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C00001 00001
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)
C00029 00008	.SSEC(Examples of AM in Action)
C00030 ENDMK
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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

Kind of expts one could perform on AM:

(i) Remove individual concept modules,
and individual heuristic rules.
How is the  performance degraded?
AM should operate even if most of its heuristic rules and/or
most of its modules excised,
so long as ↓_any_↓ parts of any of the modules remain
enabled. 
If the remaining fragment of AM is too small, however, it may not be
able to find anything interesting to do. In fact, this situation was actually
encountered experimentally, when the first few partially complete
concepts were inserted. If only some knowledge is removed,
AM may in fact keep operating without this "uninteresting collapse".
The converse situation should also hold: although still functional with any module
unplugged, the performance ↓_should_↓ be noticably degraded. 
That is, while not indispensible, each module should nontrivially help the others.
Which concepts does AM now "miss" discovering? Is the removed concept/heuristic
later discovered anyway by those which are left in AM?  This should indicate
the importance of each kind of concept and rule which AM starts with.

(ii) Vary the relative weights given to features by the criteria which judge
aesthetics, interestingness, worth, utility, etc.  See how important each factor
is in directing AM along successful routes. In other words, vary the little numbers
in the formulae (both globaal and inside heuristic rules).

(iii) Add several new concept modules (including new heuristics) and see if AM
can work in some unanticipated field of mathematics (like 
graph theory or calculus or plane geometry).

(iv) Add several new intuitions, and see if AM can develop nonmathematical
theories (elementary physics, program verification). This would also require
limiting AM's freedom to "ignore a given body of data and move on to something
more interesting".

⊗8≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡⊗*

Experiments actually performed on AM:

To date, there are 7 experiements which were performed on AM.

< Maybe these 7 expts should be grouped into a few 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

.GEOEX: PAGE;

.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