Gödel, Escher, Bach: An Eternal Golden Braid (59 page)

[ATTACCA]

Levels of Description,

and Computer Systems

Levels of Description

GODEL'S STRING G, and a Bach fugue: they both have the property that they can be understood on different levels. We are all familiar with this kind of thing; and yet in some cases it confuses us, while in others w handle it without any difficulty at all. For example, we all know that w human beings are composed of an enormous number of cells (around twenty-five trillion), and therefore that everything we do could in principle be described in terms of cells. Or it could even be described on the level c molecules. Most of us accept this in a rather matter-of-fact way; we go t the doctor, who looks at us on lower levels than we think of ourselves. W read about DNA and "genetic engineering" and sip our coffee. We seem t have reconciled these two inconceivably different pictures of ourselves simply by disconnecting them from each other. We have almost no way t relate a microscopic description of ourselves to that which we feel ourselves to be, and hence it is possible to store separate representations of ourselves in quite separate "compartments"

of our minds. Seldom do we have to fir back and forth between these two concepts of ourselves, wondering "How can these two totally different things be the same me?"

Or take a sequence of images on a television screen which show Shirley MacLaine laughing. When we watch that sequence, we know that we are actually looking not at a woman, but at sets of flickering dots on a flat surface. We know it, but it is the furthest thing from our mind. We have these two wildly opposing representations of what is on the screen, but that does not confuse us. We can just shut one out, and pay attention to th other-which is what all of us do. Which one is "more real"? It depends o; whether you're a human, a dog, a computer, or a television set.

Chunking and Chess Skill

One of the major problems of Artificial Intelligence research is to figure out how to bridge the gap between these two descriptions; how to construe a system which can accept one level of description, and produce the other One way in which this gap enters Artificial Intelligence is well illustrated b the progress in knowledge about how to program a computer to play goof chess. It used to be thought in the 1950's and on into the 1960's-that the

trick to making a machine play well was to make the machine look further ahead into the branching network of possible sequences of play than any chess master can. However, as this goal gradually became attained, the level of computer chess did not have any sudden spurt, and surpass human experts. In fact, a human expert can quite soundly and confidently trounce the best chess programs of this day.

The reason for this had actually been in print for many years. In the 1940's, the Dutch psychologist Adriaan de Groot made studies of how chess novices and chess masters perceive a chess situation. Put in their starkest terms, his results imply that chess masters perceive the distribution of pieces in
chunks
. There is a higher-level description of the board than the straightforward "white pawn on K5, black rook on Q6" type of description, and the master somehow produces such a mental image of the board. This was proven by the high speed with which a master could reproduce an actual position taken from a game, compared with the novice's plodding reconstruction of the position, after both of them had had five-second glances at the board. Highly revealing was the fact that masters' mistakes involved placing whole
groups
of pieces in the wrong place, which left the game strategically almost the same, but to a novice's eyes, not at all the same. The clincher was to do the same experiment but with pieces randomly assigned to the squares on the board, instead of copied from actual games. The masters were found to be simply no better than the novices in reconstructing such random boards.

The conclusion is that in normal chess play, certain types of situation recur-certain patterns-and it is to those high-level patterns that the master is sensitive. He thinks
on a different level
from the novice; his set of concepts is different. Nearly everyone is surprised to find out that in actual play, a master rarely looks ahead any further than a novice does-and moreover, a master usually examines only a handful of possible moves!

The trick is that his mode of perceiving the board is like a filter: he literally
does not see
bad moves
when he looks at a chess situation-no more than chess amateurs see illegal moves when they look at a chess situation. Anyone who has played even a little chess has organized his perception so that diagonal rook-moves, forward captures by pawns, and so forth, are never brought to mind. Similarly, master-level players have built up higher levels of organization in the way they see the board; consequently, to them, bad moves are as unlikely to come to mind as illegal moves are, to most people. This might be called
implicit pruning
of the giant branching tree of possibilities. By contrast,
explicit pruning
would involve thinking of a move, and after superficial examination, deciding not to pursue examining it any further.

The distinction can apply just as well to other intellectual activities -- for instance, doing mathematics. A gifted mathematician doesn't usually think up and try out all sorts of false pathways to the desired theorem, as less gifted people might do; rather, he just

"smells" the promising paths, and takes them immediately.

Computer chess programs which rely on looking ahead have not been taught to think on a higher level; the strategy has just been to use brute

force look-ahead, hoping to crush all types of opposition. But it h worked. Perhaps someday, a look-ahead program with enough brute ,gill indeed overcome the best human players-but that will be a intellectual gain, compared to the revelation that intelligence de crucially on the ability to create high-level descriptions of complex such as chess boards, television screens, printed pages, or painting

Similar Levels

usually, we are not required to hold more than one level of understanding of a situation in our minds at once. Moreover, the different descriptions a single system are usually so conceptually distant from each other tl was mentioned earlier, there is no problem in maintaining them both are just maintained in separate mental compartments. What is confusing though, is when a single system admits of two or more descriptions different levels which nevertheless
resemble
each other in some way. we find it hard to avoid mixing levels when we think about the system can easily get totally lost.

Undoubtedly this happens when we think about our psychology-for instance, when we try to understand people's motivations: for various actions. There are many levels in the human m structure-certainly it is a system which we do not understand very we But there are hundreds of rival theories which tell why people act the way they do, each theory based on some underlying assumptions about he down in this set of levels various kinds of psychological "forces" are f( Since at this time we use pretty much the same kind of language f mental levels, this makes for much level-mixing and most certain] hundreds of wrong theories. For instance, we talk of "drives"-for se power, for fame, for love, etc., etc.-without knowing where these drives come from in the human mental structure. Without belaboring the pc simply wish to say that our confusion about who we are is certainly r( to the fact that we consist of a large set of levels, and we use overlapping language to describe ourselves on all of those levels.

Computer Systems

There is another place where many levels of description coexist for a system, and where all the levels are conceptually quite close to one an( I am referring to computer systems.

When a computer program is ping, it can be viewed on a number of levels. On each level, the description is given in the language of computer science, which makes all the de descriptions similar in some ways to each other-yet there are extremely imp( differences between the views one gets on the different levels. At the 1 level, the description can be so complicated that it is like the dot-description of a television picture. For some purposes, however, this is by far the important view. At the highest level, the description is greatly
chunked
and

takes on a completely different feel, despite the fact that many of the same concepts appear on the lowest and highest levels. The chunks on the high-level description are like the chess expert's chunks, and like the chunked description of the image on the screen: they summarize in capsule form a number of things which on lower levels are seen as separate. (See Fig. 57.) Now before things become too abstract, let us pass on to the
FIGURE 57. The idea of "chunking": a group of items is reperceived as a single "chunk".

The chunk's boundary is a little like a cell membrane or a national border: it establishes
a separate identity for the cluster within. According to context, one may wish to ignore
the chunk's internal structure or to take it into account.

concrete facts about computers, beginning with a very quick skim of what a computer system is like on the lowest level. The lowest level? Well, not really, for I am not going to talk about elementary particles-but it is the lowest level which we wish to think about.

At the conceptual rock-bottom of a computer, we find a memory, a central processing unit (CPU), and some input-output (I/O) devices. Let us first describe the memory. It is divided up into distinct physical pieces, called words. For the sake of concreteness, let us say there are 65,536 words of memory (a typical number, being 2 to the 16th power). A word is further divided into what we shall consider the atoms of computer science-bits. The number of bits in a typical word might be around thirty-six.

Physically, a bit is just a magnetic "switch" that can be in either of two positions.

--- a word of 36 bits --

you could call the two positions "up" and "down", or "x" and "o", o and "0" ... The third is the usual convention. It is perfectly fine, but i the possibly misleading effect of making people think that a comp deep down, is storing numbers. This is not true. A set of thirty-six bits not have to be thought of as a number any more than two bits has i thought of as the price of an ice cream cone. Just as money can do va things depending on how you use it, so a word in memory can serve r functions. Sometimes, to be sure, those thirty-six bits will indeed repn a number in binary notation. Other times, they may represent thin dots on a television screen. And other times, they may represent a letters of text. How a word in memory is to be thought of depends eni on the role that this word plays in the program which uses it. It ma course, play more than one role-like a note in a canon.

Instructions and Data

There is one interpretation of a word which I haven't yet mentioned, that is as an instruction. The words of memory contain not only data t acted on, but also the program to act on the data. There exists a lin repertoire of operations which can be carried out by the central proce5 unit-the CPU-and part of a word, usually its first several bits-is it pretable as the name of the instruction-type which is to be carried What do the rest of the bits in a word-interpreted-as-instruction stand Most often, they tell which other words in memory are to be acted upoi other words, the remaining bits constitute a pointer to some other wor( words) in memory. Every word in memory has a distinct location, li house on a street; and its location is called its address. Memory may have "street", or many