COMMENT ⊗   VALID 00009 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	\input cmpdes
C00003 00003	\chapbegin{2}		% Here beginneth Chapter 2
C00006 00004	\sectionbegin[1]{DISCUSSION OF THE AM PROGRAM}
C00012 00005	\subsectionbegin[1.2]{Agenda and Heuristics}
C00023 00006	\sectionbegin[2]{WHAT (NOT) TO GET OUT OF THIS EXAMPLE}
C00028 00007	\SSEC{Deciphering the Example}
C00036 00008	\SSEC{The Example Itself}
C00057 00009	\SSEC{Recapping the Example}
C00062 ENDMK
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\input cmpdes
\def\draft{T}
\def\AM{{\bf AM}}
\titlepage
\newpage
\setcount0 14
\chapbegin{2}		% Here beginneth Chapter 2
\chaptertitle{EXAMPLE: DISCOVERING PRIME NUMBERS}
\rjustline{\:B Example: Discovering}
\rjustline{\:B Prime Numbers}
\vskip 13.5pc plus 6pt minus 4pt
\runningrighthead{INTRODUCTION}


This chapter will present an example of {\AM} in action, an excerpt from
the output of {\AM}, as it investigates some concepts.


After a brief discussion of {\AM}'s control structure in Section 2.1,
the reader will be told what the  point of this example is---and  is
{\sl not}.  Section  3 provides a few eleventh-hour  hints at decoding
the example.

The excerpt itself follows in Section 4.  It skips the first
half of the  session, and picks up at a point just  after {\AM} has defined  the
concept ``Divisors-of.''  Soon afterward, {\AM} defines Primes, and begins
to find interesting conjectures related to them.  The excerpt goes on
to show how {\AM} conjectured the fundamental  theorem of arithmetic and
Goldbach's  conjecture.   {\AM}  derived  the notion  of  partitioning a
collection of $n$ objects into smaller bundles, but failed to  find any
interesting  conjectures  about  that   process.    Instead,  {\AM}  was
side-tracked  into the (probably)  fruitless investigation of numbers
which can be represented as the sum of two primes in one unique way.

The final section of this  chapter will recap this example the  way a
math historian might report it.

\sectionbegin[1]{DISCUSSION OF THE AM PROGRAM}
\vskip -9pt

\subsectionbegin[1.1]{Representation}

{\AM}  is a  program  which expands  a  knowledge base  of  mathematical
concepts.    Each concept  is stored  as  a particular  kind  of data
structure, namely  as  a  collection of properties  or  ``facets''  of  the
concept.   For example, here  is a  miniature example of  a concept
\ffootnote{The right arrow (``$\→$'') is the symbol for ``implies.''
``Nos.''  is an abbreviation for  ``Numbers.'' The vertical  bar ``$\relv $'' is a
symbol for the  predicate ``divides  evenly into;'' with a slash through it
(``$\not\,\ \relv\,\ $'') we denote ``does {\sl not} divide evenly into.''
``$\otimes $'' indicates exclusive
or, and  the  symbol ``$\forall $''  is  read ``for  all.''   }:

\sectionskip
{\yskip \parskip 1pt \6
\hjust{\ \ \   NAME: {\it Prime  Numbers}}

\hjust{\ \ \   DEFINITIONS:}

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ ORIGIN: {\it Number-of-divisors-of(x) = 2 }}

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ PREDICATE--CALCULUS: $Prime(x) \↔ (\forall z) (z\relv x \→ (z=1 \otimes z=x))$}

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ ITERATIVE: $(for\  x>1): For\  i\  from \ 2 \  to\  \sqrt{x},\  i{\not\,\,\relv\,}\  x $  }


\hjust{\ \ \   EXAMPLES: $2,\  3,\  5,\  7,\  11,\  13,\  17$ }

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ BOUNDARY: $2,\  3$ }

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ BOUNDARY--FAILURES: $0,\  1$ }

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ FAILURES: $ 12$ }



\hjust{\ \ \   GENERALIZATIONS: {\it  Nos.,  Nos. with an even no. of divisors}}

\hjust{\ \ \   SPECIALIZATIONS: {\it  Odd Primes,\  Prime Pairs,\  Prime Uniquely--addables }}

\hjust{\ \ \   CONJECS: {\it  Unique factorization,\  Goldbach's conjec.,\  Extrema of No--of--divisors--of }}

\hjust{\ \ \   INTU'S: {\it A metaphor to the effect that Primes are the building blocks of all numbers}  }

\hjust{\ \ \   ANALOGIES:  }

\hjust{\ \ \   \ \ \ \ \ \ \ \ \ \ {\it  Maximally--divisible numbers are converse extremes of Number--of--divisors--of }}

\hjust{\ \ \     \ \ \ \ \ \ \ \ \ \ {\it  Factor a non--simple group into simple groups }}

\hjust{\ \ \   INTEREST: {\it  Conjectures tying Primes to Times,\  to Divisors--of,\  to related operations }}

\hjust{\ \ \   WORTH: $800$}}

\yskip

\noindent ``Creating a  new  concept'' is  a well-defined  activity: it  involves
setting  up a new  data structure like the one above,  and filling in
entries for some  of its facets  or slots.   Filling in a  particular
facet  of a  particular concept  is also  quite well-defined,   and is
accomplished  by executing a collection  of relevant heuristic rules.
This process will be described in great detail in later chapters.

\subsectionbegin[1.2]{Agenda and Heuristics}

An agenda of plausible tasks is maintained by {\AM}.   A typical task is
{\bf ``Fill-in examples of  Primes.''} The agenda may contain hundreds of
entries such as  this one.   {\AM}  repeatedly selects the top task from  the
agenda  and  tries to  carry  it  out.   This  is  the whole  control
structure! Of  course, we must still explain how {\AM} creates plausible
new tasks to place on  the agenda, how {\AM} decides which task  will be
the best one to execute next, and how it carries out a task.

If the task is {\bf ``Fill-in new Algorithms for Set-union,''} then 
{\sl satisfying} it would mean actually synthesizing some new procedures,
some new LISP code capable of forming the union of any two sets.
A heuristic rule is {\sl relevant} to a task if and only if executing
that rule brings {\AM} closer to satisfying that task. 
Relevance is determined {\sl a priori} by where the rule
is stored. A rule tacked onto the Domain/range facet of the Compose
concept would be presumed relevant to the task {\bf ``Check the Domain/range
of Insert$\circ$Delete.''}

Once  a task is  chosen from  the agenda,  {\AM} gathers  some heuristic
rules which might  be relevant  to satisfying  that task.   They  are
executed, and then {\AM} picks  a new task.  While a  rule is executing,
three kinds of actions or effects can occur:\par

\listbegin
\numlistentry[1]{Facets of  some concepts  can get filled  in (\eg, 
examples of
primes may actually be found and tacked onto the ``Examples''  facet of
the ``Primes''  concept).   A typical heuristic  rule which  might have
this effect is:}

\yskip
\indentedline[20pt]{{\it To fill in examples of X, where X is a kind of Y
(for some more general concept Y),}}

\indentedline[20pt]{{\it Check the examples of Y; some of them may be examples
of X as well.}}
\yskip

\noindent 
For the task of filling  in examples of Primes, this rule  would have
{\AM} notice  that Primes is a  kind of Number, and  therefore look over
all the known examples of Number. Some of those would be primes, and
would be transferred to the Examples facet of Primes.

\numlistentry[2]{New concepts may be created (\eg, the concept ``primes which
are uniquely representable as the sum of two other primes'' may be somehow
be  deemed worth  studying).   A typical  heuristic rule  which might
result in this new concept is:}

\yskip
\indentedline[20pt]{{\it If some (but not most) examples of X are also
examples of Y (for some concept Y),}}

\indentedline[20pt]{{\it Create a new concept defined as  the intersection of
those 2 concepts (X and Y).}}
\yskip

\noindent
Suppose {\AM} has already isolated the concept of being representable as
the sum of two primes in only one way ({\AM} actually calls such numbers
``Uniquely-prime-addable numbers'').   When {\AM} notices that some primes
are in this  set, the  above rule will  create a  brand new  concept,
defined as the set of numbers which are both prime and uniquely prime
addable.

\numlistentry[3]{New tasks may be added to the agenda (\eg, the current
activity may suggest  that the following  task is worth  considering:
{\bf ``Generalize the concept of prime numbers''}).  A typical heuristic rule
which might have this effect is:}

\yskip
\indentedline[20pt]{{\it If very few examples of X are found,}}
\indentedline[20pt]{{\it Then  add the following  task to the agenda:
{\bf ``Generalize the concept X.''}}}
\yskip

.noindent
Of course, {\AM} contains a  precise meaning for the phrase  ``very few.''
When  {\AM} looks for  primes among  examples of already-known  kinds of
numbers, it will find dozens of  non-examples for every example of  a
prime it  uncovers.  ``Very  few'' is thus  naturally implemented  as a
statistical  confidence level.\footnote{The ratio of examples found to non-examples
stumbled over lies between .001 and .05. Philosophers outraged by this may
be somewhat appeased by knowledge  that large changes in the  precise
numbers very rarely alter {\AM}'s behavior.}
\listend

The concept of  an agenda is certainly not new:  schedulers have been
around  for a long  time.  But  one important feature  of {\AM}'s agenda
scheme {\sl is} a  new idea:  attaching---and  using---a  list  of
quasi-symbolic\footnote{Each reason is an English sentence. While {\AM}
can tell whether two given reasons coincide,
it can't actually do any internal processing on them. If this lack of intelligence
had proved to be a limiting problem,
then more work would have been expended on giving {\AM} some
such abilities.} reasons  to each  task which explain  why the task  is worth
considering, why it's  plausible.   {\sl It is the  responsibility of  the
heuristic rules to include  reasons for any tasks they  propose\/}.\!
\footnote{An alternative scheme, perhaps even a bit more human-like, would be to
(perhaps only occasionally) allow a burst of poorly-motivated tasks to be
proposed, and then use some pruning criteria to weed out the obvious losers.
During this time, {\AM} could type out to the user (who otherwise would be closely
monitoring its activities)  a cute anthropomorphic phrase like ``I'm 
now sitting back and puffing on my pipe, lost in contemplation.''}
For example,  let's  reconsider  the heuristic  rule  mentioned  in (3)
above.  It really looks more like the following:\par\ninepoint

\yskip
\indentedline[20pt]{{\it If very few examples of X are found,}}
\hangindent 20pt after 0{{\it Then add the  following task to the  agenda:
{\bf ``Generalize the  concept X''}, for  the following reason: ``X's  are
quite rare; a  slightly less restrictive concept might be more interesting.''}}
\yskip

\tenpoint\noindent
If the same task is proposed by several rules, then several different
reasons for it  may be  present.  In  addition, one ephemeral  reason
also exists: ``Focus of attention.'' Any tasks which are similar to the
one last executed  get ``Focus of  attention'' as a  bonus reason.   {\AM}
uses all these  reasons, \eg   to decide  how to rank  the tasks on  the
agenda.   The  ``intelligence'' {\AM}  exhibits  is not  so much  ``what it
does,'' but rather the {\sl order} in which it arranges its agenda.\footnote{For
example, alternating a randomly-chosen task  and the ``best'' task (the
one {\AM}  chose to do) only slows the system down by a factor of 2, yet
it totally destroys its  credibility as a rational researcher
(as judged by the human user of {\AM}).   This
is   one   conclusion   of   experiment  4  (see   Section 5.4).}
{\AM} uses  the  list  of
reasons in another way: Once a task has been selected, the quality of
the reasons  is used to decide how much time  and space the task will
be permitted to absorb, before {\AM}  quits and moves on to a new  task.
This    whole    mechanism    will    be    detailed    in    Section
3.2.

\sectionskip
\sectionbegin[2]{WHAT (NOT) TO GET OUT OF THIS EXAMPLE}


The purpose of  the example which  comprises the next subsection is to
convey  a bit of  {\AM}'s flavor. After  reading through  it, the reader
should be convinced that  {\AM} is {\sl not}  a theorem-prover, nor is  it
{\sl randomly}  manipulating entries  in a  knowledge base,  nor is  it
{\sl exhaustively} manipulating  or searching.  {\AM} is carefully growing a network
of data structures representing mathematical concepts,  by repeatedly
using heuristics both (a) for guidance in choosing a task to work on next, and
(b) to provide methods to satisfy the chosen task.

The following points are important  but can't be conveyed by any lone
example:

\listbegin
\numlistentry[1]{Although  {\AM}  appears   to  have  reasonable  natural   language
abilities, this  is a  typical AI  illusion: most  of the phrases  {\AM}
types  are mere tokens,  and the syntax  which the user  must obey is
unnaturally constrained. For the sake of clarity, I have ``touched up''
some of  the wording, indentation,  syntax, etc. of what  {\AM} actually
outputs,  but left the spirit  of each phrase intact.   As the reader
becomes more familiar with {\AM}, future examples  can be ``unretouched.''
If  he wishes,  he may  glance at  Appendix 3.2,
which shows some actual listings of {\AM} in action.}

\numlistentry[2]{The reader  should  be skeptical
of  the generality  of  the
program; is  the knowledge base  ``just right'' (\ie, finely  tuned to
elicit  this one chain of  behaviors)?  The  answer is {\sl ``No''}.\footnote{The
{\sl design} of {\AM} was finely tuned so that the answer to this question
would be ``{\sl No}.'' Ponder that one!} The whole point of this project is
to show that a relatively small set of general heuristics can guide a
nontrivial discovery process.  Each activity, each task, was proposed
by some heuristic rule (like ``look for extreme cases of X'') which was
used time and time again, in many situations.  It was  not considered
fair  to insert  heuristic guidance  which  could only  ``guide'' in  a
single situation.
\listend

This  kind of generality can't be  shown convincingly in one example.
Never\-the\-less,  even within  this  small  excerpt, the  same  line  of
development which  leads to decom\-posing num\-bers  (using Times\inv ) and
there\-by dis\-co\-ver\-ing unique fac\-tori\-za\-tion,  also leads to  decomposing
numbers (using Add\inv ) and thereby dis\-covering Goldbach's conjejec\-cture.
The   same  heuristic   which  caused  {\AM}   to  ex\-pect   that  unique
factorization  will  be  useful,  also  caused  {\AM}  to  sus\-pect  that
Goldbach's conjecture will be useless.

\yskip
Let me  reemphasize that the ``point''  of this example is {\sl not} the
specific   mathematical  concepts,  nor   the  particular  chains  of
plausible reasoning {\AM}  produces, nor the  few flashy conjectures  {\AM}
spouts,  but rather  an illustration  of the {\sl kinds} of  things {\AM}
does.

\SSEC{Deciphering the Example}

Recall that  in general, each  task on the  agenda will  have several
reasons attached to it.  In the example excerpt, the reasons for each
task are  printed just  after the  task is  chosen,  and before  it's
executed.

{\AM} numbers its activities sequentially. Each time  a new task is chosen, a counter
is  incremented. The  first task in  the example  excerpt is labelled
{\strdotstr{Task\ 65}}, meaning that the example skips the first  64 tasks which
{\AM} selects and carries out.
The reason simply is that the development of simple concepts related to
divisibility will probably be more intelligible and palatable to the reader,
than {\AM}'s early ramblings in finite set theory.

In the  example itself, several irrelevant tasks  have been excised.\!
\footnote{This is  fair, despite  the results  of Experiment 5 (see
Section 5.3.5) because the remaining
tasks  clump together  in twos, threes,  etc; they  are uninterrupted
lines of research (\eg, Tasks  65-67), separated by very large  gaps
(\eg, the jump from  Task 67 to 79).}  About  half of those omitted
tasks were interesting in themselves, but all of them were tangential
or unrelated to  the development shown.   The reader can tell  by the
global  task numbering how  many were  skipped.  For  example, notice
that the excerpt jumps from Task 67 to Task 79.

To help gauge {\AM}'s  abilities, the reader may  be interested to  know
that {\AM}  defined ``Natural Numbers''  during Task  32, and ``Times''  was
defined during Task 122.  {\AM} started with no knowledge of numbers, and
only scanty knowledge of sets and set-operations. Task 2, \eg, was to
fill in examples of Sets.

The  concepts that  {\AM} talks  about  are self-explanatory---by  and
large.  Below are discussed some nonstandard ones.

{$ \underline{BAG}$}\0  is a kind of  list structure, a  bunch of elements which
are unordered, but one in  which multiple copies of the  same element
are permitted.  One may visualize  a paper bag  filled with cardboard
letters. Technically, we shall say  that a set is \4not\0  considered
to be a bag.  A bag is denoted  by enclosure within parentheses, just
as sets are within braces. So the bag containing X and four Y's might
be written (X  Y Y Y  Y), and  would be considered  indistinguishable
from the bag (Y Y Y X Y).

${\underline{Number}}$  will mean (typically) a positive integer.

${\underline{Times\minv}}$  is  a  particular  relation.   For  any  number  x,
Times\inv(x)  is a set of bags. Each  bag contains some numbers which,
when multiplied together, equal x.  For example,
Times\inv (18) = $\{$ (18) (2 9)  (2 3 3) (3 6) $\}$.  Checking,  we see that
multiplying, \eg, the numbers in the bag (2 3 3) together, we do get
$2\times 2\times 3 = 18$.  Times\inv (x)  contains all possible  such bags  (containing
natural numbers >1).

${\underline{Add\minv}}$\0 is a relation analogous to Times\inv . For any number x,
Add\inv (x)  is also  a set  of bags.    Each bag  contains a  bunch of
numbers which, when added together, equal x.  For  example, Add\inv (4)
= $\{$  (4) (1 1 1  1) (1 1  2) (1 3) (2  2)\penalty 1000$\}$.  
Add\inv (x)  contains all
possible  such bags  (containing numbers  >0); it finds  all possible
\4partitions\0 of x.

${\underline{Divisors-of}}$\0 is a more  standard relation.  For any  number x,
Divisors-of(x) is the set of all positive numbers which divide evenly
into x.  For example, Divisors-of(18) = $\{$1 2 3 6 9 18$\}$.

\yskip

Whenever
there  is  a conflict  between  ``computer science  jargon''  and ``math
jargon,'' I have opted for  the latter. So, \eg, all ``functions''  are
necessarily single-valued for each member of their domain.

{\AM} is  an \4interactive\0  computer program.   It prints  out phrases
indicating  what  it is  doing  at each  moment, and  a  human being,
referred to as the \4User,\0 watches {\AM}'s activities.  At any moment,
the user can interrupt {\AM} to ask questions or re-direct its energies.
In the  excerpt,  the user  never  re-directs {\AM},  although  he  does
occasionally  ask {\AM}  to  rename  some  concept and  to  explain  its
behavior  (\eg, just after {\strdotstr{Task\ 106}}).  For  the benefit of the
reader, I have prefaced all the user's comments by ``{\:tUSER:},'' 
and italicized
my {\sl post hoc} annotations.

\SSEC{The Example Itself}

{\eightpoint \:>  \parskip 1pt \lineskip 1pt \parindent 0pt
\threstar{Task\ 65}

Filling in examples of the following concept: ``Divisors-of.''

	3 Reasons:

 \hjust{\ \ \  (1) No known examples for Divisors-of so far.}

 \hjust{\ \ \  (2) Times, which is related to Divisors-of, is now very interesting.}

 \hjust{\ \ \  (3) Focus of attention: {\AM} recently defined Divisors-of.}

  26 examples found, in 9.2 seconds. \eg: Divisors-of(6)=$\{$1 2 3 6$\}$.


\threstar{Task\ 66}

Considering numbers which have very small sets of Divisors-of (\ie, very few divisors).

	2 Reasons:

 \hjust{\ \ \  (1) Worthwhile to look for extreme cases.}

 \hjust{\ \ \  (2) Focus of attention: {\AM} recently worked on Divisors-of.}

  Filling in examples of numbers with 0 divisors. 
      0 examples found, in 4.0 seconds. 
      Conjecture: no numbers have 0 divisors.

  Filling in examples of numbers with 1 divisors. 
      1 examples found, in 4.0 seconds. \eg: Divisors-of(1)=$\{$1$\}$.
      Conjecture: 1 is the only number with 1 divisor.

  Filling in examples of numbers with 2 divisors. 
      24 examples found, in 4.0 seconds. \eg: Divisors-of(13)=$\{$1 13$\}$.
      No obvious conjecture.  This kind of number merits more study.
      Creating a new concept: ``Numbers-with-2-divisors.''
}

\yskip

{\hangindent 20pt for 44 \it {\AM} had previously derived the notion of singleton, doubleton, etc.
Above, {\AM} was actually defining,  \eg, the set of numbers which gets
mapped into a doubleton of numbers, by the operation Divisors-of.
I took the liberty of translating this into the above form, which uses, \eg,
the numeral ``2.''}

\yskip

{\eightpoint \:>  \parskip 1pt \lineskip 1pt \parindent 0pt

  Filling in examples of numbers with 3 divisors. 
      11 examples found, in 4.0 seconds. \eg: Divisors-of(49)=$\{$1 7 49$\}$.
      All numbers with 3 divisors are also Squares.  This kind of number merits more study.
      Creating a new concept: ``Numbers-with-3-divisors.''
}

\yskip

{\hangindent 20pt for 44 \it How did {\AM} notice that conjecture? 
It took a random example of Numbers-with-3-divisiors,
say 49. Then it asked what other known concepts ``49'' was an example of. 
The two answers were: Odd-numbers and Perfect-squares. {\AM} then tested these
conjectures on the other ten examples just found. The only surviving conjecture
was that all numbers-with-3-divisors are also perfect-squares.}

\yskip
{\eightpoint \:>  \parskip 1pt \lineskip 1pt \parindent 0pt

\threstar{Task\ 67}

Considering the square-roots of Numbers-with-3-divisors.

	2 Reasons:

 \hjust{\ \ \  (1) Numbers-with-3-divisors are unexpectedly also perfect Squares.}

 \hjust{\ \ \  (2) Focus of attention: {\AM} recently worked on Numbers-with-3-divisors.}

  All square-roots of Numbers-with-3-divisors seem to be Numbers-with-2-divisors. 
	\eg, Divisors-of( Square-root(169) ) = Divisors-of(13) = $\{$1 13$\}$. 

  Formulating the converse to this statement. Empirically, it seems to be true.
	The square of each Number-with-2-divisors seems to be a Number-with-3-divisors. 
 	This is very unusual. It is not plausibly a coincidence. (Chance of coincidence is < .001)

  Boosting interestingness factor of the concepts involved:
	Interestingness factor of ``Divisors-of'' raised from 300 to 400.
	Interestingness factor of ``Numbers-with-2-divisors'' raised from 100 to 600.
	Interestingness factor of ``Numbers-with-3-divisors'' raised from 200 to 700.

\yskip

{\2USER: Call the set of numbers with 2 divisors ``Primes.''}

\yskip

\threstar{Task\ 68}

Considering the squares of Numbers-with-3-divisors.

	2 Reasons:

 \hjust{\ \ \  (1) Squares of Numbers-with-2-divisors were interesting.}

 \hjust{\ \ \  (2) Focus of attention: {\AM} recently worked on Numbers-with-3-divisors.}

\hjust{\ \ \ \ \ \ $\vdots$}


}

\yskip

{\hangindent 20pt for 44 \it This gap in the sequencing---from task 68 to task 79---eliminates some
tangential and boring tasks. See page GAPPAGE for an explanation.}

\yskip

{\eightpoint \:>  \parskip 1pt \lineskip 1pt \parindent 0pt

\hjust{\ \ \ \ \ \ $\vdots$}


\threstar{Task\ 79}

Examining Times\inv (x), looking for patterns involving its values.

  	2 Reasons:

 \hjust{\ \ \  (1) Times\inv\ is related to the newly-interesting concept ``Divisors-of.''}

 \hjust{\ \ \  (2) Many examples of Times\inv\ are known, to induce from.}

  Looking specifically at Times\inv (12), which is $\{$ (12) (2 6) (2 2 3) (3 4) $\}$.
    13 conjectures proposed, after 2.0 seconds. 
    \eg, ``Times\inv (x) always contains a bag containing only even numbers.''
    Testing the conjectures on other examples of Times\inv .

    5 false conjectures deal with even numbers. 
    {\AM} will sometime consider the restriction of Times\inv\ to even numbers.

    Only 2 out of the 13 conjectures are verified for all 26 known examples of Times\inv :

  Conjecture 1: Times\inv (x) always contains a singleton bag.
    \eg, Times\inv (12), which is $\{$ (12) (2 6) (2 2 3) (3 4) $\}$, contains (12).
    \eg, Times\inv (13), which is $\{$ (13) $\}$, contains (13).

  Creating a new concept, ``Single-times.''
    Single-times is a relation from Numbers to Bags-of-numbers.
    Single-times(x) is all bags in Times\inv (x) which are singletons.
    \eg, Single-times(12)=$\{$ (12) $\}$.
    \eg, Single-times(13)=$\{$ (13) $\}$.

  Conjecture 2: Times\inv (x) always contains a bag containing only primes.

    \eg, Times\inv (12), which is $\{$ (12) (2 6) (2 2 3) (3 4) $\}$, contains (2 2 3).
    \eg, Times\inv (13), which is $\{$ (13) $\}$, contains (13).

  Creating a new concept, ``Prime-times.''
    Prime-times is a relation from Numbers to Bags-of-numbers.
    Prime-times(x) is all bags in Times\inv (x) which contain only primes.
    \eg, Prime-times(12)=$\{$ (2 3 3) $\}$.
    \eg, Prime-times(13)=$\{$ (13) $\}$.



\threstar{Task\ 80}

Considering the concept ``Prime-times.''

	2 Reasons:

 \hjust{\ \ \  (1) Conjecs about Prime-times will tell much about Primes and Times\inv .}

 \hjust{\ \ \  (2) Focus of attention: {\AM} recently defined Prime-times.}

  Looking specifically at Prime-times(48), which is $\{$ (2 2 2 2 3) $\}$.
    4 conjectures proposed, after .6 seconds. 
    \eg, ``x is never inside any member of Prime-times(x).''
    Testing them on other examples of Prime-times.

    Only 1 out of the 4 conjectures are verified for all 23 known examples of Prime-times:

  Conjecture 1: Prime-times(x) is always a singleton set. 
    That is, Prime-times is a function, not just a relation.
    \eg, Prime-times(48), which is $\{$ (2 2 2 2 3) $\}$, is a singleton set.
    \eg, Prime-times(47), which is $\{$ (47) $\}$, is a singleton set.
    This holds for all 17 known examples of Prime-times. (Chance of coincidence is .0001)
    This fails for 2 of the boundary cases (extreme numbers): 0 and 1.
    Conjecture is amended: Each number >1 is the product of a unique bag of primes.
    I suspect that this conjecture may be very useful.

}

\yskip

{\hangindent 20pt for 44 \it How did {\AM} know this? One of the (unfortunately few!) meta-heuristics in {\AM}
said the following: 
``When using the {\sl 
`look at the inverse of extreme items under
the operation f'} rule,
Tack the following note onto the Interest facet of the new concept which is created:
{\sl `Conjectures involving this concept and  f (or f\inv ) are natural, interesting, and
probably useful.'}''
Now the concept PRIMES was defined using the `extrema'
heuristic rule, with f=Divisors-of. When PRIMES was first created, the meta-rule
we just presented tacked the following note onto Primes.Interest:
{\sl ``Conjectures involving Primes and division (or multiplication) 
are natural, interesting, and
probably useful.''} Thus the unique factorization conjecture triggers this feature,
whereas Goldbach's conjecture wouldn't. }

{\eightpoint \:>  \parskip 1pt \lineskip 1pt \parindent 0pt

\yskip

{\2USER: Call this conjecture  ``Unique factorization conjecture.''}

\vfill

\hjust{\ \ \ \ \ \ $\vdots$}

\vfill

}

{\hangindent 20pt for 44 \it To show that {\AM} isn't really always right on the mark, the next sequence of
tasks includes a crime of omission (ignoring the concept of Partitions) and a
false start (worrying about numbers which can be represented as the sum of two
primes in precisely one way).
Notice the skip here; 3 tasks have been omitted.}

\yskip


{\eightpoint \:>  \parskip 1pt \lineskip 1pt \parindent 0pt

\threstar{Task\ 84}

Examining Add\inv (x), looking for patterns involving its values.

  	2 Reasons:
 \hjust{\ \ \  (1) Add\inv\ is analogous to the newly-interesting concept ``Times\inv .''}

 \hjust{\ \ \  (2) Many examples of Add\inv\ are known, to induce from.}

  Looking specifically at Add\inv (6), which is $\{$ (1 1 1 1 1 1) (1 1 1 1 2) (1 1 1 3) (1 1 2 2)
		 (1 1 4) (1 2 3) (1 5) (2 2 2) (2 4) (3 3) (6)$\}$.
    17 conjectures proposed, after 3.9 seconds. 
    \eg, ``Add\inv (x) always contains a bag of primes.''
    Testing them on other examples of Add\inv .
    Only 11 out of the 17 conjectures are verified for all 19 known examples of Add\inv :
    3 out of the 11 conjectures were false until amended.

  Conjecture 1: Add\inv (x) never contains a singleton bag.

  Conjecture 2: Add\inv (x) always contains a bag of size 2 (also called a ``pair'' or a ``doubleton'').
    \eg, Add\inv (6) contains (1 5), (2 4), and (3 3).
    \eg, Add\inv (4) contains (1 3), and (2 2).

  Creating a new concept, ``Pair-add.''
    Pair-add is a relation from Numbers to Pairs-of-numbers.
    Pair-add(x) is all bags in Add\inv (x) which are doubletons (\ie, of size 2).
    \eg, Pair-add(12)=$\{$ (1 11) (2 10) (3 9) (4 8) (5 7) (6 6) $\}$.
    \eg, Pair-add(4)=$\{$ (1 3) (2 2) $\}$.

  Conjecture 3: Add\inv (x) always contains a bag containing only 1's.

\hjust{\ \ \ \ \ \ $\vdots$}

  Conjecture 10: Add\inv (x) always contains a pair of primes.
    This conjecture is false. Conjecture is amended:
    ``Add\inv (x) usually (but not always) contains a pair of primes.''
    \eg, Add\inv (10) contains (3 7), and (5 5).
    \eg, Add\inv (4) contains (2 2).
    \eg, Add\inv (11) does not contain a pair of primes.

  Creating a new concept, ``Prime-add.''
    Prime-add is a relation from Numbers to Pairs-of-numbers.
    Prime-add(x) is all bags in Add\inv (x) which are pairs of primes.
    \eg, Prime-add(12)=$\{$ (5 7) $\}$.
    \eg, Prime-add(10)=$\{$ (3 7) (5 5) $\}$.
    \eg, Prime-add(11) = $\{$  $\}$


\hjust{\ \ \ \ \ \ $\vdots$}

\threstar{Task\ 106}

Considering the set of numbers for which Prime-add is defined (has non-empty value).

	1 Reason:

 \hjust{\ \ \  (1) Prime-add often has non-empty value. Worth isolating that case.}

  Warning: no task\ on the agenda has an interestingness value above 200!!!

  Creating a new concept ``Prime-addable.''
    Prime-addable is a kind of Number.  x is Prime-addable iff Prime-add(x) is non-empty.
    Will spend 5.0 seconds filling in examples of Prime-addable.
    18 examples found.   Here are some of them:  4 5 6 7 8 9 10 12 13 14 17 16 18.
    Empirically, all even numbers are also in this set (ignoring boundary cases: 0 2 4 6).
    So conjectured.  Danger: must examine boundary cases: the numbers 0, 2, 4, and 6.
    Two exceptions noticed. The only exceptions are the smallest boundary cases: 0, 2.
    Conjecture is amended: All even numbers >2 are the sum of two primes.

  Warning: I expect this conjecture will be cute but useless.

\yskip

{\2USER: Why?}

\yskip

Because the concept of ``Primes'' originally arose as extreme cases of Divisors-of.
  The more closely an operation F is related to the concept Divisors-of, the more
  natural will be any conjecture involving both that operation F and Primes.
  \eg, conjectures involving both Primes and Times will be natural and useful.
  But this conjecture, which involves Primes and Add\inv , will be cute but useless,
  since the relation ``Add\inv '' is unrelated to the relation ``Divisors-of.''

End of explanation.

\vfill

\yskip

{\2USER: Call this conjecture ``Goldbach's conjecture.''}

\yskip

\threstar{Task\ 107}

Considering the set of numbers for which the relation Prime-add is single-valued.

	3 Reasons:

 \hjust{\ \ \  (1) Prime-add often has singleton value. Worth isolating that case.}

 \hjust{\ \ \  (2) Restricted to this set, Prime-add would be a function.}

 \hjust{\ \ \  (3) Focus of attention: {\AM} recently worked on Prime-add.}

  Creating a new concept ``Uniquely-prime-addable.''
    ``Uniquely-prime-addable'' is a kind of Number. 
    x is Uniquely-prime-addable iff Prime-add(x) is a singleton.
    Will spend 10.0 seconds filling in examples of Uniquely-prime-addable.
    11 examples found. Here are some of them:  4 5 7 8 9 12 13.
    No obvious conjecture derived empirically.

  Will forget ``Uniquely-prime-addable numbers,'' if no Ties found in near future.
}

\tenpoint \parindent 19pt
\SSEC{Recapping the Example}


Let's once again eavesdrop  on a mathematician, as he  describes to a
colleague what {\AM} did.

This   example  was   preceded  by   the  momentous   discoveries  of
multiplication and division. Several interesting properites of  these
operations were  noticed.  
The  first task  which was illustrated  ({\strdotstr{Task\ 65}})
involves exploring the concept of ``divisors of a number'' (meaning all
positive integers which divide  evenly into the given  number). After
tiring of finding examples  of this relation, {\AM} investigates extreme
cases: that is, it wonders which  numbers have very few or very  many
divisors.

{\AM} thus  discovers Primes  in a  curious way.   Numbers with  0 or  1
divisor  are  essentially nonexistent,  so  they're not  found  to be
interesting. {\AM} notices that numbers  with 3 divisors always seem  to
be squares  of numbers  with 2  divisors (primes).   This  raises the
interestingness of several concepts, including primes.  Soon ({\strdotstr{Task
79}}), another  conjecture involving primes  is noticed: Many  numbers
seem to factor into primes. This causes a new relation to be defined,
which associates to a number x,  all prime factorizations of x.   The
first question {\AM}  asks about this  relation is ``is it  a function?.''
This  question  is the  full  statement of  the  unique factorization
conjecture: the fundamental theorem of arithmetic.  {\AM} recognized the
value of this relationship, and assigned it a high interestingness rating.

In a  similar manner, though with  lower hopes, it noticed  some more
relationships  involving primes, including Goldbach's conjecture.  {\AM}
quite correctly predicted that this would turn out to  be cute but of
no future use mathematically.

The  last activity  mentioned  ({\strdotstr{Task\ 107}})  shows  {\AM} examining  a
rather nonstandard concept: ``numbers which can be written as the  sum
of  a  pair  of  primes,  in  only  one   way.''    These  are  termed
``uniquely-prime-addable'' numbers.   It was mildly unfortunate that {\AM}
gave   up   on   this   concept   before   noticing   that   p+2   is
uniquely-prime-addable,  for any  prime number  p, and  that in  fact
these  are the only odd  uniquely-prime-addable numbers. The session
was  repeated once,  with  a  human  user telling  {\AM}  explicitly  to
continue  studying   this  concept.     {\AM}  did  in   fact  construct
``Uniquely-prime-addable-odd-numbers,''   and    then   notice    this
relationship. Here  we  see an  example of  unstable equilibrium:  if
pushed slightly this way, {\AM} will get very interested and spend a lot
of time working on this kind of number. Since it doesn't have all the
sophistication (\ie, compiled hindsight) that we have, it can't know
instantly
whether what it's doing will be fruitless.