COMMENT ⊗   VALID 00014 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00003 00002	.NSECP(Overview)
C00004 00003	.SSEC(Three-page Summary of the Project)
C00006 00004	. SSSEC(Detour: Analysis of a discovery)
C00010 00005	. SSSEC(What AM does: Syntheses of discoveries)
C00018 00006	. SSSEC(Results)
C00024 00007	. SSSEC(Conclusions)
C00026 00008	.SSEC(Ways of viewing AM as some common process)
C00028 00009	. SSSEC(AM as Hill-climbing)
C00032 00010	. SSSEC(AM as Heuristic Search)
C00043 00011	. SSSEC(AM as a Mathematician)
C00049 00012	.SSEC(Fifteen-page Summary of the entire project)
C00051 00013	.SSEC(Guide to reading the remainder of the thesis)
C00053 00014	. SSSEC(Varied Readership of this thesis)
C00058 ENDMK
C⊗;
.NSECP(Overview)

.SSEC(Three-page Summary of the Project)

<<Edit this, making it follow the general outline of the thesis>

Scientists  often face  the difficult  task  of formulating  research
problems which  must be soluble yet nontrivial.   In any given branch
of science, it is usually  easier to tackle a specific given  problem
than  to   propose  interesting   yet  managable  new   questions  to
investigate.   For example, contrast ⊗4solving⊗* the Missionaries and
Cannibals problem with  the more ill-defined  reasoning which led  to
⊗4inventing⊗* it.

This   thesis  is  concerned   with  creative  theory   formation  in
mathematics: how to  propose interesting new  concepts and  plausible
hypotheses connecting them.  The experimental  vehicle of my research
is   a  computer   program  called  ⊗2AM⊗*   (for  ⊗2↓_A_↓⊗*pprentice
⊗2↓_M_↓⊗*athematician), which  carries  out some  of  the  activities
involved in  mathematical research: noticing simple  relationships in
empirical  data, formulating  new definitions  out of  existing ones,
deciding carefully what to explore next,
evaluating the overall worth of new concepts.

. SSSEC(Detour: Analysis of a discovery)

Before  discussing  how  to  ⊗4synthesize⊗*  a new  theory,  consider
briefly how to ⊗4analyze⊗* one, how to construct a plausible chain of
reasoning which terminates in a given discovery.   One can do this by
working  backwards,  by  reducing the  creative  act  to simpler  and
simpler creative acts.   For example, consider  the concept of  prime
numbers.  How  might one be led  to define such a  notion? Notice the
following plausible strategy:

.ONCE INDENT 9,9,9 SELECT 6

"If  f is a function which transforms  elements of A into elements of
B, and B is ordered, then consider just those members  of A which are
transformed  into  ⊗4extremal⊗*  elements  of  B.   This  set  is  an
interesting subset of A."

When f(x) means "divisors of x", and the ordering is "by length", this
heuristic says to consider  those numbers which have a  minimal$$ The
other  extreme, numbers with  a MAXIMAL  number of factors,  was also
proposed  by AM  as  worth  investigating.    This  led  AM  to  many
interesting questions, including one of its few 
"new-to-Mankind" mathematical results
so   far. This is discussed in detail in Appendix {[2]MAXDIV}.
$ number  of factors  -- that
is,  the primes.  So  this rule  actually ⊗4reduces⊗*  our  task from
"proposing the  concept  of prime  numbers"  to the  more  elementary
problems of "discovering ordering-by-length" and "inventing divisors-of".

But suppose  we know  this general  rule: ⊗6"If  f is  an interesting
function, consider its inverse."⊗* It reduces the task of discovering
divisors-of to  the  simpler  task of  discovering  multiplication$$
Plus noticing that multiplication is associative and commutative. $.
Eventually, this task reduces to the discovery of very basic notions,
like substitution, set-union, and equality.   To explain how a  given
researcher might  have made a  given discovery,  such an analysis  is
continued  until  that  inductive task  is  reduced  to "discovering"
notions which the researcher already knew, which were his conceptual primitives.

. SSSEC(What AM does: Syntheses of discoveries)

Suppose a  large collection of  these heuristics  has been  assembled
(e.g., by  analyzing a great  many discoveries, and writing  down new
heuristic  rules  whenever  necessary).   Instead  of  using  them to
⊗4explain⊗* how  a given  idea might  have evolved,  one can  imagine
starting from a basic  core of knowledge and "running" the heuristics
to ⊗4generate⊗* new concepts.

Such syntheses are precisely what AM does.  The program consists of a
large corpus primitive mathematical concepts, each with a few associated
heuristics$$
Situation/action rules which function as local "plausible move generators".
Some suggest tasks for the system to carry out, some suggest ways of
satisfying a given task, etc. $.
AM's activities all
serve  to expand  AM  itself,
to enlarge  upon  a  given  body  of  mathematical
knowledge.  To cope with the enormity of the potential "search space"
involved, AM  uses its  heuristics as  judgmental  criteria to  guide
development in  the most  promising direction.   It appears  that the
process  of inventing worthwhile new$$ Typically,  "new" means new to
AM, not to Mankind; and "worthwhile" can only be judged in hindsight.
$ concepts  can be  guided successfully using  a collection of  a few
hundred such heuristics.

Each concept is represented as a ⊗6Being⊗* [Lenat], a frame-like data
structure with  25 different  facets or  slots. The  types of  facets
include:   ⊗6Examples,  Definitions,  Generalizations,  Domain/Range,
Analogies,  Interestingness,⊗*  and many others.    The
⊗6Beings⊗* representation provides a convenient scheme for organizing
the  heuristics; for  example, the  following strategy fits  into the
⊗4Examples⊗* facet of the ⊗4Predicate⊗* concept: "If, empirically, 10
times as many elements  ⊗4fail⊗* some predicate P, as ⊗4satisfy⊗* it,
then some ⊗4generalization⊗*  (weakened version) of  P might be  more
interesting than P".   AM considers  this suggestion after  trying to
fill  in examples of  any predicate$$ In  fact, after  AM attempts to
find examples of SET-EQUALITY,  so few are  found that AM decides  to
generalize that predicate.   The result is a new  predicate which means
"Has-the-same-length-as" -- i.e., a rudimentary precursor to Numbers.  $.

AM is  initially given a collection of 115 core concepts, with only
a few facets filled in for each.  Its sole activity is to choose some
facet  of some  concept, and  fill in  that particular  slot.   In so
doing, new  notions  will  often  emerge.    Uninteresting  ones  are
forgotten, mildly interesting ones are kept as  parts of one facet of
one  concept,  and very  interesting  ones are  granted  full concept
status. Such new  ⊗6Beings⊗* have  dozens of blank  parts, hence  the
space of  possible actions  (blank slots to  fill in)  grows rapidly.
The  same  heuristics are  used both  to  suggest new  directions for
investigation, and to limit attention: both to grow and to prune.

. SSSEC(Results)

The particular  mathematical domains in  which AM operates  depend on
the  choice of initial concepts.   Currently, AM  begins with nothing
but a scanty  knowledge of  concepts which Piaget  might describe  as
⊗4prenumerical⊗*:  Sets, substitution,  operations, equality,  and so
on.     In  particular,   AM  is  not  told   anything  about  proof,
single-valued functions,  or numbers.   With this  basis, AM  quickly
discovered$$ "Discovering" a concept  means that (1) AM recognized it
as a distinguished entity (e.g.,  by formulating its definition)  and
also (2) AM decided it was worth investigating (either because of the
interesting way  it was formed, or  because of surprising preliminary
empirical results). $ elementary numerical concepts (corresponding to
those we  refer to as  natural numbers, multiplication,  factors, and
primes)  and  wandered  around in  the  domain  of  elementary number
theory.    Although  it  was  never  able  to  ⊗4prove⊗*  the  unique
factorization theorem, AM actually did ⊗4conjecture⊗* it.

AM was  not able to discover any  "new-to-Mankind" mathematics purely
on its own, but ⊗4has⊗* done so when working as a co-researcher  with
a human⊗A1⊗*.   AM noticed  simple new concepts  which mathematicians
had overlooked, but  AM by itself was not able to precisely formulate
and prove interesting  statements about those  concepts.  I  conclude
that a synergetic AM--human  combination can sometimes produce better
research than either could alone.

Everything that  AM does can be viewed as testing the underlying body
of heuristics.   Gradually, this knowledge becomes  better organized,
its implications clearer.   The resultant body of detailed heuristics
may be the  germ of  a more  efficient programme  for educating  math
students than the  current dogma$$ Currently, the educator  takes the
very best work any mathematician has ever done, polishes it until its
brilliance is  blinding, and  presents it  to the  student to  induce
upon. Many individuals (e.g., Knuth  and Polya) have pointed out this
blunder.    A  few  (e.g., Papert  at  MIT,  Adams  at  Stanford) are
experimenting  with   more   realistic  strategies   for   "teaching"
creativity.  $.

Another   benefit   of   actually   constructing   AM  is   that   of
⊗4experimentation⊗*: one can vary the  concepts AM starts with,  vary
the  heurisitics available,  etc.,  and  study  the effects  on  AM's
behavior.   Several  such experiments were  performed.   One involved
adding a couple dozen new concepts from an entirely new domain: plane
geometry.  AM busied  itself exploring elementary geometric concepts,
and  was almost as productive  there as in its  original domain.  New
concepts were defined, and new conjectures formulated$$ A  new -- but
bizarre -- result was obtained there: any angle (between 0 and 180↑o)
can be approximated to  within one degree, as  the sum of two  angles
drawn from the set α{0↑o, 1↑o, 2↑o, 3↑o, 5↑o, 7↑o, 11↑o,..., 179↑oα},
i.e., the set of angles of prime size.  If our culture and technology
were different, this might  have been an important, familiar  result.
Incidentally, much  sharper results and  much more general  ones were
also obtained. $. Other experiments indicated that AM was more robust
than anticipated;  it withstood  many kinds of  "de-tuning".   Others
demonstrated  the tremendous impact  that a  few key  concepts (e.g.,
Equality) had on AM's behavior.   Several more experiments have  been
planned for the near future.

. SSSEC(Conclusions)

AM is forced to judge ⊗4a priori⊗* the value  of each new concept, to
quickly lose interest  in concepts which aren't going to develop into
anything.  Often, such judgments can only be based on hindsight.  For
similar reasons,  AM has difficulty formulating  new heuristics which
are  relevant to the new  concepts it creates.   Heuristics are often
merely  compiled  hindsight.   While  AM's  "approach"  to  empirical
research may be used in other scientific domains, the main limitation
(reliance on hindsight) will probably  recur.  This prevents AM  from
progressing indefinitely far on its own.

Before this  ultimate limitation  was reached,  AM demonstrated  that
selected  aspects  of creative  discovery  in  elementary mathematics
could  be adequately  represented  as  a  heuristic  search  process.
Actually constructing a computer model  of this activity has provided
an  experimental  vehicle  for  studying  the  dynamics  of plausible
empirical inference.

.SSEC(Ways of viewing AM as some common process)

This section will provide  a few metaphors: some hints  for squeezing
AM  into paradigms the reader  might be familiar with.   For example,
the existence of heuristics in AM is quite similar to the presence of
domain-specific information in any knowledge-based system.

Consider  assumptions,  axioms,  definitions,   and  theorems  to  be
syntactic  rules  for the  language  that we  call  Mathematics. Thus
theorem-proving, and the whole  of textbook mathematics, is  a purely
syntactic process.  Then  the heuristic rules used by a mathematician
(and by AM)  would correspond  to the  semantic knowledge  associated
with these more formal methods.

Just   as   one   can   upgrade   natural-language-understanding   by
encorporating  semantic knowledge,  AM is only  as successful  as the
heuristics it knows.

Three more ways of "viewing"  AM as something else will be  provided:
(i) AM as a hill-climber
[designed for computer science readers],
(ii)  AM as a heuristic search program [designed for A.I. readers], and
(iii) AM as a mathematician [for readers with a good math background].

. SSSEC(AM as Hill-climbing)

Let's  draw an  analogy between  developing  mathematics and  another
process with which  you're probably more familiar: hill-climbing.  We
may visualize AM  as exploring a space  using an evaluation  function
which imparts to it a topography.

Consider  AM's core  of very  simple knowledge.   By  compounding the
known  concepts and methods,  AM can extend this  foundation a little
wherever  it wishes.    The  incredible variety  of  alternatives  to
investigate  includes all known  mathematics, much  trivia, countless
deadends, and so on.  The  only "successful" paths near the core  are
the  narrow  ribbons  of  known  mathematics   (perhaps  with  a  few
undiscovered other slivers).

How  can  AM  walk  through this  immense  space,  with  any hope  of
following  the   few,   slender   branches   of   already-established
mathematics  (or some  equally  successful new  fields)?  AM must  do
hill-climbing: As  new concepts are formed, decide how promising they
are, always  explore the  currently most-promising  new concept.  The
evaluation  function is quite  nontrivial, and  this research  may be
viewed  as  an  attempt  to  study  and  explain  and  duplicate  the
judgmental  criteria  people  employ.   Attempts  at  codifying  such
"mysterious"  emotive   forces  as  intuition,  aesthetics,  utility,
richness, interestingness, relevance...  indicated that  a large  but
not unmanagable collection of heuristic rules should suffice.

The important  visualization to make  is that with  proper evaluation
criteria,  AM's planar  mass of interrelated  concepts is transformed
into a  breath-taking  relief map:  the  known lines  of  development
become mountain ranges, soaring  above the vast flat plains of trivia
and inconsistency below.

Occasionally an isolated  hill is discovered  near the core;$$  E.g.,
Conway's  numbers, as  described  in  Knuth's ↓_Surreal  Numbers_↓  $
certainly  whole ranges lie  undiscovered for long  periods of time$$
E.g., non-Euclidean  geometries  $, and  the  terrrain far  from  the
initial core is not yet explored at all.

. SSSEC(AM as Heuristic Search)

As the title of this section -- and this thesis -- proclaims, AM is a
kind  of  "heuristic search"  program.   That  must mean  that  AM is
exploring  a  particular  "space,"  using  some  informal  evaluation
criteria to guide it.

The flavor  of search  which is  used here  is that  of progressively
enlarging  a tree. Heuristics  are used  to decide which  node of the
tree to expand next, and to produce from that  node a few interesting
successor nodes. To do mathematical research well, I claim that it is
necessary and  sufficent  to  have good  methods  for  proposing  new
concepts from  existing ones, and  for deciding how  interesting each
candidate (concept, node) is.

AM explores mathematics  by selectively enlarging itself: AM ⊗4is⊗* a
body of mathematical knowledge (concepts, plus the wisdom to use them
effectively).   To see this, we  must explain what the  nodes of AM's
search  space  are,  what  the  operators  or  links  are, where  the
heuristic  information  comes into  play,  and  what  the  evaluation
function is.

AM's  space can  be  considered to  consist  of all  nodes which  are
consistent, partially-filled-in concepts. That is, a primitive "legal
move" for AM would be  to (i) enlarge some facet of  some concept, or
(ii)  create a new, partially-complete  concept. Consider momentarily
the size of this space.  Since there is no constraint on what the new
concepts  can  be, and  no  informal  knowledge for  quickly  finding
entries  for a desired  facet, a blind "legal-move"  program would go
nowhere --  slowly!   One shouldn't  even  call the  activity such  a
program would be doing "math research."

The heuristic  rules are used as little  "plausible move generators".
They suggest what  facet of what  concept to enlarge  next, and  they
suggest specific new concepts to create. The only activities which AM
will  consider doing  are those  which have  been motivated  for some
specific good reason. The validity  of that last statement of  course
depends  on  the  validity  of  the  heuristic  rules.  This  is  the
programmer's responsibility.

AM  has  a definite  algorithm  for rating  the nodes  of  its space.
Namely,  the   heuristic   rules   provide  enough   information   to
meaningfully  order the  tasks on  the agenda  list.   Yet AM  has no
specific goal criteria: it  can't quit just  because a dynamite  task
has been proposed. AM goes on forever$$ Technically, forever is about
100,000 list cells and a couple cpu hours. $.

Consider  Nilsson's  description  of  depth-first  searching, and  of
breadth-first searching. He has us  maintain a list of "open"  nodes.
Repeatedly, he  plucks the top  one and  expands it. In  the process,
some  new  nodes may  be  added  to the  Open  list. In  the  case of
depth-first searching, they are  added at the top;  for breadth-first
search, they  must be added at  the bottom. For  heuristic search, or
"best-first" search, they are evaluated in some numeric way, and then
"merged" into the already-sorted list of Open nodes.

.ONCE TURN ON "{}"

This process is very  similar to the ⊗4agenda⊗* mechanism  AM uses to
manage  its  search. This  will  be discussed  in  detail  in Chapter
{[2]AGENDA}.  The agenda is a  list of plausible tasks for AM to  do,
plus supporting reasons for each task.   When a task is suggested for
some  reason, it  is added to  the agenda.   A task  may be suggested
several times,  for different reasons.   A  global priority value  is
assigned to  each task, based on  the combined value  of its reasons.
The control structure  of AM is  simply to select  the task with  the
highest prioirty,  execute it, and  select a new  one.   The "agenda"
appears  to  be a  very  well-suited data  structure  for  managing a
"best-first" search process.

Similar control  structures were  used in  Dendral$$ The  "Predictor"
part of  DENDRAL. See [MI4  ref.]. $ and  KRL [reference].   The main
difference  is  that in  AM,  symbolic  reasons are  used  (albeit in
trivial token-like  ways) to  decide whether --  and how  much --  to
boost the priority of a task when it is suggested again.


There are  several difficulties and anomalies in  forcing AM into the
heuristic search paradigm.   AM's  heuristics are  used as  plausible
move generators; if  they are all  removed, AM would have  nothing to
do.  In tradtional  heuristic searches,  the heuristics  are separate
from a "legal  move generator", hence could  all be eliminated:  they
merely help constrain a single generator.

Another anomaly  is that the operators  which AM uses to  enlarge and
explore  the space  of concepts are  themselves mathematical concepts
(e.g., some  heuristic rules  result in  the creation  of new  rules;
"Compose" is  both a  concept and an  operation which results  in new
concepts).  Thus  AM should be  viewed as a  mass of knowledge  which
enlarges ⊗4itself⊗*  repeatedly.   As  far as  I  know, all  previous
computer  programs  kept  the  information  they  "discovered"  quite
separate from the knowledge they used to make discoveries$$ Of course
this  is  typically because  the  two  kinds  of knowledge  are  very
different:  For  a  chess-player,  the  first  kind  is  "good  board
positions," and the  second is "strategies  for making a good  move."
So-called "learning programs"  typically learn one specific task, not
how to become better learners. etc. $.

Let me reemphasize  that AM  has no well-defined  target concepts  or
target relationships.  Rather, its "goal criteria" -- its sole aim --
is  to  maximize  the  interestingness  level  of the  activities  it
performs, the priority ratings  of the top tasks  on the agenda.   We
don't care what AM does -- or misses -- so long as it spends its time
on plausible tasks.  There is no fixed set of theorems that AM should
discover (AM is thus not like a typical ⊗4problem-solver⊗*), no fixed
set  of  traps  AM should  avoid  (AM  is  thus very  different  from
⊗4game-players⊗* like chess programs).

For example, there is  no stigma attached to  the fact that AM  never
discovered  real numbers$$  There  are many  "nice"  things which  AM
didn't -- and can't -- do: e.g., devising ↓_geometric_↓ concepts from
its initial simple  set-theoretic knowledge.   See the discussion  of
the limitations of AM,  Section {[2]DIFSECNUM}.{[2]DIFSSECNUM}. $; it
was  rather  surprising  that  AM  managed  to  discover  ⊗4natural⊗*
numbers!    Even  if  it  hadn't  done  that,  it   would  have  been
acceptable$$ Acceptable  to whom? Is there  really a domain-invariant
criterion  for  judging  the  quality   of  AM's  actions?  See   the
discussions in Section {[2]EVALU}.1. $ if AM  had simply gone off and
developed ideas in set theory.

. SSSEC(AM as a Mathematician)

Before diving into the  depths of AM, 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 relations (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
naively established conjectures  like "a set can never be a member of
itself", and procedures for making chains of new sets ("insert  a set
into itself").   No sophisticated set  theory (e.g., diagonalization)
was ever done.

After this  initial period of exploration, AM decided that "equality"
was  worth   generalizing,  and  thereby   discovered  the   relation
"same-size-as".  "Natural numbers"  were based on this, and soon most
simple arithmetic operations were defined. Since addition arose as an
analog to union, and multiplication as an analog to cross-product, it
came  as quite  a surprise  when  AM noticed  that they  were related
(namely, N+N=2xN).   AM later  re-discovered multiplication in  three
other ways.   One of  these was as repeated  addition. Exponentiation
was defined as repeated multiplication. Unfortunately, AM never found
any of its obvious properties, hence lost all interest in it.

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. Perfect squares and perfect foruth-powers were isolated.

.ONCE TURN ON "{}"

The associativity and commutativity of multiplication indicated  that
it could accept  a BAG (a multiset) of numbers  as its argument. This
led to the notion of factoring a 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.   Motivated by AM,  an explicit characterization  of these
numbers  was  proposed,  which appears  to  be  "new-to-Mankind".   A
similar (but slightly different) result was later located  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  Appendix
{[2]MAXDIV}.  $ 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  mastered, progress  in number  theory was
arrested.

.SSEC(Fifteen-page Summary of the entire project)

<<This section is not written yet. Sorry. >

<potential organization: mirror the overall organization of the thesis itself>

Include the following points on Motivation (why is this worthwhile?):

.B

	Inherent interest of getting a handle on the task (sci. creativity)
		Personal belief that discovery can be (ought to be) demystified
		Potential for learning, from the system, more about the process 
			of sci. concept formation, thy. formation, chance discovery
			(do experiments on the implementations: eg, vary AM's heurs)
	Potential usefulness of the implementations themselves (including AM)
		Aids to research; i.e., ultimately: new discoveries.
		Potential to education: like Mycin, extract heurs. and teach them
	All the usual bad reasons:
		"Look ma, no hands" + maternal drives + ego + thesis drives +... 
	Historical: 
		Need task with no specific goal, to test BEINGs ideas.
		Disenchantment with theorem-provers that plod along, in contrast
			to the processes which my model of math demands: intu, need,
	                aesth., multiple reprs, proposing vs proving, fixed task.
	
.E

.SSEC(Guide to reading the remainder of the thesis)

<<This guide is not written yet. Sorry. >

.B

	i) Overall organization of the thesis
	ii) Plans for what to read (and in what order), depending on your interests
		Plan for those interested in the AI ideas
		Plan for those interested in the systems ideas
		Plan for those interested in mathematics
	iii) Pre-requisites and how to satisfy them, for each chapter
		For those with little pure mathematics in their background
		For those with little computer science background
		For computer scientists with little contact to AI before
	    <either organized by "type" of reader, or by chapter/section>

.E

. SSSEC(Varied Readership of this thesis)

<<Should this ssection -- and this A/C/M scheme -- be used or not?>

⊗7¬(A∨C∨M)⊗*

This thesis  -- and its  readers --  must come to  grips with a  very
interdisciplinary  problem.   For the reader  whose background  is in
Artificial  Intelligence,  most  of  the  system's  actions   --  the
"mathematics" it does  -- may seem inherently  uninteresting. For the
mathematician,  the  word "LISP"  signifies  nothing beyond  a speech
impediment  (to Artificial  Intelligence  types  it also  connotes  a
programming impediment). If I  don't describe "LISP" the first time I
mention it, a large fraction of potential readers will never  realize
that potential. If I ⊗4do⊗* stop to  describe LISP, the other readers
will  be bored.   The standard solutions  are to  sacrifice one fixed
community in favor of the other, or to be entertaining enough to keep
both of them around. In this work, a third alternative will be taken.
Sections  will be  tagged with  descriptors like  "⊗7M⊗*" (indicating
that the section will be of interest to those who enjoy mathematics),
or "⊗7¬A⊗*" (the  section will be a waste of  time for those familiar
with Artificial Intelligence). The labels will consist of the letters
⊗7A⊗* (for  ↓_A_↓I),  ⊗7M⊗* (for  ↓_M_↓ath), and  ⊗7C⊗* (for  general
↓_C_↓omputer  science), connected by standard  logical symbols: ⊗7¬⊗*
(negation), ⊗7∧⊗* (and), ⊗7∨⊗* (or).  For example, since  the current
paragraph is  labelled ⊗7¬(A∨C∨M)⊗*, the  reader can assume it  is of
little interest to anyone.

.ONCE TURN ON "{}"

In addition,  there are two glossaries of terms. Appendix {[2]GLOS}.1
contains capsule descriptions of about 100  mathemtical terms, ideas,
notations,  and  jokes. Appendix  {[2]GLOS}.2  renders the  analogous
service for  Artificial  Intelligence  jargon  and  computer  science
concepts.