⊗5↓_4a. Glossary of Math Terms_↓⊗*


Cardinality:  the concept  of "number".    Two sets  are of  the same
cardinality if they have the same number of elements.

Composition of two relations R and S: This is a new  relation denoted
R⊗7o⊗*S,  and  defined  as  R⊗7o⊗*S(x) =  R(S(x)).  So  R⊗7o⊗*S  maps
elements of  the domain of S into elements of the range of R.  Notice
that if R and S are both functions, then so is R⊗7o⊗*S. The intuitive
picture of this  process is to operate on x  with the relation S, and
⊗4then⊗* apply R to the results.

Function: an operation f which associates, to each element x of  some
set D,  an element f(x)  of some set  R. D and  R are the  domain and
range of f.  Notice that a function may be considered a single-valued
relation.

Integers: positive and negative whole numbers; i.e. ...,-2, -1, 0, 1,
2,...

Map: used  as a verb,  this word indicates  the action of  applying a
function  or a relation; e.g.,  we say that  ⊗4squaring⊗* maps 7 into
49.  Used as a noun, it is a synonym for function.

Mathematical concept: this  is taken to  mean all the  constructions,
definitions,  conjectures,  operations,  structures,  etc.    that  a
mathematician deals with. Some examples: Set-intersection, Sets,  The
unique factorization theorem, every entry listed in this glossary.

Mathematical  intuition: this  is  the mental  imagery  which can  be
brought  to  bear.   Typically,  we  transform  the  situation to  an
abstract, simplified one, manipulate  it there, and re-translate  the
results into the original notation.  For example, our intuition about
"ordering" may involve the image of marks on a yardstick. We can then
answer   questions   involving    ordering   rapidly,   using    this
representation.   Three  features of  the intuitive  image  should be
noted: (i) it is  typically fast and simple,  (ii) it is opaque,  one
cannot introspect  too  easily on  "why it  works", and  (iii) it  is
fallible, occasionally leading to wrong results.

Mathematical research:  The fundamental idea here is that mathematics
is an ⊗4empirical⊗*  science, just as much  as chemistry or  physics.
In  doing  research,  the  ultimate  goal is  the  creation  of  new,
interesting  theories, but  the techniques  used include  looking for
patterns in empirical data, inducing new  conjectures, modelling some
aspects of the real world, etc. Although the final product looks like
a smooth, formal  development, magically flowing  from postulates  to
lemmas to theorems, the actual research process involved untold blind
alleys,  rough guesses,  and  hard work.   (analogy:  The  process of
painting is rarely itself artistic.)

Mathematical theory: to qualify as a theory, we must have (i) a basis
of undefined primitive terms, (ii) definitions involving these, (iii)
axioms   involving  all   the  primitives  and   defined  terms  (iv)
conjectures  and  theorems relating  these  terms.    To  be  at  all
worthwhile, however, the theory must also meet the fuzzy requirements
that (v) there is some correspondence between the primitives and some
"real-world"   concepts,  between   the   axioms  and   some   "real"
relationships, and (vi)  some of the theorems are unexpected, hard to
prove, elegant, interesting, etc.

Natural numbers: non-negative integers; i.e., 0, 1, 2, 3,...

Ordering: the  concept of "before" and "after".  This distinguishes a
list from a bag  (multiset).  The formal  axioms for ordering  simply
state the obvious properties of the intuitive image of a list.

Prime numbers: natural  numbers which have  no divisors other  than 1
and themself; e.g., 17, but ⊗4not⊗* 15 (=3x5). Primes are interesting
because of the myriad times they crop up in diverse theorems  -- from
the Chinese  Remainder Theorem (solving systems  of linear congruence
equations), to  the Law of Quadratic Reciprocity, to Fermat's Theorem
(for all integers  n, for all  primes p, n↑p  is congruent to n  (mod
p)).  The "secret" of  their value lies in the fact that all integers
can be factored  ⊗4uniquely⊗* into  a set  of prime  divisors.   This
"Unique  Factorization  Theorem"  lets  us   reduce  questions  about
integers to questions about primes.

Relation: an operation which associates, for each element of some set
D, a set of elements  E = {e↓1, e↓2,...} of  some set R. D and R  are
the  domain and  range of  the  relation. For  example, the  realtion
"⊗6≤⊗*" associates  to 5 the set of numbers {5, 6, 7, 8,...} -- i.e.,
all integers which 5 is less than or equal to.   The domain and range
of this relation are the integers.

⊗5↓_4b. Glossary of AI Terms_↓⊗*


ACTORs: A modular form of representation,  useful for distributing of
the  task  of  ⊗4control⊗* among  several  components  in a  computer
program. Each ACTOR is a black box, with no parts or slots, but which
does have  some assertions (a  "contract") which  he must honor.   It
merely  responds to a fixed  set of messages,  by sending out certain
messages  of  his own.    These  are  delivered  via  a  bureaucracy.
Recursive sending is permitted.

BEINGs: A modular form of representation of knowledge as a collection
of   cooperating   experts.      Each    module   is   a   list    of
Question/Answering-program pairs, where the set of questions is fixed
for all  the Beings in the system. When any  Being has a question, he
broadcasts it to the entire system, and some Being who  recognizes it
will  take over  control  and try  to answer  it  by running  ⊗4his⊗*
appropriate  Answering-program. In the process  of running this, some
new questions may arise. Notice that Beings distribute responsibility
for control and for static  knowledge.  The advantages of having each
BEING composed of the same structure, the same names for its "slots",
are  (i)  efficient  communication  between  Beings,  and  (ii)  easy
creation of and "filling out" of brand new Beings.

Cooperating Knowledge Sources: Very often, in tackling a problem, one
receives some hints and some constraints from very different sources,
phrased  in  very  different  languages,  often addressing  different
representations of the problem.  For example, in trying understand  a
human speaker, our memory of the previous discussion and knowledge of
the  speaker may narrow down the possible  ⊗4meanings⊗* of what he is
saying. Our ears, of course, register the precise acoustic wave-forms
he  is  uttering.  Our  English  vocabulary forces  us  to  interpret
imperfect signals as real words.   Our eyes see his gestures and  his
lip movements,  and  give us  more information.  All these  different
sources of information  must be used, and yet they all are talking in
different "languages" to us.   The most  trivial solution is to  keep
all the sources  independent, and keep working until one  of them can
solve  the  problem all  by itself.   A  much  better solution  is to
transform all their babblings into one  canonical representation, one
single language. There are in  fact no more profound ideas around yet
on this "interfacing" problem.

FRAMEs: A modular representation of knowledge.  Each module is a list
of Feature/Value pairs. The ⊗4value⊗* represents a default assumption
which  can be relied  on until/unless  new information comes  in abut
that feature.  Each frame has whatever ⊗4features⊗*  (called "slots")
seem  appropriate.    Whenever  a situation  S  is  encountered,  the
frame(s)  for  S are  activated.   As  new information  rolls  in, it
replaces  the default  information  in  various slots.    Notice  the
emphasis on distributing static knowledge (⊗4data⊗*), not necessarily
control, in such a system.

Heterarchy: This term refers to the control structure of a computer
program. The typical hierarchical structure is one in which a function
calls a subroutine, which processes and then returns a value to that function.
A program is viewed as a tree structure, with lines indicating "calling".
Heterarchical structuring views the whole program as a collection
of equal partners, an unstructured set of functions. 
"Control" is viewed as a spotlight,
which can be flicked from one function to another. The functions can
affect who does or doesn't get control next, but there is no
guarantee who will get control, or that control will revert back to
some function which once had it. Aside from the lure of its democratic flavor,
it is clearly a natural way to represent cooperating knowledge modules.

Modular Representations of Knowledge  in AI Systems: (1)  Definition:
Knowledge is partitioned into packets (called modules, frames, units,
experts,   actors)   along  lines   of:   different  applicabilities,
expertise, purpose,  importance,  generality, etc.    Each packet  is
structurally similar to all the  rest.  (2) Advantages: By having the
knowledge discretized, pieces  can be  added and/or  removed with  no
trouble.   The  knowledge  of  the  system is  easily  inspected  and
analyzed.  The  structural similarity  yields  several  advantages: a
simple control  system  suffices  to  "run" all  the  knowledge,  the
modules  can intercommunicate  easily,  new modules  can be  inserted
without  knowing precisely "who else" is already  in the system.  (3)
Examples: Some modular schemes (and their  program incarnations) are:
Actors  (Plasma),  Frames, Beings  (PUP6),  Production Systems  (PSG,
Dendral, Mycin), Predicate  Calculus.  (4)  Relation to  "Cooperating
Knowledge Sources" Although  modular representation is a  natural way
to  implement  cooperating knowledge  sources, the  two  concepts are
distinct. For example, Hearsay uses opaque modules, which  do ⊗4not⊗*
have similar structures, who communicate  via a global blackboard. In
general, if the modules are to have non-standard structures, then the
inter-communication media  must be  a  simple scheme  (like a  global
assertional data base, a blackboard).


⊗5↓_5. Documentation_↓⊗*

.BEGIN INDENT 0,3,0 PREFACE 0

1. Thesis Proposal: SU-AI file SYS4[TLK,DBL]

2. This supplementary file: SU-AI file SUP[HPP,DBL]

3. The text of the tutorial: SU-AI file TUT[HPP,DBL]

4. The text of the planning talk: SU-AI file DET[HPP,DBL]

5. The system itself: SUMEX files <LENAT> TOP6, CON6, and UTIL6.
   To run: get into LISP, load <LENAT>L, follow instructions.

6. The use of BEINGS representation in AM is described in the paper:
   ⊗4Duplication of Human Actions by an Interacting Community of Knowledge Modules⊗*,
   Proceedings of the Third International Congress of Cybernetics and Systems,
   Bucharest, Roumania, August, 1975.

7. An English-like description of the heuristics for each facet of each
   concept can be perused as SU-AI file GIVEN[TLK,DBL].

.END