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sn#163084 filedate 1975-10-07 generic text, type C, neo UTF8
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C00001 00001
C00002 00002 .NSEC(OVERVIEW)
C00010 00003 .NSEC(IDEAS)
C00016 00004 .SSEC(A Proposed System)
C00020 00005 .SSEC(The ENVIRONMENT)
C00038 00006 .NSEC(INITIAL KNOWLEDGE and REPRESENTATION)
C00043 00007 .SSEC(Initial Knowledge: Level 2)
C00051 00008 .SSEC(Representation: Level 3)
C00057 00009 We have dwelt on BEINGs so much, the reader is now entitled to hear about the
C00078 00010 .SSEC(Initial Knowledge: Level 3)
C00079 00011 .NSEC(COMMUNICATION)
C00092 ENDMK
C⊗;
.NSEC(OVERVIEW)
.SSEC(SUMMARY)
The methods of mathematical creativity are being studied.
A taxonomy of theory formation and of elementary mathematics is developed, then
embodied in a programmed system able to do simple research, form interesting
"mini-theories" and study their consequences.
The fundamental organizational unit is the ⊗4BEING⊗*, abbreviated β. This is
merely a collection of knowledge about a certain topic, organized as the answers to a
fixed set of a couple dozen questions about that topic. In answering a query, one
individual piece of knowledge (a part of a Being) might have to call on several others.
The control is implicit in the collection of Beings which exist: each β has a Recognition
part, the answer to "Are you relevant to this situation...?", whose task is to determine
when to seize control and when to yield it.
One unusual feature of the system will be a powerful "intuitive" ability
to analogize with part of the real world. The system may perform experiments on
this simulated Nature, and receive valid results, but the actual code which
represents the environment is ⊗4opaque⊗*.
⊗7For example, a model of a seesaw might exist, and the
system could play around at varying the weights on each side and their distance from
the fulcrum, and the seesaw function would explain which side sank and how fast.
This might be useful in getting an intuition about multiplication, substitution,
or symmetry.⊗*
The initial knowledge in the system will consist of (i) specific facts about
mathematics, reasoning, programming, and communication, (ii) strategies
for filling out parts of incomplete β's, (iii) opaque functions
which simulate parts of Nature, and (iv) opaque judgment criteria
for aesthetics, interest, utility, complexity, etc.
The specific facts are
organized into 4 families of Beings; each family initially has about 35 β's, and
each β has about 20 parts. The families are:
Static (eg, sets), Active (eg, relation),
Static-Meta (eg, analogy), and Active-Meta (eg, prove).
For uniformity, the strategies form a fifth family of β's, called Archetypical β's.
⊗7A strategy BEING is simply a collection of facts for
dealing with a particular
type of BEING part; the "Examples" β contains suggestions for filling in the
Examples part of any BEING.⊗*
The quantity of this corpus appears large (about 3000 β-parts to
encode, each as a little LISP program), and it is of some interest to hope that
the very same techniques which lead to discovering new mathematical knowledge
later on might be able to "grow" this knowledge base from a much smaller core --
say a collection of 100 β's with only a few parts filled in for each.
The first activity of such a system, then, would be ⊗4contemplative⊗*: the
interaction with the user would be minimal.
General strategies would interact with observations, and
new concrete facts about its world would emerge, along with some new
specific tactics.
The system would also combine its intuitions to form plausible conjectures and,
where all terms have formal definitions, try to prove them.
The activities in this period are
universal, not limited to any single
domain of mathematics.
The user is considered slow and dangerously contradicatory, hence not a good channel
to obtain data in general.
But as the known information swells, the need for guidance also grows.
At some point the
system may simply be swamped by a multitude of equally-mediocre alternatives to
investigate.
It will then (abeit reluctantly) request direction from
a human user, in what is to be an ⊗4assimilative⊗* phase.
These teachings should be the
core definitions of a specific field, and of course should be based on what is
already mastered. The first experiences could be in set theory, Boolean algebra,
abstract algebra, logic, or
arithmetic. This will probably be the level finally attained by the actual system.
One higher mode of interaction is conceivable: that of a colleague in research.
In conjunction with a
human adviser, the system would propose and explore interesting new relationships,
decide which creations to name, explore the intuitive meanings of statements, etc.
Hopefully, the reader has balked, complaining that this sounds just like the earlier
phases. In fact, the system will not ring a bell and suddenly switch its activities;
it has no way of knowing that its discovery of PLUS is not new to Mankind.
The driving and pruning forces in all phases are the same: use
aesthetics and utility judgments to fill out parts of incomplete BEINGs.
⊗7If the guidance of the human turns out to be
important, however, then it will come as no surprise if
the flavor of the interactions changes as the system enters a realm unfamiliar
to the user.⊗*
.NSEC(IDEAS)
Throughout all of science, one of the most important issues is that of
theory formation: how to extend, when to define, what to examine next,
how to recognize potentially related concepts, how to tie such concepts
together productively,
how to use intuition, how to choose, when to give up and try another
approach. These questions are difficult to answer precisely, even in a
single domain. Problems with natural language, with experimental
apparatus, and
with subjects which are complex yet poorly-structured,
all becloud the answers. By restricting the domain of attention to
⊗4mathematics⊗*, we hope to avoid these difficulties.
A ⊗4solution⊗* to this task would mean successfully
accounting for the ⊗4driving⊗* and the ⊗4pruning⊗* forces which
result in interesting
mathematical theories being developed. Success
could be measured in operational terms, by applying these forces to
various domains of mathematics, and comparing the results to what is
already known in those fields.
The ideas explored here are that:
(i) These forces are (in decreasing order of importance) aesthetics/interestingness,
intuition, utility, analogy, inductive inference (based on empirical evidence),
and deductive inference (formal methods).
(ii) Each of these forces is useful both in generating new conjectures, and
in assessing their acceptability.
(iii) If the essence of these ideas can be factored out into an explicit set
(of rules, predicates, BEINGs, programs...), then they can be used to
develop almost any branch of mathematics, at almost any level.
(iv) A protocol was taken, and indicates that the researcher must have a very
good set of strategies, organize them carefully, and use them wisely
to avoid getting bogged down in barren
pursuits. Some of this wisdom must pertain to precisely what is to be
remembered/recorded: a surfeit is bewildering, a shortage dangerous.
(v) Each mathematical concept should be represented in several ways,
including declarative, operational, exemplary (especially boundary
examples), and intuitive.
(vi) A large foundation of intuition, spanning several mathematical and real-
world concepts, is prerequisite to sophisticated behavior in ⊗4any⊗*
branch of mathematics. It is not "cheating" to demand some intuitive
concept of sets, before studying number theory, nor to demand some
intuitive understanding of counting before studying set theory, provided the
intuition is ⊗4opaque⊗* (can be used but not inspected in detail)
and fallible.
The more serious attack on the reliance upon divinely-provided
intuitive abilities is
that the creators might stack the deck: might contrive just the right intuitions to
drive the worker toward making the "proper" discoveries. The rebuttal is two-pronged:
first, one must assume the integrity of the creators; they must strive not
to anticipate the precise uses that will be made of the intuition functions. Second,
regardless of how contrived it was, if a small set of intutition models were found
which are sufficient to drive a researcher to disocver a significant part of
mathematics, that alone would be an interesting discovery
(educators would like to ensure
that children understood this core of images, for example).
(vii) The vast amount of necessary initial knowledge can be
generated from a much smaller
core of intuition and definite facts, using the same collection of
strategies and wisdom which also drive the discovery and the development
(those outlined above in (i)-(iv)).
(viii) The more basic the initial core concepts, the more chance there is that the
research will go off in directions different from humans, the more
chance it will be a waste of time, and the more valid the test of the search-pruning
forces.
.SKIP 3
.SSEC(A Proposed System)
Let us consider now what would be the characteristics of
a man-machine system which could be used experimentally.
The system would have about a hundred packets of information, each of which deals
with a small concept related to the foundations of mathematics, techniques for
research, etc. Inside each packet is an organized cluster of
specific facts, intuition, strategies, knowledge of how to
use the facts and the strategies, and an ability to estimate
the interest of the packet's topic and its sureity.
Each such knowledge module will be called a ⊗4BEING⊗*, abbreviated β, and each
unit of its contents will be called a ⊗4part⊗*.
The system would think to itself awhile, producing primarily intuitive "universal"
relationships. Since these activities don't utilize any alien authority,
this ⊗4contemplative⊗* stage can be programmed and run
even before any natural communication system is designed.
The overall control flow would be a series of Complete(P,B) calls, in which some
part P of some β B would be worked on, filled out more, etc.
The driving/pruning forces would each time select the next (P,B) pair.
During the course of such
completions, new β's might be called for (split off rich parts of already-exisiting
β's). One huge savings for the creators would be that the system should be able to
fill in examples of each β itself; much of this phase will in fact be doing just
that. Many mini-theorems will arise as a result of filling out examples of
Relations, Compositions, Conjectures, Theorems, etc.
Eventually, the system's model of the user would indicate that his
guidance, though slow and errorful, was preferable to continue this wandering
development. The system might ask for specific information relating to the
concepts it had discovered the best intutive "theorems" about, or might simply
request tutoring in any domain of the user's choosing.
The human user's
first major task would be to input a body of concepts about a specific domain
(for each concept, he should provide definitions, examples, intuitive pictures,
etc.) Then the system will begin exploring that domain, using its
(hopefully universal) body of mathematical strategies. Occasionally, the
user may interact with the system. Occasionally, the system may do
something interesting. The following ideas are fairly concrete, dealing
with such a programmed, runnable system.
.SSEC(The ENVIRONMENT)
.QP2←PAGE
COMPLETE(P,B) means fill in material in part P of BEING B.
@21. Locate P and B.⊗*
If P is unknown but B is known, ask B.ORDERING
and up↑*(B).ORDERING. Also,
there may be special information
stored in some part(s) of B earlier, by other BEINGs, which make them more or less
promising to work on now.
[ up↑*(B).P means the set of BEINGs named P, B.P,
(B.Ties.Up).P,
((B.Ties.Up).Ties.Up).P,
etc. ]
If B is unknown but P is known, ask P and ask each β about interest of filling in P.
Each β runs a quick test to see if it is worth doing a detailed examination.
Sometimes the family of B will be known (or at least constrained).
If neither is known, each β must see how rele. it is; the winner decides on P.
If there is more than one β tied for top recognition, then place the results
in order using the environment function ORD, which examines the Worth components
of each, and by using the value of the most promising part to work on next for each
BEING. The frequent access to the (best part, value) pair for each BEING means that
its calculation should be quick; in general, each β will recompute it only when new
info. is added to some part, or at rare intervals otherwise.
After ranking this list, chop it off at the first big break in values, and print it out
to the user to inspect. Pause WAIT seconds, then commence work on the
first in the list.
WAIT is a parameter set by the user initially. ⊗7(0 would mean go on unless user
interrupts you, infinity would mean always wait for user's reply, etc.)⊗*
When you finish, don't throw the list away until after the
next B is chosen, since the list might immediately need to be recomputed!
If the user
doesn't like the choice you've made, he can interrupt and switch you over.
A similar process occurs if P is unknown, (except the list is never saved).
@22. Collect pointers to helpful information: ⊗*
Create a (partialy ordered) plan for dealing with part P of BEING B (abbreviated B.P).
This includes the P.FILLIN part, and in fact any existing up↑*(B).P.FILLIN, and
also some use of the representation, defn, views, dom/range parts of the P BEING.
Consult ALGORITHMS and FILLIN parts of B and all upward-tied β's to B.
@23. Decide what must be done now⊗*;
which of the above pieces of information is "best". Tag it as having been tried.
If it is precisely = one currently active goal, then forget it and go to 3.
@24. Carry out the step.⊗* (Evaluate the interest of any new BEING when it is created)
Notice that the step might in turn call for accessing and (rarely) filling
in parts of other BEINGs. This activity will be standard heirarchical calling.
As parts of other BEINGs are modified, update their (best part, value) estimate.
@25. When done, update.⊗*
Update statistics in B, P, and current situation. (worth and recog parts)
If we are through dealing with B.P (because of higher interest entity ∃,
or because the part is filled in enough for now) goto 1; else goto 3.
If you stop because of higher interest entity, save the plan for P.B inside P.B.
.BEGIN W(1) NARROW 5,0
ACCESS(K,P,B) means access pieces of knowledge K from part P of BEING B.
1. Locate each argument
Typically given K. Find P' by asking archetypes, B' by asking all BEINGs.
By iterating through this loop, the sets P' and B' will become singletons.
As they become smaller, more individualized effort can be spent on distinguishing the choice.
2. Interpret the material in part P of BEING B.
Use the representation part of P.
3. Match K to this pattern, and try to extract it directly.
Often this will entail evalling or applying B.P.
Evaluation is viewed as just one technique for processing a clump of knowledge, B.P,
and extracting the precise bit K which is desired.
4. If the accession fails, consider P.VIEWS, consider setting up a message, consider
giving up. Let the interest of the current goal (activation energy) be your guide.
CURRENT SITUATION is a vector of weights and features of the recent behavior of the system.
.FILL
The Environment also maintains a list of records
and statistics of the recent past activities, in a structure called CS,
for "Current Situation".
Each Recognition grouping part is prefaced by a vector of numbers which are
dot-multiplied into CS, to produce a rapid rough guess of relevance.
Only the best performers are examined more closely for relevance.
The representation of each CS component is (identification info, motivation,
safety, interest, work done so far on it, final result or outlook). The
actual components might be:
.NOFILL
Recent Accesses. For each, save (B, P, contents of subpart used).
Recent Fillins. Save (B, P, old contents which were altered).
Current Hierarchical History Stack. Save (B, P, why).
Recent Top-level B,P pairs.
A couple significant recent but not current hierarchical (B,P,why) records.
A backward-sorted list of the most interesting but currently-deferred (B,P) fillins.
A few recent or collossal fiascos (B, P, what, why this was a huge waste).
ORD(B,C) Which of the recognition-tied BEINGs B,C is potentially more worthwhile?
.FILL
This simple ordering function will probably examine the Worth vectors, perhaps
involving the sum of weighted factors, perhaps even cross-terms such as
(probability of success)*(interest rating).
.SELECT 6; NOFILL; NARROW 3,0
PLAUSIBILITY(z) How believable is z? INTEREST(z) How interesting is z?
each statement has a probability weight attached to it, the degree of belief
this number is a fn. of a list of justifications
Polya's plausibility axioms and rules of inference
if there are several alternate justifs., it is more plausible
if some consequences are verified, it is more plaus.
if an analogous prop. is verified, it is more plaus.
if the consequences of analogue are verif., it is slightly more plaus.
the converses of the above also hold
believe in those things with high enough prob. of belief (rely on them)
this level should fluctuate just above the point of belief in contradictions
the higher the prob., the higher the reliability
the amt. one bets should be prop. to the reliability
the interest increases as the odds do
Zadeh: p(∧) is min, p(⊗6∨⊗*) is max, p(¬) is 1-.
Hintikka's formulae (λ, αα)
Carnap's formulas (λ)
p=1 iff both the start and the methods are certain ←← truth
p=0 iff both start is false and method is false-preserving ←← falsity
p is higher as the plausibility is higher, and as the interest is lower
if ∃ several alternative plaus. justifs., p is higher
don't update p value unless you have to
update p values of contradictory props.
update p values of new props
maybe update p value if it is a reason for a new prop
empiricism, experiment, random sampling, statistics
true ideas will be "verified" in (consistent with) any and all experiments
false ideas may only have a single exceptional case
a single exception makes a universal idea false
nature is fair, uniform, nice, regular; coincidences have meaning
more plaus. the more cases verified
more plaus. the more diff. types of cases verified
central tendency (mean, mode, median)
standard deviation, normal distribution
other distributions (binomial, Poisson, flat, bimodal)
statistical formulae for significance of hypothesis
regularity, order, form, arrangement
economy of description means regularity exists
aesthetic desc (ana. to known descs. elsewhere)
each part of desc. is organized regularly
the parts are related regularly
Below, αα means ⊗4increases with increasing⊗* (proportionality), and
αα↑-↑1 means ⊗4decreases with increasing⊗* (inversely proportional).
Perhaps one should distribute these morsels among the various concerned β's:
Completeness of an analogy αα safety of using it for prediction
Completeness of an analogy αα↑-↑1 how interesting it is
How expected a relationship is αα↑-↑1 how interesting it is
How intuitive a conjec/relationship is αα↑-↑1 how interesting it is
How intuitive a conjec/relationship is αα how certain/safe it is
How superficial something is αα how intuitive it is
How superficial something is αα how certain it is
How superficial something is αα↑-↑1 how interesting it is
Perhaps included here should be universally applicable algorithms for applying these rules
to choosing the best strategies, as a function of the situation.
One crude estimate of interest level is the interest component of the current β's
Modify this estimate in close cases using the above relations
Generally, choose the most specific strategies possible
If the estimated value of applying one of these falls too low, try a more general one
Rework the current β slightly, if that enables a much more specific strategy to be used
Locate specific concepts which partially instantiate general strategies
The more specific new strategies are associated with the specific info. used
Once chosen, use the strategies on the most promising specific information
If a strat. falters: Collect the names of the specific, needed but blank (sub)parts
Each such absence lowers int. and raises cost, and may cause switch to new strategy
If too costly, low int, store pointer to partial results in blank parts
The partial results maintain set of still-blank needed parts
Competing goals: On the one hand, desire to maximize certainty,
safety, complete analogies, advance the level of intuition.
On the other hand, desire to maximize interestingness, find poss. and poten. interesting ana.
find unexpected, nonsuperficial, and unintuitive relationships.
If an entity is used frequently, it should be made efficient.
Conversely, try to use efficient entities over nearly
equivalent (w.r.t. given purpose) but inefficient ones.
If an entity is formally justified but resists intuitive comprehension, its use is
dangerous but probably very interesting; ibid for intuitive but unprovable.
Resolve choices in favor of aesthetic superiority
Maximize net behavior
Maximize desired effects
In this case, prefer hi interest over hi safety.
Generally preferred to the folowing case.
Minimize costs, conserve resources
In this case, prefer safety to interest.
Locate the most inefficient, highest-usage entity, and improve or replace it
Use: If time/space become a problem, worry about conservation until this relaxes.
.END
.NSEC(INITIAL KNOWLEDGE and REPRESENTATION)
This section proposes a corpus of information, some of which will be carefully
constructed, and all of which should
be present in the system before the user approaches it.
This presentation will be repeated at several levels of detail, so that
the reader will obtain a global view before going into detail.
The deeper the level, the more definite the assumptions which are needed in
order to fill out the knowledge. Even at the descriptive level in this
document, some representation decisions had to be tentatively assumed.
The theme of a BEINGs system is to distribute the understanding of knowledge among
all the parts of all the modules. Thus there will be many different ways in which
the system can claim to understand something. For example, it might be able to carry
out some activity (Algorithms), to formally discuss that activity (Definition), to
relate it to other activities it knows about (Ties), and even to give vivid intuitive
imagery to aid in visualizing the essence of the activity (Intuition).
Some of the knowledge present
initially will be stored in each of these forms.
The actual ways to represent the knowledge, especially
intuitive knowledge, are of some interest. As before, the presentation
will be repeated at a few different levels of detail.
Since the representation must be known before the knowledge can be understood
in that format, details about our representations precede details about the
core of facts and strategies initially supplied to the system.
.SSEC(Representation: Level 1)
The two broad categories of knowledge are definite and intuitive. To represent
the former, we employ (i) rules and assertions, (ii) BEINGs grouped into families,
and (iii) opaque Environment functions. To represent the latter, we employ
(i) abstract rules, (ii) pictures and examples, and (iii) opaque Environment functions.
.SSEC(Initial Knowledge: Level 1)
The following is a sketch of how the top level of knowledge in the system
is organized. Each node in the right lower section is both a BEING and the
prototypical representative of a family of BEINGs.
The Environment node stands for a collection of opaque background system functions.
.B7
⊂ααααααααααα⊃
~ Knowledge ~
%αααααπααααα$
~
⊂αααααααααααααπααααααααααααααβαααααααααααααααααπαααααααααααααπαααααααα⊃
↓ ↓ ↓ ↓ ↓ ↓
Environment Active-Meta Static-Meta Active Static Parts
.QP←PAGE
.E
.SSEC(Initial Knowledge: Level 2)
.ONCE TURN ON "{"
Below are diagrams of the knowledge present under each of the six major categories
of knowledge, as pictured in Initial Knowledge, Level 1, on page {QP}.
The first sketch indicates the major structures and functions in the
environment which the BEINGs see. Notice that the intuitive simulations don't
appear here; they are distributed among the INTU parts of all the BEINGs.
.B7
⊂ααααααααααααα⊃
~ Environment ~
%αααααααπααααα$
~
⊂ααααααααααααπααααααααααβαααααααααπααααααααααα⊃
↓ ↓ ↓ ↓ ↓
Interest Control Belief Choice Current-Situation
.E
The next five trees show the individual
BEINGs present in each of the five families of BEINGs.
Each node is a β; almost all the β's envisioned are present in the sketches below.
.B7
⊂ααααααααααααα⊃
~ Active-Meta ~
%αααααααπααααα$
~
⊂αααααααααααααααααααααααααααααβαααααααααααααααααααααααααααα⊃
↓ ↓ ↓
Infer Test Communicate
⊂ααααπαα∀ααααπααααααααπαααααααα⊃ ~ / \
↓ ↓ ↓ ↓ ↓ ~ / \
Find Guess Analogize Conserve Examine ~ With other BEINGs With the user
~ / \
⊂ααααααπααα∀αααπαααααααα⊃ Translation User-Model
↓ ↓ ↓ ↓ / \
Disprove Debug Assume Prove Into-English From-English
/ \ ~
Constructive Indirect ⊂ααααααααβααααααααπαααααααααπαααααααα⊃
↓ ↓ ↓ ↓ ↓
Natural-Deduc. Backward Indirect Existential Univ.
↓ / \
Cases Constructive Indirect
.E
One point to notice is that testing and inferring activities (above)
are separated from the
⊗4by-products⊗*
of testing and inferring
(below), namely conjectures, proofs, and counterexamples.
The former are things to do, the latter are objects which are static.
One can use a theorem, e.g., without remembering or caring how it was proved.
.B7
⊂ααααααααααααα⊃
~ Static-Meta ~
%ααααααπαααααα$
~
⊂ααααααααααααααααααααααααβααααααααααααααααααααααααπααααααααααααααααα⊃
↓ ↓ ↓ ↓
.ONCE TURN ON "α"
Non-justifiable Quasi-justified Fully-justified Math
/ | \ / | \ / | \ ~
Assumption Message Contradiction Analogy Bug Conjecture Proof Theorem Counterex. ~
~
Mathematical Theory, Basis, Formal System ←ααα$
.E
Although conjectures are far removed from belief (in the tree), the environmental
routines permeate throughout temporal and arboreal space. Belief and interest
are constantly being evaluated.
.B7
⊂αααααααα⊃
~ Active ~
%ααααπααα$
~
⊂αααααααααααααααααααααααααααααααβααααααααααααααααααααααααααααααα⊃
↓ ↓ ↓
Operation Property Relation
~ ~ ~
~ Ordered / \
.ONCE TURN ON "α"
~ ⊗7Equals Member Contain Equivalent Ordering Quantification⊗*
~
/ \
.ONCE TURN OFF "@"; TURN ON "α"
⊗7Compose Insert Delete Convert Subst Rule ∨ ∧ Unite ∪ Common-parts ∩ Setdifference⊗*@
.APART
.GROUP
⊂αααααααα⊃
~ Static ~
%ααααπααα$
~
⊂αααααααααααααααααααααααααααααααβααααααααααααααααααααααααααααααα⊃
↓ ↓ ↓
Primitive Containers Structures Assertions
~ ~ ~
⊂αααα∀ααααπααααααα⊃ ⊂ααααπαααβααααπααα⊃ Axioms
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
Ord.pair Variable T,F Hist List Oset Bag Set
.APART
.GROUP
⊂αααααααααααααααααααααααααααααααααααααα⊃
~ Parts (Archetypical Strategy BEINGs) ~
%αααααααααααααααααπαααααααααααααααααααα$
~
⊂ααααααααααααααααααααπααααα∀ααααααααπααααααααααααααα⊃
↓ ↓ ↓ ↓
Recognition Alter Act Info
~ / / \ ~
⊂αααααπαα∀ααπαααα⊃ / / \ ⊂ααααβααααπαααπαααα⊃
↓ ↓ ↓ ↓ / / \ ↓ ↓ ↓ ↓ ↓
Changes Final Past Iden / Interpret Change Defn Intu Ties Exs Contnts
/ ~ ~
/ ⊂αααπααααλ εααααααπααααααπααααααα⊃
/ ↓ ↓ ↓ ↓ ↓ ↓ ↓
/ Check Repr Views Bdy-ops Fillin Struc Algorithms
/
⊂αααααααααπααααααα∀∀παααααααααπαααααααπαααααπαααααααπαααααααπααααααα⊃
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
Genlzations Speclzations Boundary Dom/Range Order Worth Interest Justif Operations
.E
.SKIP TO COLUMN 1
.SSEC(Representation: Level 3)
At the moment, this section may appear to be a bizarre collection of data
too specific to be placed anywhere else.
The first item is: Which parts might a BEING have? Below, we list all the possible
parts, and give a brief description of what questions each can handle.
.BEGIN W(6) NARROW 2,0
⊗4RECOGNITION GROUPING⊗*
CHANGES Is this rele. to producing the desired change in the world?
FINAL What situations is this β rele. to bringing about?
PAST Where is this used frequently, to advantage?
IDEN {not}{quick} {fast} tests to see if this β is {not} currently referred to
⊗4ALTER GROUPING⊗*
GENERALIZATIONS What is this a special case of? How to make this more general.
SPECIALIZATIONS Special cases of this? What new properties exist only there?
BOUNDARY What marks the limits of this concept? Why exactly there?
DOMAIN/RANGE {not} Set of (what one can{'t} apply it to, what kind of thing one {never} gets)
ORDERING(Complete) What order should the parts be concentrated on (default)
WORTH Aesthetic, efficency, complexity, ubiquity, certainty, analogic utility, survival basis
INTEREST What special factors make this type of BEING interesting?
JUSTIFICATION Why believe this? Formal/intu. For thms and conjecs. What has been tried?
OPERATIONS Properties associated with β. What can one do to it, what happens then?
⊗4ACT GROUPING⊗*
CHANGE subgrouping of parts
BOUNDARY-OPERATIONS {not} Ops rele. to patching {messing}up not-bdy-entities {bdy-entities}
FILLIN How to initially fill it in, when and how to augment what is there already.
STRUCTURE Whether, When, How to retructure (or split) this part.
ALGORITHMS How to compute this function. Related to Repr.
INTERPRET subgrouping of parts
CHECK How to examine and test out what is already there.
REPRESENTATION How should entities of type β be structured internally? Contents' format.
VIEWS (e.g., How to view any Active as an operator, function, relation, property, corres., set of tuples)
⊗4INFO GROUPING⊗*
DEFINITION Several alternative formal definitions of this concept. Can be axiomatic, recursive.
INTU Analogic interp., ties to simpler objects, to reality. Opaque.
TIES Alterns. Parents/offspring. Analogies. Associated thms, conjecs, axioms, specific β's.
EXAMPLES {not} {bdy} Includes trivial, typical, and advanced cases of each type.
CONTENTS What is the value stored here, the actual contents of this entity.
⊗1**********************************************************************⊗*
.END
The next items of interest are which parts each BEING must have.
In PUP6, each β had (theoretically) exactly
the same set of parts. Here, each ⊗4family⊗* will have the same set.
For each possible part, we list below those families having that part:
.BEGIN W(7); TABS 30,40,50,62,74; TURN ON "\" GROUP
@2 ↓_Part Name_↓\Static\Active\Static Meta\Active Meta\Archetypical
.TABS 34,44,57,69,80
.INDENT 6
⊗6RECOGNITION GROUPING⊗*
CHANGES\X\X\X\X\X
FINAL\X\X\X\X\X
PAST\X\X\X\X\X
IDEN {not}{quick}\X\X\X\X\X
⊗6ALTER GROUPING⊗*
GENERALIZATIONS\X\X\X\X
SPECIALIZATIONS\X\X\X\X
BOUNDARY\X\X\X\
DOMAIN/RANGE {not}\\X\\X
ORDERING(Complete)\\X\X\X
WORTH\X\X\X\X
INTEREST\X\X\X\X
JUSTIFICATION\\\X\
OPERATIONS\X\X\X\X
⊗6ACT GROUPING⊗*
CHANGE subgrouping of parts
BOUNDARY OPERATIONS {not}\X\X\X\
FILLIN\\\\\X
STRUCTURE\\\\\X
CHECK\\\\\X
ALGORITHMS\\X\\X\
INTERPRET subgrouping of parts
REPRESENTATION\X\X\X\X\
VIEWS\\X\X\\
⊗6INFO GROUPING⊗*
DEFINITION\X\X\X\X\X
INTU\X\X\X\X\X
TIES\X\X\X\X\X
EXAMPLES {not} {bdy}\X\X\X\X\
CONTENTS\X\X\X\X\
⊗1**********************************************************************⊗*
.END
We have dwelt on BEINGs so much, the reader is now entitled to hear about the
other representations.
Since they are more conventional, there is less need to
delve into their details.
The rules are arranged in pools, with several independent
pointer systems to locate rules relevant in various ways. The functions
are compiled Interlisp code, perhaps using CLISP and some of the QLISP features.
Even the terminology here suggests the importance of BEINGs over these formalisms:
the rules are mere parts of β's, and the functions are merely the
⊗4environment⊗*, the background for the BEING activities.
************************************************************************************
↓_INTUITION FOR A SET:_↓
Let us now deal with the "square" representation for a set in more detail.
A set S is characterized as a rectangle in the Cartesian plane; the opaque
intuition function knows about numerical equality and inequality, hence about
borders of such sets. The notions
of intersection, union, complement, setdifference, disjointness, projection
onto each axis, etc. are also intuitively available. Notice that the
sophisticated operations required (e.g., projection) will exist as opaque
functions, totally inaccessable to the rest of the system. This is worth
rejustifying: is fair to write a LISP program (which uses the function
TIMES) whose task is to synthesize code for the function TIMES, so long as
the program does not have access to, does not even know about its use of
TIMES.
This "square" representation is not well suited to all concepts
involving sets.
For that reason,
the system will simultaneously maintain several of the other forms of
intuitive storage mentioned previously. Consider, for example, the
possibility of fuzzy rules, which can latch onto almost anything
and produce some type of result (but with low certainty). That is, they
operate at a higher level of abstraction than definite rules, by ignoring
many details. Another possibility is the use of examples. If a small set of
them can be found which is truly representative of a concept, then future
references to that concept can be compared to these examples. This may
sound very crude, but I believe that people rely heavily (and
successfully!) on it.
Euler, to overcome language problems when lecturing a princess of
Sweden, devised the use of circles to represent sets. Venn and others
have frequently adopted this image. For a machine, it seems more a
propos to use a rectangle, not a circle. Consider the lattice of
integral points in two dimensions. Now a set is viewed as a rectangle
-- or a combination of a few rectangles -- in this space. This makes it
hard to get any intuition about continuity or boundary or openness, but
works fine for the discrete sets which are dealt with in logic,
elementary set theory, arithmetic, number theory, and algebra. It is
probable that the system will therefore not be tried in the domains of
real analysis, geometry, topology, etc. with only this primitive notion
of space and confinement. Specificly, a set in this world is an
ordered pair of pairs of natural numbers. Projection is thus trivial
in LISP (CAR or CADR), as is test for intersection, subset, etc.
Notice that these require use of numbers, ordering, sets, etc., so the
functions which accomplish them must be opaque. The interaction
with the rest of the system will be for these pictures to suggest and
reinforce and veto various conjectures. They serve to generate
empirical evidence for the rest of the system.
To avoid gerrymandering, it might be necessary to view a set as a list
(of arbitrary length) of ordered pairs; an absent pair can be assumed to be
some default pair. That is, a set is a simplex in Hilbert space; each set has
infinite dimension, but differs from any other in only finitely many of them.
How should the system choose which intuitive representation(s) of a set to use?
Some considerations are:
What operations are to be done to this set
(e.g., ⊗6ε⊗*, ⊂, ∩, ∪, ⊗6≡⊗*, =, ',...)? The representations differ in cost of
maintenance and in the ease with which each of these operations can be
carried out.
How artificial is the representation for the given set?
Some will be quite natural, e.g., if the set is a nest then use the
pointer structure; if the set is a relation over the small set AxB, then use the
lattice points representation.
How much is "given away" by the model? This is a
question of fairness, and means that the system-writers must build in
opacity constraints and/or make the intuitive operations faulty.
We shall do both.
How compatible is each representation with the computer's
physiology? Thus it is
almost impossible to represent pictures or blobs directly, but very
suitable to store algebraic equations defining such geometric images.
Does the representation suggest a set theory with basic elements
which are non-sets; with an infinite model; with any special desirable or
undesirable qualities? For example, the geometric representation
seems to demand the concept of continuity, which the system probably
won't ever use in any definite way.
************************************************************************************
There are about 125 β's in the proposed core,
and each one of them should have an intuition almost
as rich as that for SETS, above. Space precludes delving into each one; some few
lines about each β's intuition is present in the document "⊗4GIVEN KNOWLEDGE⊗*".
.SKIP 2
↓_REVIEW OF THE PARTS GROUPINGS_↓
In case the reader wants to see the breakdown of the parts again, they are
reviewed below, group by group. The particular families are not mentioned,
since most of the parts occur in most of the five families anyway.
During system runtime, a part is filled in or extended
whenever a new idea becomes explicit.
The proximate driving force of the system is the urge to ⊗4complete⊗*
each BEING. The true drivers are the judgmental criteria functions.
The four pictures below indicate the four main parts groupings, which in turn
reflect the four reasons for calling on a BEING or a part of one:
to see if it is relevant, to modify itself in some way, to deal with a
supplied argument (some part of some other BEING), or simply to answer a question
(accessable information). Under each category are several distinct parts and
in some cases further groupings of parts. Each grouping is itself a BEING; each
part is also represented by one archetypical BEING. In any given case, however,
the value stored in part of a BEING is simply some rules, pointers, numbers, etc.
The exact format of, e.g., part P of BEING B is specified in the REPRESENTATION
part of the archetypical BEING
whose name is P. In case some special information exists for dealing with B.P,
there may be another relevant archetypical BEING, whose name would actually be B.P.
.B7
⊂ααααααα⊃
~ RECOG ~
%αααπααα$
~
⊂αααααπαα∀ααπααααα⊃
↓ ↓ ↓ ↓
Changes Final Past Iden
.E
The RECOG grouping is concerned with handling the following types of questions:
Are you relevant to effecting this change in the world..., Can you bring about this
state of the world..., How successful were you in situations similar to the
current one...,
Can you recognize this phrase...
These four types of questions are handled respectively
by the CHANGES, FINAL, PAST-USE, and IDEN parts.
.B7
⊂ααααααα⊃
~ ALTER ~
~ self ~
%αααπααα$
~
~
⊂αααααααπαααααααααααβαααααααααααπααααααααπαααααααα⊃
↓ ↓ ↓ ↓ ↓ ↓
Generalize Specialize Boundary Ordering Worth Ops
↓ / \
.ONCE TURN ON "α"
Dom/Range Interest Justification
.E
The ALTER grouping is concerned with handling the following types of questions:
What is the boundary of the current concept? Why does it exist; why can't you
relax some constraint and generalize yourself? Is there anything interesting
happening when you specialize yourself; how ⊗4can⊗* you specialize yourself?
How incomplete are you; what part should be attended to next? Are you worth
surviving; why, what good are you?
What factors make a β like you interesting/uninteresting?
What can (can't) be done to you?
These types of questions are handled respectively
by the Boundary, Generalize, Specialize, Ordering, Worth, Interest, and
Operations parts.
.B7
⊂ααααααα⊃
~ ACT w.~
~ other ~
εαααααααλ
/ \
/ \
/ \
/ \
/ \
Interpret Change
/ ~ \ ~
Representation Views Check ⊂αααα∀αααπαααααααααααααπαααααααααααααα⊃
↓ ↓ ↓ ↓
Structure Fillin Bounday-operations Algorithms
.E
The ACT grouping is concerned with handling the following types of questions:
How can this entity be pulled across your boundary? (Boundary operators part).
Most of the rest of the questions deal with BEINGs which
represent a part: whether to check to see if this part might be too "full";
if so, ⊗4how⊗* to check this; if indicated, how interesting should
the subpart(s) be before actually doing something? to act, do we split or
merely restructure? (Structure part)
What is the format of a typical one of you? (Representation part).
How much of this has been filled in so far?
How do I doublecheck this information?
How do I fill in
some more? (Check, Fillin).
In general, there are two kinds of requests here. One is for actually changing
a part whose name is the name of this BEING (use the Change subgrouping). The other
kind of job is simply one of interpreting some aspect of such a part
(the Interpret subgrouping of parts).
.B7
⊂ααααααα⊃
~ INFO ~
%αααπααα$
~
~
⊂ααααααααααπαααααααβαααααααπααααααααα⊃
↓ ↓ ↓ ↓ ↓
.ONCE TURN ON "α"
Definition Intuition Ties Examples Contents
~
⊂αααααααααπααααααααααααβαααααααααααααα⊃
↓ ↓ ↓ ↓
Analogues Family Alternatives Related-objects(thms, conjecs, axioms)
.E
The INFO grouping is concerned with handling types of questions dealing with
ubiquitous facts about this BEING. These include categories which are
needed by more than one of the preceding three groupings, those needed in
several different ways, those which other BEINGs might want to inspect, etc.
The names of the parts in the picture are self-explanatory.
.SKIP 5
A scheme for organizing the pointer systems for RULES now follows.
Each rule will have several types of pointers, to indicate relevant
rules. One set might be as follows:
.BEGIN W(1) INDENT 7
ABSOLUTE The rules pointed to here should definitely be examined.
SUCCESS If this rule succeeds, then look at these anyway.
FAILURE If this rule fails, by a little, then look at these. (More descriptive, perhaps).
EXTEND If a more comprehensive result is desired
CONTRACT If a more restricted, simpler result is desired.
WORTH What is this rule's expense of execution? Its chance of success?
Point to cheaper rules/functions; point to costlier rules/BEINGS.
INTU Point to abstract intuitive rules relevant to this rule.
DEF Point to less abstract rules which are related to this one.
.E
Notice that the rule parts are simpler, fewer, and more uniform than the set
of BEING parts. A simple pool of unstructured rules might be all that is needed
(situation-action productions). That is, each rule is executable, and has some of
the above 8 supplementary pointers filled in. The drive to fill in the pointers of
Rules is much lower than the drive to fill in parts of BEINGs.
Conceivably, the system might not even have such pointers attached unless the need
specificly arises. The structure of a part of a Rule is considered opaque, to prevent
any regress here, and to permit the rules to be coded for speed and compiled.
A major fraction of the environment will consist of absolutely opaque
functions, coded for maximum efficiency,
which perform "primitive" functions absent in
INTERLISP but desirable for our system.
The precise representation of the efficient functions is not important,
since they are completely opaque to the rest of the system. Access to a
compiler should probably be permitted; once the system has an algorithm
to do something, there is no reason why it shouldn't be allowed to point
to a compiled routine for the same algorithm.
⊗7Indeed, most humans who use a compiler don't really understand or care about how it
works. Even those who ↓_do_↓ understand it will typically just
extract a few general
do's and don'ts and tricks,
and not keep recalling pieces of the compiler's code.⊗*
.SKIP 3
.SSEC(Initial Knowledge: Level 3)
For each BEING, we now present a brief summary of the value stored in each of its
parts.
If a part name is absent, it is expected that this will ⊗4NEVER⊗* be filled in for
this particular BEING. If the name is present but there is no value, then the
system might need to (and would, then) fill this part in sometime.
.SELECT 4
See GIVEN KNOWLEDGE document, please, for this information.
In there you will find a few lines of information about each part of each of the
(roughly 125) BEINGs planned to be given to the system initially.
.SELECT 1
.NSEC(COMMUNICATION)
The work in this area consists of collecting English words and
grammatical constructions, of the kind found in various mathematics texts.
The next step is to exhaustively categorize all
words and phrases, and tie each one in to a BEING or a specific part. Also, some
fixed language scheme for communicating intuitive information must be devised.
Another ability which must be present is a DWIM-like recovery facility, tailored to
the kinds of errors one makes when discussing mathematics. For example, if someone
mentions "3+4", when + is defined only for rational numbers (a diffenret symbol
is emplyed for integers), then the error should be resolved by this simple bit of
psychology: "If an operation is applied incorrectly, and its real domain is in a very
closely analogous system, then map it back to find out which operator was really
meant, and warn the speaker to be more precise in the future."
A third aspect is that of acclimatization to individual vocabulary and terminology.
For example, is a function from A to B necessarily defined on all of A? One way to
acquire the user's preferences is during analysis of an error (as above); another way
is of course to allow the user to name the β himself (e.g., give him examples and
intuition parts). His specific choices will go into the IDEN parts of the relevant
β's; if there is any possibility of contradiction with standard usage, the entry will
be tagged with the user's name, for future reference. Of course a single user may refer
to the identical concept by more than one name, but the system should never permit
him to refer to two different things by the same name. In such a case, if the user
stands firm on the new entity, allow him to rename the older entity.
.NOFILL
.GROUP
.SSEC(Categories of Languages)
English ↔ BEINGs
Standard Math Notation
IMPLICATION
SPECIFICATION
COMBINATION
OPERATION
DEFINITION
KNOWN RELATIONS
ENTITIES
Fixed Formats for Quasi-English Meta-Comments, Questions, Hints
ACTIVITIES
RESTRICTED CONCEPTS
INTELLECTUAL PROCESSING
TIME AND SPACE REFERENCES
INDEFINITES
QUESTIONS
Fixed Language for Communicating Intuitive Concepts
BEINGs ↔ BEINGs
The whole idea of BEING parts; especially: representation part of archetypical β's.
Language for Intuitive Communication
Language for Communication via Inference from Examples
.APART
.SSEC(Standard Math Notation)
.BEGIN W(7); FILL RETAIN; INDENT 0,6,0
IMPLICATION
IF ... THEN ...
IMPLIES
IFF
IF
ONLY IF
IS IMPLIED BY
THEREFORE
THUS
SUPPOSE ... THEN
LET ... THEN
THEN
SO
HENCE
IN ORDER TO...
IT SUFFICES THAT
NECESSITY
SUFFICIENCE
→
←
↔
WHENEVER
WHEN
CAUSE, CAUSALITY, BECAUSE
ENTAILMENT
SPECIFICATION
SUCH THAT
SATISFYING
WITH
WHERE
SOME
THE
A/AN
ALL
EVERY
NO ... IN
⊗6∀⊗*
∃
FIXED
VARIABLE
ANY
EACH
MOST
THERE EXISTS
WHICH
THAT
THIS
OTHER
ABOUT
COMBINATION
AND
OR
∧
⊗6∨⊗*
NOT
⊗6¬⊗*
ALSO
BUT
OPERATION
RELATION
PREDICATE
f/g/h
DO
APPLY
COMPUTE
OPERATE
PRODUCE
ACCORDING
CORRESPOND
ALGORITHM
<silent imperative>
COMPOSITION
o
MAP
TAKE
SEND
PULL
IMAGE
RANGE
DOMAIN
f:D→R
PREIMAGE
UNDEFINED
DEFINED
f(a,b,c)
CLOSED
DEFINITION
DEFINE
CALL
=df
NOTATION FOR ...
REFER TO...
NAME
KNOWN RELATIONS
EQUALITY
=
IS/ARE
INEQUALITY
ORDERING
GREATER
LESS
SUBSET
⊂
⊃
CONTAINS
INCLUDES
MORE
INTERSECTS
∩
UNION
∪
APPEND
BETWEEN
INSIDE
OUTSIDE
INCLUSION
EXACTLY
COMPLEMENT
SETDIFFERENCE
+,-,x for sets
CONS
CAR
CDR
FIRST
LAST
ALL BUT
JOIN
PREVIOUS
PRECEDE
SUCCEED
FOLLOWING
NEXT
NEAR
FAR
CLOSE
ANALOGOUS
ENTITIES
ATOM
ELEMENT
CONSTANT
VARIABLE
SET
TUPLE
BAG
MEMBER
⊗6ε⊗*
THING
ENTITY
OBJECT
IDENTIFIER
NAME
LABEL
VALUE
.SKIP 2
.SSEC(Fixed Formats for Quasi-English Meta-Comments and Questions)
ACTIVITIES
DO...
CONSIDER...
USE
LOOP
REPORT
DISTINGUISH... AND/FROM ...
EXPLAIN
DISCUSS
GET
RESTRICTED CONCEPTS
ELLIPSIS
ETC.
AND SO ON
...
pronouns
SIMILARLY
ANALOGY
SIMPLIFY
REDUCE
FAILURE
SUCCESS
INTELLECTUAL PROCESSING
THINK
CONCENTRATE
CONSIDER
ATTEND
ASSUME
SOLVE
PROVE
SEE
HYPOTHESIS
PROBLEM
SOLUTION
INVESTIGATE
DISCOVER
UNDERSTAND
TIME AND SPACE REFERENCES
EARLIER
LATER
BEFORE
AFTER
THEN
NOW
NEVER
ALWAYS
HERE
THERE
UNDER
ANYWHERE
NOWHERE
INDEFINITES
SHOULD
WOULD
COULD
MIGHT
POSSIBLE
PROBABLE
PLAUSIBLE
BEAUTY
POTENTIAL
CAN
forms of TO BE
OUGHT
CONFUSION
DEFINITE/INDEFINITE
CERTAIN/UNCERTAIN
TRANSLATE
DIFFICULTY
PLEASURE
SO
UNIQUE
EXISTENCE
QUESTIONS
WHAT x
WHY/WHY NOT x
HOW
WHEN
.END
.SSEC(Fixed Languages for Intuitive Communication)
No good new ideas have yet been found. At the moment, the plan is as follows:
.FILL
Each intuition will be an opaque function, which simulates some real-world situation.
The caller must specify as much as possible about the situation, after which the
function takes over and produces a description of what happens and/or the final state
of the world afterwards. The caller and the function together should know enough to
provide the caller with the specific piece(s) of information desired. Often, the
kind of data provided will clue the intuition function as to what is wanted in
return; often, the caller will know specifically what he wants back. Thus there may
not need to be any "language" in the normal sense of the word
(just some default schedule for calling).
Similarly, any BEING can
communicate any information by encoding it into examples and letting the receiver
decode it by inference from those examples. In that case, though, one must ensure a
universal sort of inference mechanism, perhaps an Infer-from-examples BEING with
whom it is easy for everybody to communicate directly.
Of course this is a very slow, inefficient mode of communication, and much information
may be lost or distorted.
An example: the old seesaw intuition. The function S simulates a seesaw, with
person p of weight w sitting on left or right side of seesaw, d distance from
the center, with person q of weight.... etc., for any number of people, and
also says which way the seesaw was originally(left,right,balanced),
which way it was finally,
and finally how quickly it moved from the start to the end. Any number of these
parameters may be left unspecified; the function will make an effort to provide
ranges for them, and/or examples of them. It is important to notice that the
function itself is not permitted to "give away" the fact that, eg., the names are
completely irrelevant, and that interchanging all lefts↔rights is equivalent.
That is, the function's ⊗4insides⊗* may know this when they compute the value, but
no BEING can ever access that information; the most he can do is look at lots of
examples and infer that invariance from them.
The actual code will compute L = the sum of (w x d) for each person on the left side,
R = sum of weight times distance from center for each person on the right side,
and the final activity is:
.BEGIN NOFILL INDENT 6
If L=R, then same as initial state, else Maximum(L,R).
If the state changed, the speed is proportional to the difference between L and R.
.END
The above are inverted easily in case the final change is given and a proposed
configuration of sitters is the desired unknown.