22C:096
Computation, Information, and Description

Department of Computer Science
The University of Iowa

Lecture Notes




Last modified: 14 January 1997


Foundations: discrete symbols carrying information

Begin at the beginning, ...
This seemingly simple advice from the King in Alice's Adventures in Wonderland is remarkably difficult to follow. There are lots of different beginnings: historical, ontological, teleological, taxonomical, psychological, .... For this course, there are beginnings in the Theory of Computation, in the development of computing technology, in the philosophical literature that uses computational concepts, etc. What is the right beginning?

In this course, we will explore the impact of technical ideas that are classified as Theory of Computation on a number of topics outside of computer science. The title is taken from the fact that we will provide particular insight into the intuitive meanings of the concepts, computation, information, and description. But, right now I want to start at the ontological beginning of the material, or as close to it as I can get.

Ontological roots: discrete collections of symbols

The essential roots of all the material in this course lie in certain uses of discrete collections of symbols. For our purposes, a "symbol" is something that we intend to use, either by itself or in combination with other symbols, to represent something else. A collection of symbols is "discrete" if each symbol is clearly distinguishable from each other. The letters of the Roman alphabet form a discrete set of symbols. The visible colors might be used as symbols, but they are not discrete, since they shade continuously into one another. On the other hand, a specific selected set of distinguishable colors, such as red, green, and blue, may be used as discrete symbols. From now on, when we mention "symbols," we assume that they are discrete unless we state otherwise.

In particular, we will study the use of discrete symbols to convey information, and the manipulation of such symbols in computations. You might think that we now must learn the details of particular ways of encoding information into symbols, such as the English language. Not so. We can fill up our semester studying the remarkably rich insights into the structure of symbol manipulations that derive merely from the fact that the symbols are used to convey information, without considering the content of that information.

Given that we will use symbols to convey information, we are at liberty to choose symbols that are particularly easy to manipulate for such purposes. This is true because, information has a sort of transitive property. Suppose that we already have a method of conveying information using a particular 10-ton granite rock, the water in the Pacific Ocean, and the Eiffel tower as our symbols. We encode information into the particular west-to-east order in which we arrange these symbols. With such a language, we probably won't say very much, since the cost of expressing ourselves is immense. But, we may choose three other symbols, such as the letters A, B, and C, and agree that A refers to the rock, B to the water in the Pacific Ocean, and C to the Eiffel tower. We may further agree that, when these letters are written on a flat surface, their left-to-right order represents an intended west-to-east order for the rock, water, and tower. Now, whatever information could be conveyed through the rock/water/tower language may be conveyed through the A/B/C language with a lot less work.

Restating the conclusion of the previous paragraph, if a symbol refers to another symbol, and the latter symbol refers to a thing (which might or might not be a symbol itself), then we may use the first symbol to refer to the final thing. This sort of connection between two relations is sometimes called a "transitive" property. The transitive property of reference allows us to replace one set of symbols with another set that is more convenient to manipulate. We will look more carefully at the precise requirements for such replacements of symbols later.

Given a lot of freedom to choose sets of symbols, we typically choose those that are the easiest to use for our particular purposes. When we wish to communicate instantaneously with other people, using only the resources of our own bodies, we usually choose spoken symbols, such as the phonemes of our spoken languages, or body gestures, such as those used in a sign language. When we wish to communicate with many people at a distance, including some people who will be ready to receive our messages in the distant future, we often choose a printed alphabet. When we wish to process information rapidly and automatically, we usually choose electrical voltages as symbols. When we wish to transmit large amounts of information rapidly between electronic computers, we may choose electrical voltages or pulses of light as symbols. Mother nature has chosen a particular set of organic chemicals as the symbols in which life reproduces and evolves itself. The details of our choices of symbols depend on the resources available for manipulating the symbols, including sending/receiving messages and computing, and on the particular requirements of speed, quantity, transmission over a distance, preservation over time, reliability, etc.

Four particular properties of symbols are almost always present when symbols are chosen in order to convey information: the symbols are easy to

Depending on the details, and on our point of view, we might regard creation as a special case of copying from some standard template. We might regard copying and erasing as special cases of rearranging, where an unlimited supply of copies of symbols may be moved between a region where their meanings are interpreted and a region where they are hidden or ignored.

The exception proves the rule: money

This much-abused phrase is often quoted as an excuse for inaccuracy, or even a glorification of error. In the origins of the phrase, "prove" did not mean "demonstrate correct," as it typically does in mathematical discourse today. Rather it meant "test." "The proof of the pudding is in the eating," and "180 proof alcohol" both use "proof" to mean "test." Exceptions test rules by challenging their consequences. Rules pass the exception test when they are seen to be valid, or in some cases just useful, in spite of the challenge.

A dollar bill is a symbol. It represents a certain amount of value, and may be traded for that value in a wide variety of different goods, from bubble gum to surgical services. Dollar bills are easy to rearrange in one's wallet, and particularly easy to rearrange out of one's wallet. But, they have been carefully constructed to be difficult to create and copy, at least by anyone other than the Bureau of Engraving and Printing. We may erase dollar bills fairly easily, but it is illegal to do so deliberately, and they have been constructed to make accidental erasure unlikely. Why are money symbols chosen in violation of the principles for choosing symbols to convey information? Because money conveys more than information: it conveys an incentive to provide value as well. That incentive does not usually survive free creation and copying.

Owners of information who wish to sell it are in a difficult position. Information is most useful when it is conveyed by symbols that are easy to create, rearrange, copy, and erase. But, the ease of copying makes it hard to exchange information for money. Ease of rearrangement makes it difficult to judge the performance of contracts, to satsify warranties, and to distinguish the true owner of an arrangement of symbols. So, some information vendors go to great lengths to engineer symbols that are hard to copy and rearrange. Others find ways to earn money from the timely delivery of information, or custom presentation of information, instead of the information itself. This mismatch between monetary symbols and informative symbols presents a fundamental dilemma to productive commerce in the information age. The requirements for serving society through the creation and distribution of information appear to conflict radically with the requirements for making a buck from such services. The Free Software Foundation has an interesting ideology regarding information commerce, expressed in the famous GNU Manifesto.

Teleology strikes back

Although I tried above to trace the ontological roots of my topic, I found that those roots lie in the concept of discrete collections of symbols used to convey information. A thing becomes a symbol, not by what it is, but by the way in which we intend to use it. The phrase, "used to convey information" refers very clearly to intended use, rather than inherent essence. Even the concept of discreteness has to do with our ability to distinguish things effectively, as much as with the inherent nature of the things that we distinguish. So, the ontological roots of computation, description, and information appear to be teleological in nature. That's OK.

Essentially all of the concepts that we study in this course will have teleological roots: they will distinguish things by their use more than by their essence. There is nothing wrong with such concepts, but we need to pay close attention to the difference between essential qualities and teleological ones, particularly because so much of the rhetoric of science and even of philosophy emphasizes a search for the inherent essence of things, rather than the structure that comes from intention and use. Paradoxically (but not antinomially), we will find that some of these teleologically based concepts appear to be among the most stable and objective concepts available to our intellects.


Maintained by Michael J. O'Donnell, email: [] odonnell@cs.uiowa.edu