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C00002 00002 .NSECP(Results)
C00004 00003 .SSEC(What AM Did)
C00005 00004 . SSSEC(AM as a Mathematician)
C00014 00005 . SSSEC(AM as an Explorer)
C00016 00006 . SSSEC(AM as a Program)
C00017 00007 .SSEC(Experiments with AM)
C00026 00008 .SSEC(Examples of AM in Action)
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.NSECP(Results)
This chapter opens by summarizing what AM "did". Section 1 gives a
fairly high-level description of the major paths which were explored,
the concepts discovered along the way, the relationships which were
noticed, and ones which "should" have been but but weren't.
The next section continues this exposition by presenting the results
of experiments which were done with (and ⊗4on⊗*) AM.
.ONCE TURN ON "{}"
A second purpose of this chapter is to provide some concrete
illustrations for the disconnected ideas of the last few chapters. So
Section {SECNUM}.3 contains several examples of AM in action. By
reading through these traces, one may gain a better feel for how the
mechanisms explained in Chapters {[2] AGENDA}-{[2] KNOWL} work
together.
.ONCE TURN ON "{}"
Chapter {[2] EVALU} will draw upon these examples -- and others given
in the appendices -- to form conclusions about AM. Several meta-level
questions will be tackled (e.g., What are AM's limitiations?).
In the final chapter, all the ideas which are common to mathematics
research, and ultimately to empirical investigation in general, are
isolated and packaged together as models or theories of those
activities.
.SSEC(What AM Did)
After an overview (the way a modern math historian might write it),
we'll present the paths which AM followed.
. SSSEC(AM as a Mathematician)
Let's take a moment to discuss the totality of the mathematics which
AM carried out. Like a contemporary historian summarizing the work
of Euclid, we shall not hesitate to use current terms, and criticize
by current standards.
AM began its investigations with scanty knowledge of a few
set-theoretic concepts (sets, equality of sets, set operations).
Most of the obvious set-theory relationships (e.g., de Morgan's laws)
were eventually uncovered; since AM never fully understood abstract
algebra, the statement and verification of each of these was quite
obscure. AM never derived a formal notion of infinity, but it did
create a procedure for making arbitrarily long chains of new sets
("insert a set into itself"). On the other hand, AM naively
established conjectures like "a set can never be a proper subset of
itself", showing it had no real comprehension of inifinite sets. No
sophisticated set theory (e.g., diagonalization, ordinals) was ever
done.
After this initial period of exploration, AM decided that "equality"
was worth generalizing, and thereby discovered the relation
"same-size-as". Cardinality was based on this, and soon most simple
arithmetic operations were defined. Addition arose as an analog to
union. Multiplication arose in four separate ways: as an analog to
cross-product, as repeated addition, as iterated substitution$$ Take
two sets A and B. Replace each element of A by the set B. Remove one
level of parentheses by taking the union of all elements of the
transfigured set A. Then that new set will have as many elemens as
the product of the lengths of the two original sets. $, and by
studying the cardinality of power sets$$ The size of the set of all
subsets of S is 2↑S. Thus the power set of A∪B has length equal to
the ↓_product_↓ of the lengths of the power sets of A and B
individually (assuming A and B are disjoint). $. So multiplication
immediately captured AM's attention as an interesting concept.
Soon after defining multiplication, AM investigated the process of
multiplying a number by itself: squaring. The inverse of this turned
out to be interesting, and led to the definition of square-root.
Although AM was very close to discovering irrationals at this point,
it turned aside and was content to work with integer-square-root.
Raising to fourth-powers, and fourth-rooting, were discovered at this
time.
.ONCE TURN ON "{}"
AM also worked on defining a meaningful inverse operation for
multiplication. This led to both division and to factoring. The
associativity and symmetry of multiplication indicated that it could
accept a BAG (a multiset) of numbers as its argument, so factoring
was taken to mean finding all bags of numbers whose product equalled
the given number. Minimally-factorable numbers turned out to be what
we call primes. Maximally-factorable numbers were also thought to be
interesting at the time, and in fact an unusual $$ These are the
so-called "highly-composite" numbers of Ramanujan. As far as the
author and his committee know, this is the first such explicit
characterization of these numbers, hence is probably
"new-to-Mankind". A similar (but slightly different) result has
recently been noticed in [Hardy] (p. 93). Since the purpose of the
thesis is not to derive "new" mathematics, discussion of this result
will be minimized in this document. A short discussion will be
provided in Section {[2] MAXDIVSEC}.{[1] MAXDIVSSEC}, on page {[3]
MAXDIVPAGE}. $ characterization of such numbers was discovered.
AM conjectured the fundamental theorem of arithmetic (unique
factorization into primes) and Goldbach's conjecture (every even
number >2 is the sum of two primes) in a surprisingly symmetric way.
The unary representation of numbers gave way to a representation as a
bag of primes (based on unique factorization), but AM never thought
of exponential notation. Since the key concepts of modulus and
exponentiation were never discovered, progress in number theory was
arrested.
When a new base of geometric concepts was added, AM began finding
some more general associations. In place of the strict definitions
for the equality of lines, angles, and triangles, came new
definitions of concepts we refer to as Parallel, Equal-measure,
Similar, Congruent, Translation, Rotation, plus many which have no
common name (e.g. the relationship of two triangles sharing a common
angle). A cute geometric interpretation of Goldbach's conjecture was
found$$ Given all angles of a prime number of degrees,
(0,1,2,3,5,7,11,...,179 degrees), then any angle between 0 and 180
degrees can be approximated (to within 1 degree) as the sum of two of
those angles. If our culture and our technology were different, this
result might have been a well-known one. $. Lacking a geometry
"model" (an analogic representation like the one Gelernter emplyed),
AM was doomed to failure with respect to proposing geometric
conjectures.
Similar restrictions due to poor "visualization" abilities would crop
up in topology. The concepts of continuity, infinity, and measure
would have to be fed to AM before it could enter the domains of
analysis. More and more drastic changes in its initial base would be
required, as the desired domain gets further and further from simple
set theory.
. SSSEC(AM as an Explorer)
This section will list all the paths which AM followed, explain why,
and indicate where they led. Along the way, some concepts were
created which were interesting to ⊗4us⊗* (in the smug wisdom of
millenia of hindsight) but which AM never bothered to develop. These
will be noted, and a stab will be made to apologize for AM$$ The
typical excuse is that AM was distracted at that moment by some even
more interesting task. $. A few exciting moments occurred when AM became
interested in a concept which had been ignored by humans. One instance
of this led to an unusual characterization of numbers with an abnormally
large number of divisors; another time, AM found an "application" of
Goldbach's conjecture. In other such unexpected fixations, the concept
has still not proven to be anyhting other than "cute" (e.g., triangles
are related by R iff they share a common side, and they are similar).
<< List those paths!!! >
. SSSEC(AM as a Program)
Considering AM a computer program...
.SSEC(Experiments with AM)
.EXPTSSEC: SSECNUM;
.EXPTPAGE: PAGE;
The following points are covered for each experiement:
.BN
λλ How it was thought of. Why did it come to mind.
λλ What will be gained by it. The implications of some possible
outcomes.
λλ How the experiment was set up. What preparations/modifications
had to be made. How much time (man-hours) it took.
λλ Description of what happened.
λλ How did this differ from normal? From what was expected?
λλ Conclusions. What have we really learned from this experiment.
Does it suggest any new ones? Does it imply anything about how an
AM-like system would benefit from a better machine? a different
domain? Anything about math or teaching of math?
.E
In all, there are six experiements which were performed on AM.
< Maybe these 7 expts should be grouped into a feew related packets of expts>
<< Write this up nicely >
.B APART
1) Set the interestingness factor of all concepts to 200 initially.
Result: occasional wanderings, but still bursts of creative driving.
Cardinality in about 3 times as many cycles.
Conclusion: the int. factors of the concepts are useful for deciding
what to do in close situations, or where few good reasons exist,
but even 1 good reason is far more influential -- and rightly so!
2) Pick a random candidate to do next, but maintain INTHRESH as it is
(so the average job-list length is about 20). Also, leave the
interestingness factors of the concepts as they are normally (0-1000).
Result: on the average, it will take about 20 times as long to get to
a given job. On the other hand, several "good" jobs are sprinkled
around in the queue, so the performance is cut only by a small factor.
(timewise). On the other hand, behavior is much less focused, rational.
Typically, a "good" cand will be chosen, having reasons all of which
were true 10 cycles ago -- and which are clearly superior to those of
the last 10 Cands! This is what is so annoying to human onlookers.
Result: Since AM was frequently working on a low-value task, it was unwilling
to spend much time or space on it. So the mean time alotted per task
fell to about 15 seconds (from the typical 30 secs). Thus, the "losers"
were dealt with quickly, so the detriment to performance was softened.
In fact, many of these "failed" almost instantly (meaningless ones).
Conclusion: Picking (on the average) the 20th-best candidate impedes prgress
by a factor less than 20 (about 7), but it dramaticly degrades the
"sensibleness" of AM's behavior, the continuity of its actions.
Humans place a big value on absolute sensibleness, and believe that
doing something silly 50% of the time is MUCH worse than half as
productive as always doing the next most logical task.
Conclusion: having 20 multi-processors simultaneously execute the top 20
jobs will result in a gain of about 7 in the rate of "big" discoveries.
That is, not a full factor of 20, nor no gain at all.
3) Pick a random candidate to do next, and adjust INTHRESH so that no
candidate ever is excluded from the job-list, and set all ints. to 200.
Result: Many "explosive" tasks were chosen, and the number of new concepts
increased rapidly. As expected, most of these were real "losers".
There seemed no rationality to AM's sequence of actions, and it was quite
boring it watch it floundering so. The typical length of the agenda was
about 500, and AM's performance was "slowed" by at least a couple orders
of magnitude. A more subjective measure of its "intelligence" would say
that it totally collapsed under this random scheme.
Conclusion: Having 500 processors simultaneously execute all the jobs on
the agenda would increase AM's performance only by a factor of 10 or so.
The truly "intelligent" behavior is AM's plausible sequencing of tasks.
4) Modify the global formula assigining a priority value to each job. Let it still
be a function of the reasons for the job, but trivialize it:
let the priority of a job be defined as simply the number of reasons it has.
(normalize by multiplying by 100, and cut-off if over 1000).
This raisies the new question of what to do if several jobs all have the
same priority. I suppose the answer is to execute them in stack-order
(most recent first), since this is what AM will do anyway.
5) Eliminate "Equality", and see what AM does.
The reason for doing this is that AM discovered Cardinality via the
technique of generalizing the relation "Equality"-of-2-sets. What will
happen if we eliminate this path? Will AM rederive Equality? Will it get
to Cardinality via another route? Will it do some set-theoretic things?
6) General classes of expts: modify/add/eliminate certain concepts;
modify certain heuristics;
modify the strategy for choosing the next job/ value assigned to jobs.
7) Big expt: GEOMETRY. Add a new base of concepts to the ones already there,
incl. Point, Line, Angle, Triangle, Equality of pts/lines/angles/triangles.
Results: fairly good behavior. Derived congruence, similarity of triangles.
Derived the idea of timberline in several ways.
Use for Goldbach's conjecture: any angle (0-180 degrees) can be built up
(to within 1 degree) as the sum of two angles of prime degrees (<180).
.E
.SSEC(Examples of AM in Action)
.B APART
<Go over the examples in varying levels of detail. Occasionally, give a
"snapshot" of the new jobs, concepts, facet entries, etc.>
Consider the example of discovering cardinality
Initial behavior of the system
Proposing unique factorization
Noticing a "real" number theory conjecture
Geometry example: congruence and similarity.
.E