Gödel, Escher, Bach: An Eternal Golden Braid (111 page)

The I-Mode and the M-Mode Again

An intelligent program would presumably be one which is versatile enough to solve problems of many different sorts. It would learn to do each different one and would accumulate experience in doing so. It would be able to work within a set of rules and yet also, at appropriate moments, to step back and make a judgment about whether working within that set of rules is likely to be profitable in terms of some overall set of goals which it has. It would be able to choose to stop working within a given framework, if need be, and to create a new framework of rules within which to work for a while.

Much of this discussion may remind you of aspects of the
MU
-puzzle. For instance, moving away from the goal of a problem is reminiscent of moving away from
MU
by making longer and longer strings which you hope may in some indirect way enable you to make
MU
. If you are a naive "dog", you may feel you are moving away from your "
MU
-bone" whenever your string increases beyond two characters; if you are a more sophisticated dog, the use of such lengthening rules has an indirect justification, something like heading for the gate to get your
MU
-bone.

Another connection between the previous discussion and the
MU
puzzle is the two modes of operation which led to insight about the nature of the
MU
-puzzle: the Mechanical mode, and the Intelligent mode. In the former, you are embedded within some fixed framework; in the latter, you can always step back and gain an overview of things. Having an overview is tantamount to choosing a representation within which to work; and working within the rules of the system is tantamount to trying the technique of problem reduction within that selected framework. Hardy's comment on Ramanujan's style-particularly his willingness to modify his own hypotheses-illustrates this interplay between the
M
-mode and the
I
-mode in creative thought.

The Sphex wasp operates excellently in the
M
-mode, but it has absolutely no ability to choose its framework or even to alter its
M
-mode in the slightest. It has no ability to notice when the same thing occurs over and over and over again in its system, for to notice such a thing would be to jump out of the system, even if only ever so slightly. It simply does not notice the sameness of the repetitions. This idea (of not noticing the identity of certain repetitive events) is interesting when we apply it to ourselves. Are there highly repetitious situations which occur in our lives time and time again, and which we handle in the identical stupid way each time, because we don't have enough of an overview to perceive their sameness? This leads back to that recurrent issue, "What is sameness?" It will soon come up as an Al theme, when we discuss pattern recognition.

Applying Al to Mathematics

Mathematics is in some ways an extremely interesting domain to study from the Al point of view. Every mathematician has the sense that there is a kind of metric between ideas in mathematics-that all of mathematics is a network of results between which there are enormously many connections. In that network, some ideas are very closely linked; others require more elaborate pathways to be joined. Sometimes two theorems in mathematics are close because one can be proven easily, given the other. Other times two ideas are close because they are analogous, or even isomorphic. These are two different senses of the word "close" in the domain of mathematics. There are probably a number of others. Whether there is an objectivity or a universality to our sense of mathematical closeness, or whether it is largely an accident of historical development is hard to say.

Some theorems of different branches of mathematics appear to us hard to link, and we might say that they are unrelated-but something might turn up later which forces us to change our minds. If we could instill our highly developed sense of mathematical closeness-a "mathematician's mental metric", so to speak-into a program, we could perhaps produce a primitive "artificial mathematician". But that depends on being able to convey a sense of simplicity or "naturalness" as well, which is another major stumbling block.

These issues have been confronted in a number of AI projects. There is a collection of programs developed at
MIT
which go under the name «
MACSYMA
", whose purpose it is to aid mathematicians in symbolic manipulation of complex mathematical, expressions. This program has in it some sense of "where to go"-a sort of

"complexity gradient" which guides it from what we would generally consider complex expressions to simpler ones. Part of
MACSYMA's
repertoire is a program called "
SIN
", which does symbolic integration of functions; it is generally acknowledged to be superior to humans in some categories. It relies upon a number of different skills, as intelligence in general must: a vast body of knowledge, the technique of problem reduction, a large number of heuristics, and also some special tricks.

Another program, written by Douglas Lenat at Stanford, had as its aim to invent concepts and discover facts in very elementary mathematics. Beginning with the notion of sets, and a collection of notions of what is "interesting" which had been spoon-fed into it, it "invented" the idea of counting, then the idea of addition, then multiplication, then-among other things-the notion of prime numbers, and it went so far as to rediscover Goldbach's conjecture! Of course these "discoveries" were all hundreds-even thousands-of years old. Perhaps this may be explained in part by saying that the sense of

"interesting" was conveyed by Lenat in a large number of rules which may have been influenced by his twentieth century training; nonetheless it is impressive. The program seemed to run out of steam after this very respectable performance. An interesting thing about it was that it was unable to develop or improve upon its own sense of what is interesting. That seemed another level of difficulty up-or perhaps several levels up.

The Crux of Al: Representation of Knowledge

Many of the examples above have been cited in order to stress that the way a domain is represented has a huge bearing on how that domain is "understood". A program which merely printed out theorems of TNT in a preordained order would have no understanding of number theory; a program such as Lenat's with its extra layers of knowledge could be said to have a rudimentary sense of number theory; and one which embeds mathematical knowledge in a wide context of real-world experience would probably be the most able to

"understand" in the sense that we think we do. It is this'
representation of knowledge
that is at the crux of Al.

In the early days it was assumed that knowledge came in sentence-like packets", and that the best way to implant knowledge into a program was to develop a simple way of translating facts into small passive packets of data. Then every fact would simply be a piece of data, accessible to the programs using it. This is exemplified by chess programs, where board Positions are coded into matrices or lists of some sort and stored efficiently in memory where they can be retrieved and acted upon by subroutines.

The fact that human beings store facts in a more complicated way was Known to psychologists for quite a while and has only recently been rediscovered by AI workers, who are now confronting the problems of "chunked" knowledge, and the difference between procedural and declarative types of knowledge, which is related, as we saw in Chapter XI, to the difference between knowledge which is accessible to introspection and knowledge which is inaccessible to introspection.

The naive assumption that all knowledge should be coded into passive pieces of data is actually contradicted by the most fundamental fact about computer design: that is, how to add, subtract, multiply, and so on is not coded into pieces of data and stored in memory; it is, in fact, represented nowhere in memory, but rather in the wiring patterns of the hardware. A pocket calculator does not store in its memory knowledge of how to add; that knowledge is encoded into its "guts". There is no memory location to point to if somebody demands, "Show me where the knowledge of how to add resides in this machine!"

A large amount of work in Al has nevertheless gone into systems in which the bulk of the knowledge is stored in specific places-that is, declaratively. It goes without saying that some knowledge has to be embodied in programs; otherwise one would not have a program at all, but merely an encyclopedia. The question is how to split up knowledge between program and data. Not that it is always easy to distinguish between program and data, by any means. I hope that was made clear enough in Chapter XVI. But in the development of a system, if the programmer intuitively conceives of some particular item as data (or as program), that may have significant repercussions on the system's structure, because as one programs one does tend to distinguish between data-like objects and program-like objects.

It is important to point out that in principle, any manner of coding information into data structures or procedures is as good as any other, in the sense that if you are not too concerned about efficiency, what you can do in one scheme, you can do in the other.

However, reasons can be given which seem to indicate that one method is definitely superior to the other. For instance, consider the following argument in favor of using procedural representations only: "As soon as you try to encode features of sufficient complexity into data, you are forced into developing what amounts to a new language, or formalism. So in effect your data structures become program-like, with some piece of your program serving as their interpreter; you might as well represent the same information directly in procedural form to begin with, and obviate the extra level of interpretation."

DNA and Proteins Help Give Some Perspective

This argument sounds quite convincing, and yet, if interpreted a little loosely, it can be read as an argument for the abolishment of
DNA
and
RNA
. Why encode genetic information in
DNA
, when by representing it directly in proteins, you could eliminate not just one, but
two
levels of interpretation? The answer is: it turns out that it is extremely useful to have

the same information in several different forms for different purposes. One advantage of storing genetic information in the modular and data-like form of
DNA
is that two individuals' genes can be easily recombined to form a new genotype. This would be very difficult if the information were only in proteins. A second reason for storing information in
DNA
is that it is easy to transcribe and translate it into proteins. When it is not needed, it does not take up much room; when it is needed, it serves as a template. There is no mechanism for copying one protein off of another; their folded tertiary structures would make copying highly unwieldy. Complementarily, it is almost imperative to be able to get genetic information into three-dimensional structures such as enzymes, because the recognition and manipulation of molecules is by its nature a three-dimensional operation.

Thus the argument for purely procedural representations is seen to be quite fallacious in the context of cells. It suggests that there are advantages to being able to switch back and forth between procedural and declarative representations. This is probably true also in AI.

This issue was raised by Francis Crick in a conference on communication with extraterrestrial intelligence:

We see on Earth that there are two molecules, one of which is good for replication [
DNA
] and one of which is good for action [proteins]. Is it possible to devise a system in which one molecule does both jobs, or are there perhaps strong arguments, from systems analysis, which might suggest (if they exist) that to divide the job into two gives a great advantage, This is a question to which I do not know the answer.14

Modularity of Knowledge

Another question which comes up in the representation of knowledge is modularity. How easy is it to insert new knowledge? How easy is it to revise old knowledge? How modular are books? It all depends. If from a tightly structured book with many cross-references a single chapter is removed, the rest of the book may become virtually incomprehensible. It is like trying to pull a single strand out of a spider web-you ruin the whole in doing so.

On the other hand, some books are quite modular, having independent chapters.

Consider a straightforward theorem-generating program which uses
TNT
's axioms and rules of inference. The "knowledge" of such a program has two aspects. It resides implicitly in the axioms and rules, and explicitly in the body of theorems which have so far been produced. Depending on which way you look at the knowledge, you will see it either as modular or as spread all around and completely nonmodular. For instance, suppose you had written such a program but had forgotten to include
TNT
's Axiom I in the list of axioms. After the program had done many thousands of derivations, you realized your oversight, and inserted the new axiom. The fact that you can do so in a trice shows that the system's implicit knowledge is modular; but the new axiom's contribution to the explicit knowledge of the system will only be reflected after a long time-after its effects have "diffused" outwards, as the odor of perfume slowly diffuses in a room when the bottle is broken. In that sense the new knowledge takes a long time to be incorporated.