The following long quote is from David Marr (1982), Vision, pp. 19 to 24.
"UNDERSTANDING COMPLEX INFORMATION PROCESSING SYSTEMS
A representation is a formal system for making explicit certain entities or types of information, together with a specification of how the system does this. And I shall call the result of using a representation to describe a given entity a description of the entity in that representation (Marr and Nishihara, 1978).
For example, the Arabic, Roman, and binary numeral systems are all formal systems for representing numbers. The Arabic representation consists of a string of symbols drawn from the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the rule for constructing the description of a particular integer n is that one decomposes n into a sum of multiples of powers of 10 and unites these multiples into a string with the largest powers of 10 and unites these multiples into a string with the largest powers on the left and the smallest on the right. Thus, thirty-seven equals 3 x 10(1) + 7 x 10(0), which becomes 37, the Arabic numeral system's description of the number. What this description makes explicit is the number's decomposition into powers of 10. The binary numeral system's description of the number 37 is 100101, and this description makes explicit the number's decomposition into powers of 2. In the Roman numeral system, thirty-seven is represented as XXXVII.
This definition of a representation is quite general. For example, a representation for shape would be a formal scheme for describing some aspects of shape, together with rules that specify how the scheme is applied to any particular shape. A musical score provides a way of representing a symphony; the alphabet allows the construction of a written representation of words; and so forth. The phrase "formal scheme" is critical to the definition, but the reader should not be frightened by it. The reason is simply that we are dealing with the information-processing machines, and the way such machines work is by using symbols so stand for things--to represent things, in our terminology. To say that something is a formal scheme means only that it is a set of symbols with rules for putting them together--no more and no less.
A representation, therefore, is not a foreign idea at all--we all use representations all the time. However, the notion that one can capture some aspect of reality by making a description of it using a symbol and that to do so can be useful seems to me a fascinating and powerful idea. But even the simple examples we have discussed introduce some rather general and important issues that arise whenever on chooses to use one particular representation. For example, if one chooses the Arabic numeral representation, it is easy to discover whether a number is a power of 10 but difficult to discover whether it is a power of 2. If one chooses the binary representation, the situation is reversed. Thus, there is a trade-off; any particular representation makes certain information explicit at the expense of information that is pushed into the background and may be quite hard to recover.
This issue is important, because how information is represented can greatly affect how easily it is to do different things with it. This is evident even from our numbers example: It is easy to add, to subtract, and even to multiply if the Arabic or binary representations are used, but it is not at all easy to do these things--especially multiplication--with Roman numerals. This is a key reason why the Roman culture failed to develop mathematics in the way the earlier Arabic cultures had.
An analogous problem faces computer engineers today. Electronic technology is much more suited to a binary number system than to the conventional base 10 system, yet humans supply their data and require the results in base 10. The design decision facing the engineer, therefore, is should one pay the cost of conversion into base 2, carry out the arithmetic in a binary representation, and then convert back into decimal numbers on output; or should one sacrifice efficiency of circuitry to carry out the operations directly in a decimal representation? On the whole, business computers and pocket calculators take the second approach, and general purpose computers take the first. But even though one is not restricted to using just one representation system for a given type of information, the choice of which to use is important and cannot be taken lightly. It determines what information is made explicit and hence what is pushed further into the background, and it has a far-reaching effect on the ease and difficulty with which operations may subsequently be carried out on that information.
Process:
The term process is very broad. For example, addition is a process, and so is taking a Fourier transform. But so is making a cup of tea, or going shopping. For the purposes of this book, I want to restrict our attention to the meanings associated with machines that are carrying out information-processing tasks. So let us examine in depth the notions behind one simple such device, a cash register at the checkout counter of a supermarket.
There are several levels at which one needs to understand such a device, and it is perhaps most useful to thinks in terms of three of them. The most abstract is the level of what the device does and why. What it does is arithmetic, so our first task is to master the theory of addition. Addition is a mapping, usually denoted by +, from pairs of numbers into single numbers; for example, + maps the pair (3, 4) to 7, and I shall write this in the form (3 + 4) --> 7. Addition has a number of abstract properties, however. It is commutative: both (3 + 4) and (4 + 3) are equal to 7; and associative: the sum of 3 + (4 + 5) is the same as (3 + 4) + 5. Then there is the unique distinguished element, zero, the adding of which has no effect: (4 + 0) --> 4. Also, for every number there is a unique "inverse", written (-4) in the case of 4, which when added to the number zero gives: [4 + (-4)] --> 0.
Notice that these properties are part of the fundamental theory of addition. They are true no matter how the numbers are written--whether in binary, Arabic, or Roman representation--and no matter how the addition is executed. Thus part of this first level is something that might be characterized as what is being computed.
The other half of this level of explanation has to do with the question of why the cash register performs addition and not, for instance, multiplication when combining the prices of the purchased items to arrive at the final bill. The reason is that the rules we intuitively feel to be appropriate for combining the individual prices in fact define the mathematical operation of addition. These can be formulated as constraints in the following way:
It is a mathematical theorem that these conditions define the operation of addition, which is therefore the appropriate computation to use.
This whole argument is what I call the computational theory of the cash register. Its important features are (1) that it contains separate arguments about what is computed and why and (2) that the resulting operation is defined uniquely by the constraints it has to satisfy. In the theory of visual processes, the underlying task is to reliably derive properties of the world from images of it; the business of isolating constraints that are both powerful enough to allow a purpose to be defined and generally true of the world is a central theme of our inquiry.
In order that a process shall actually run, however, one has to realize it in some way and therefore choose a representation for the entities that the process manipulates. The second level of the analysis of a process, therefore, involves choosing two things: (1) a representation for the input and for the output of the process and (2) and algorithm by which the transformation may actually be accomplished. For addition, of course, the input and output representations can both be the same , because they both consist of numbers. However this is not true in general. In the case of a Fourier transform, for example, the input representation may be the time domain, and the output, the frequency domain. If the first of our levels specifies what and why, this second level specifies how. For addition, we might choose Arabic numerals for the representations, and for the algorithm we could follow the usual rules about adding the least significant digits first and "carrying" if the sum exceeds 9. Cash registers, whether mechanical or electronic, usually use this type of representation and algorithm.
There are three important points here. First, there is usually a wide choice of representation. Second, the choice of algorithm often depends rather critically on the particular representation that is employed. And third, even for a given fixed representation, there are often several possible algorithms for carrying out the same process. Which one is chosen will usually depend on any particularly desirable or undesirable characteristics that the algorithms may have; for example, one algorithm may be much more efficient than another, or may be slightly less efficient but more robust (that is, less sensitive to slight inaccuracies in the data on which it must run). Or again, one algorithm may be parallel, and another, serial. The choice, then, may depend on the type of hardware or machinery in which the algorithm is to be embodied physically.
This brings us to the third level, that of the device in which the process is to be realized physically. The important point here is that, once again, the same algorithm may be implemented in quite different technologies. The child who methodically adds two numbers from left to right, carrying a digit when necessary, may be using the same algorithm that is implemented by wires and transistors of the cash register in the neighborhood supermarket, but the physical realization of the algorithm is quite different in these two cases. Another example: Many people have written computer programs to play tic-tac-toe, and there is more or less a standard algorithm that can not lose. This algorithm has in fact been implemented by W.D. Hillis and B. Silverman in quite different technology, in a computer made out of tinker toys, a children's wooden building set. The whole monstrously ungainly engine, which actually works, currently resides in a museum at the University of Missouri in St. Louis.
Some styles of algorithm will suit some physical substrates better than others. For example, in conventional digital computers, the number of connections is comparable to the number of gates, while in a brain, the number of connections is much larger ( x 104 ) than the number of nerve cells. The underlying reason is that wires are rather cheap in biological architecture, because they can grow individually and in three dimensions. In conventional technology, wire laying is more or less restricted to two dimensions, which quite severely restricts the scope for using parallel techniques and algorithms; the same operations are often better carried out serially."