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.ASEC(Glossary of Technical Terms)
The "jargon" of a field facilitates communication among practitioners
of that field, but it too often excludes novices. I have tried to
soften the impact of each "buzz-word" when it was first used, but the
reader may need to frequently refresh his memory about the meanings
of certain terms.
This glossary is divided into two sections. The first contains
primarily mathematics terms, strangely biassed because it just covers
what is referenced in this thesis. The second glossary, of computer
science and Artificial Intelligence terms, suffers from the same
tunnel vision. They may suffice for reading this document, but they
are certainly ⊗4not⊗* meant to be used for more general purposes.
. ASSEC(Glossary of Math Terms)
.TURN ON "↑↓_" TURN OFF "{}" SELECT 1
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 special kind
of relation.
For a ⊗4relation⊗* f (on DxR) to be called a ⊗4function⊗*, f must satisfy two
important constraints: (i) it must be always-defined on its domain; that is,
for all domain elements xεD, f(x) must exist. (ii) f must be single-valued;
that is, f(x) must be a singleton.
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.
. ASSEC(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).