true-false system. Here, then, was the best of all possible number systems: the lowly, apparently far-from-sophisticated binary notation. As one writer exclaimed sadly, a concept which had been hailed as a monument to monotheism ended up in the bowels of a robot!
_The Binary System_
It is believed from ancient writings that the Chinese were aware of the binary or two-valued system of numbers as early as 3000 B.C. However, this fact was lost as the years progressed, and Leibnitz thought that he had discovered binary himself almost 5,000 years later. In an odd twist, Leibnitz apprised his friend Grimaldi, the Jesuit president of the Tribunal of Mathematics in China, of the religious significance of binary 1 and 0 as an argument against Buddhism!
A legend in India also contains indications of the power of the binary system. The inventor of the game of chess was promised any award he wanted for this service to the king. The inventor asked simply that the king place a grain of wheat on the first square of the board, two on the second, and then four, eight, and so on in ascending powers of two until the sixty-four squares of the board were covered. Although the king thought his subject a fool, this amount of wheat would have covered the entire earth to a depth of about an inch!
We are perhaps more familiar with the binary system than we realize. Morse code, with its dots and dashes, for example, is a two-valued system. And the power of a system with a base of two is evident when we realize that given a single one-pound weight and sufficient two-pound weights we can weigh _any_ whole-numbered amounts.
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.
[Illustration:
_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.