Computers as Universal Computational Devices

It may seem strange that an introductory textbook begins by describing how com-puters work. After all, mechanical engineering students begin by studying physics, not how car engines work. Chemical engineering students begin by studying chemistry, not oil refineries. Why should computing students begin by studying computers?

The answer is that computers are different. To learn the fundamental prin-ciples of computing, you must study computers or machines that can do what

computers can do. The reason for this has to do with the notion that computers are universal computational devices. Let's see what that means.

Before modern computers, there were many kinds of calculating machines.

Some were analog machines—machines that produced an answer by measuring some physical quantity such as distance or voltage. For example, a slide rule is an analog machine that multiplies numbers by sliding one logarithmically graded ruler next to another. The user can read a logarithmic "distance" on the second ruler. Some early analog adding machines worked by dropping weights on a scale.

The difficulty with analog machines is that it is very hard to increase their accuracy.

This is why digital machines—machines that perform computations by manipulating a fixed finite set of digits or letters—came to dominate comput-ing. You are familiar with the distinction between analog and digital watches. An analog watch has hour and minute hands, and perhaps a second hand. It gives the time by the positions of its hands, which are really angular measures. Digital watches give the time in digits. You can increase accuracy just by adding more digits. For example, if it is important for you to measure time in hundredths of a second, you can buy a watch that gives a reading like 10:35.16 rather than just 10:35. How would you get an analog watch that would give you an accurate read-ing to one one-hundredth of a second? You could do it, but it would take a mighty long second hand! When we talk about computers in this book, we will always mean digital machines.

Before modern digital computers, the most common digital machines in the West were adding machines. In other parts of the world another digital machine, the abacus, was common. Digital adding machines were mechanical or elec-tromechanical devices that could perform a specific kind of computation: adding integers. There were also digital machines that could multiply integers. There were digital machines that could put a stack of cards with punched names in alphabetical order. The main limitation of all of these machines is that they could do only one specific kind of computation. If you owned only an adding machine and wanted to multiply two integers, you had some pencil and paper work to do.

This is why computers are different. You can tell a computer how to add numbers. You can tell it how to multiply. You can tell it how to alphabetize a list or perform any computation you like. When you think of a new kind of computation, you do not have to buy or design a new computer. You just give the old computer a new set of instructions (or program) to carry out the computation. This is why we say the computer is a universal computational device. Computer scientists believe that anything that can be computed, can be computed by a computer provided it has enough time and enough memory. When we study computers, we study the fundamentals of all computing. We learn what computation is and what can be computed.

The idea of a universal computational device is due to Alan Turing. Tur-ing proposed in 1937 that all computations could be carried out by a particular kind of machine, which is now called a Turing machine. He gave a mathemat-ical description of this kind of machine, but did not actually build one. Digital computers were not operating until 1946. Turing was more interested in solving a philosophical problem: defining computation. He began by looking at the kinds of actions that people perform when they compute; these include making marks

1.6 Computers as Universal Computational Devices

on paper, writing symbols according to certain rules when other symbols are present, and so on. He abstracted these actions and specified a mechanism that could carry them out. He gave some examples of the kinds of things that these machines could do. One Turing machine could add two integers; another could multiply two integers.

Figure 1.4 provides what we call "black box" models of Turing machines that add and multiply. In each case, the operation to be performed is described in the box. The data on which to operate is shown as input to the box. The result of the operation is shown as output from the box. A black box model provides no information as to exactly how the operation is performed, and indeed, there are many ways to add or multiply two numbers.

Turing proposed that every computation can be performed by some Turing machine. We call this Turing's thesis. Although Turing's thesis has never been proved, there does exist a lot of evidence to suggest it is true. We know, for example, that various enhancements one can make to Turing machines do not result in machines that can compute more.

Perhaps the best argument to support Turing's thesis was provided by Turing himself in his original paper. He said that one way to try to construct a machine more powerful than any particular Turing machine was to make a machine U that could simulate all Turing machines. You would simply describe to U the particular Turing machine you wanted it to simulate, say a machine to add two integers, give U the input data, and U would compute the appropriate output, in this case the sum of the inputs. Turing then showed that there was, in fact, a Turing machine that could do this, so even this attempt to find something that could not be computed by Turing machines failed.

Figure 1.5 further illustrates the point. Suppose you wanted to compute g - + / ) • You would simply provide to U descriptions of the Turing machines to add and to multiply, and the three inputs, e, / , and g. U would do the rest.

In specifying U, Turing had provided us with a deep insight: He had given us the first description of what computers do. In fact, both a computer (with as much

^ADD> 7*MUL

Figure 1.5 Black box model of a universal Turing machine

memory as it wants) and a universal Turing machine can compute exactly the same things. In both cases you give the machine a description of a computation and the data it needs, and the machine computes the appropriate answer. Computers and universal Turing machines can compute anything that can be computed because they are programmable.

This is the reason that a big or expensive computer cannot do more than a small, cheap computer. More money may buy you a faster computer, a monitor with higher resolution, or a nice sound system. But if you have a small, cheap computer, you already have a universal computational device.