We now take a stored program for granted; what present-day all-digital computers do, step-by-step, is determined by a program, or sequence of instructions, written in some language that is convenient for the programmer, but translated into a more basic language that is directly interpretable by the machine’s hardware – the interconnected gates described earlier. In all-digital machines it is thus easy to distinguish between the control, which appears as code of one kind or another, and the computation itself, which is directed by that code takes place in the special little chip called the central processing unit (CPU).
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Steiglitz sketches the conceptual and engineering breakthroughs that brought us into the age of digital computers. It’s relatively non-technical, which touching on some rather advanced ideas in hardware design and information theory. There are six main landmarks on the road to digital computers outlined.
First, you need physical devices that can detect, and send, and repair weak or noisy signals.
Second, you need to stop or send signals (bits) in a controlled, reliable, fast, and small manner, which transistors are capable of doing. This allow you to conduct small logical operations and to store memory (both of these with extremely small error rates). You can then layer these tiny operations and memories into complex programs.
Third, to allow for more and more layers of these logical operations, you need smaller, faster, and cheaper transistors and other hardware, and Moore’s Law has seen a predictable growth in the capacities of chips throughout time which allows for faster, more complex programs.
Fourth, you need a way to convert continuous, analog signals into discrete, digital signals. Nyquist-Shannon sampling theorem shows the possibilities and bounds for this process. It’s possible. Essentially any analog information can be converted to digital, and vice versa. This is how music, sound waves, can be turned into bits, and back into sound waves.
Fifth, we need a way to stop noise from ruining our signals (calls getting too fuzzy, video getting too blurry, etc.). Shannon’s noisy coding theorem shows that sufficient bandwidth allows us to send essentially noise-free communication, which is how cell phone calls across the globe are possible.
Sixth, we needed a way to put all of this together into a deterministic machine that operates under known, predictable rules and can reprogrammed to be put to various uses, and the Turing Machine expressed this idea.
Ideas per Page: Quite high. Each of the six topics outlined above are entire fields of study
Related Books: ?
Recommend to Others: Only if interested in topic. Probably too dry for vast majority of people
Reread Personally: No
Quotes:
… Voltages above the halfway point of 2.5 V are pushed toward 5 goals and those below 2.5040 V… For an error to occur, there must be some point in a circuit where the noise is larger than 2.5 V, and in the usual electronic circuits, where the average noise excursion might be measured in millions of old, the chances of this happening are infinitesimally small. 23
Digital circuits are made from analog parts 24
For example, an AND gate has 2 inputs and one output; the output is ON when and only when both inputs are ON. 24
… Let us consider how vacuum tube can manipulate information in binary form… The flow of current is controlled by the grid voltage, just as the flow of water through a faucet is controlled by the handle. 30, 31
The next step is to put together 3 kinds of gates that represent 3 fundamental operations of logic: the not date, and gate, and or gate. The not gate has one input and one output, the and and or gates have 2 inputs and one output. 33
The chips on the computer I’m typing have a clock speed of 3.4 GHz, which means the gates on the chip determines their outputs from their inputs 3.4 billion times a second. 34
If you look at the doorbell as a logical device, when it’s ON (circuit contact in the closed position), it moves to the OFF position (opening the contact) and vice versa. This is the physical implementation of a logical paradox: ON implied OFF and OFF implies ON.… What happens if we do this is that the gate alternates between the output being ON and OFF, the period of oscillation being determined by the time it takes for signals round-trip from input to output input. The physical manifestation of a contradiction is thus perpetual vacillation, or oscillation, which, in fact, can be used as a measure of time to “clock” computer logic. 35
[to create memory in a computer] In this case connect two NOT gates in tandem and return the output of the second NOT gate to the input of the first. It’s now easy to see that this double NOT gate has two stable, consistent states: the first gate can be ON, the second OFF, or vice versa. In each case the second gate output returns a value to the first gate input that is consistent with its output. And the two-gate pair will maintain its state until it is forced into the opposite state. 35
The p-n junction encapsulates the magic of semiconductors. To make a vacuum-tube diode, we need to provide a vacuum where electrons can move freely (in one direction only), and a hot source of electrons. Now we have a way of creating a diode in a solid material without a hot filament. This is the point: we have replaced vacuum-tube technology with solid-state electronics. 58
A “microdot” is a microphotograph the size of the typographic dot, such as the one that and this sentence. Microdots can be sprinkled throughout the letter, for example, in place of ordinary printed dots, so that information can be transmitted undetected by unknowing eyes. They can then be read with the microscope by a recipient who knows where to look. Aficionados of the spy-novel genre know this is a standard trick, in every good spy’s repertoire. 63
If I flip a coin once, and I don’t tell you the result, I introduce a certain amount of uncertainty in your head. If I then tell you the result, I remove that uncertainty. We say I have given you some information. It is a simple, but important, insight that information is the removal of uncertainty. How much information have I given you? In the simple case of flipping a fair coin, this is easy: we say that the information in the result of flipping a fair coin is one bit. If we flip the coin twice, we said information about the result is two bits—provided that the second flip is not affected in any way by the first flip. 91-2
Some codes are designed to allow just the detection of errors, with no thought of correcting them. We can accomplish this with the simple and well-known device of a parity bit. Suppose we are sending blocks of three bits. We can add a fourth bit to each block that makes the total number of ones even (say). If we then received a block of four bits with an odd number of ones, we know that an error in transmission must have occurred. We would under the circumstance of no idea which of the bits is in error or, in fact, whether or not three errors might have occurred. The best we can do is discard the block, and, if we can, ask for a retransmission.
This scheme is effective for small blocks, but the block length gets long, it becomes more and more likely that an even number of errors will occur, such events will escape detection. The size of the channel error probability, E, therefore limits how long we can make the blocks. But shorter blocks mean that we are sending a larger fraction of bits to check parity, and this results in a slower overall retransmission. 99
Now suppose we make two gears that mesh, one with 19 teeth, and another with 235 teeth. If the gears are engaged and one is cranked, the shaft of the 19-tooth gear will turn 235 times for every 19 terms of the 235-tooth gear. Thus, we can consider revolutions of the 19-tooth gear as counting months (moon around the earth) and revolutions of the 235-tooth gear as counting years (earth around the sun). With this we have built an analog computer that nears the motions of the sun and moon, and in this way displays their positions in orbit and the relative phases. 108
At some point in your exploration, at the bottom of the valley, you might find yourself quite convinced that you have reached the ultimate lowest point. Everywhere you look, the ground is higher. But of course, there might be another valley, beyond your range of vision, that is lower. You have no way of discovering that without leaving your valley, which might be an excursion that is too risky or time-consuming.
This situation comes up all the time when computer scientists study problem-solving techniques; in general, more information about the problem itself is needed to guarantee that particular valley is actually the lowest possible. The terminology for this phenomenon is that the short-cited best is a local minimum as opposed to a global minimum.
In the same way, the physical principle of least surface tension can only find improvements of the solution that are in its neighborhood, reachable by incremental variations the current solution. The soap-film analog computer finds only local minima, a fact that is confirmed by damp experimentation. 123
We now take a stored program for granted; what present-day all-digital computers do, step-by-step, is determined by a program, or sequence of instructions, written in some language that is convenient for the programmer, but translated into a more basic language that is directly interpretable by the machine’s hardware – the interconnected gates described earlier. In all-digital machines it is thus easy to distinguish between the control, which appears as code of one kind or another, and the computation itself, which is directed by that code takes place in the special little chip called the central processing unit (CPU). But how does this distinction between control and computation work out for machines that are all or partly analog? We illustrate some answers with examples in the following section, starting with the opposite of the digital computer, the all-analog machine. 132-3
Paraphrasing: there are 6 main things that have occurred in the shift from analog to digital computation: standardizing and restoring signals; making valves, which allow us to do small logical operations and store memory with a low error rate; Moore’s Law, which is allowed for greater chip density and speed; Nyquist sampling principle, which shows that we can convert analog processes into digital representations; Sharon’s noisy coding theorem, which shows that we can achieve noise free digital communication if we have sufficient bandwidth; the turn machine which is just the idea of determinate programs run on digital machines 186
The Digital Sauna 