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00900 .TURN ON "↑↓_π{"
01000 .TURN ON "⊗" FOR "%"
01100 .MACRO B ⊂ BEGIN VERBATIM GROUP ⊃
01200 .MACRO E ⊂ APART END ⊃
01300 .TABBREAK
01400 .PAGE FRAME 54 HIGH 75 WIDE
01500 .PAGE FRAME 54 HIGH 77 WIDE
00100 .EVERY FOOTING(,⊗6{DATE},)
00200 .GROUP SKIP 5
00300 .BEGIN CENTER RETAIN
00400 ⊗7THEORY
00500 FORMATION:⊗*
00600
00700 ⊗6Some Stray Initial Thoughts on⊗*
00800
00900 ⊗2A SYSTEM WHICH CAN
01000 DEVELOP MATHEMATICAL
01100 CONCEPTS INTUITIVELY⊗*
01200 .END
01300 .GROUP SKIP 10
01400 .NOFILL
01500 ⊗3DOUG LENAT
01600
01700 AVRA COHN
01800
01900
02000 STANFORD UNIVERSITY
02100 ARTIFICIAL INTELLIGENCE LABORATORY
02200
02300 ⊗*
02400
02500
02600 FIRST SKETCH
02700
02800 ⊗4Not for distribution⊗*
00100 .NEXT PAGE
00200 .ONCE CENTER
00300 ⊗2CONTENTS⊗*
00400
00500 .GROUP SKIP 10
00600
00700 .NOFILL
00800 1. Overall motivating ideas
00900 2. Task: Level 1: Global Considerations
01000 3. Task: Level 2: How to Learn Arithmetic (once you can count)
01100 4. Ideas for the internal structure of the system
01200 5. Sample Protocol Sessions
01300 6. Task: Level 3: The actual internal structures interacting
01400 7. The formal communication language
01500 8. Bibliography
01600 .FILL
00100 .EVERY HEADING(⊗3MATH THEORY FORMATION⊗*,,⊗4Doug Lenat and Avra Cohn⊗*)
00200 .EVERY FOOTING(⊗6{DATE},,page {PAGE})
00300 .NEXT PAGE
00400 ⊗21. MOTIVATING ISSUES⊗*
00500
00600 It's easy to convince oneself of the primacy of intuition (over formal
00700 methods) in mathematical activity - in math considered as a type of
00800 intelligent behavior rather than as a finished product. Often, formal-
00900 ization, proof and axiomatization are imposed after the fact of discovery
01000 and creation of mathematics, and the formalisms are apparently unrelated
01100 to the essential intuitions. They give no hint of the thought processes
01200 that went into finding the proofs or choosing the axioms - guessing,
01300 inferring, considering seemingly unrelated fragments, extracting relevant
01400 features, noting analogies, conjecturing, planning for solutions, judging
01500 according to experience, working backwards, changing approaches, restating
01600 goals, etc.
01700
01800 More to the point yet not very helpful, are descriptions of what mathematical
01900 intuition is like (feels like):
02000
02100 The problem is laid aside, so far as the conscious mind is
02200 concerned...Apparently during this period unconscious mental
02300 activity concerned with the problem continues; for suddenly
02400 an insight relating to the problem - perhaps a conplete
02500 solution - erupts into consciousness... - Skemp
02600
02700 You cannot compel helpful ideas to appear. I take my problem
02800 earnestly. I put it to myself. I set it to myself. I realize
02900 it keenly. I become absorbed in my problem. I am waiting
03000 for a helpful idea; will it come? Perhaps at once, perhaps
03100 after some time, perhaps not at all...They [helpful ideas] are
03200 our masters and they are capricious and self-willed. They may
03300 flash upon us unexpectedly... - Polya
03400
03500 Such descriptions are valid, of course; most people can confirm having had
03600 like experiences. But they mislead by giving the impression that there
03700 isn't or cannot be an explanation or mechanism for the intuitive process.
03800 They give no hint of how a math learning and doing system could be implemented
03900 on computers - perhaps worse, they give no hint as to how people learn and
04000 do math.
04100
04200 More suggestive and helpful are descriptions of the intuitive process that
04300 allude to the importance of the organization of knowledge. What is needed
04400 for ideas to appear and occur, or for new information to be understood is
04500 not only that the requisite facts and techniques are present, but that they
04600 are accessible and available when they are appropriate. Surprisingly,
04700 there seems to be little discussion of the form and nature of mental
04800 organization in the literature on math learning and teaching (where one
04900 might expect to find it).
05000
05100 We establish contact with an extensive layer of our formerly acquired
05200 knowledge, some streak of which might be useful now... - Polya
05300
05400 Of course, the more extensive your knowledge is and the better
05500 it is organized the more chance you have to find what you need.
05600
05700 It [generalization] makes possible conscious, controlled and accurate
05800 accomodation of one's existing schemas, not only in response to
05900 the demands for assimilation of new situations as they are encountered,
06000 but ahead of these demands, seeking or creating new examples to fit
06100 the enlarged concept. - Skemp
06200
06300 Once a person is able to analyze new material for himself, he can
06400 fit it onto his own schemas in the ways most meaningful to himself
06500 which may or may not be the same ways as it was presented.
06600
06700
06800 The conception of the project is to build a system that learns and does
06900 mathematics by creating and maintaining and using "good" internal
07000 organizations of it knowledge. The self-organizing feature implies
07100 automaticity, and hence the usefulness of automatic programmming.
07200 Information, ideas and concepts should be stored with an eye for
07300 future use and connections with the existing structures, and accessed in
07400 an intelligent way.
07500
07600 The sorts of behavior envisioned (evidences of good internal organization,
07700 and behavior that could be called intuitive) are:
07800
07900 Assimilation of new information in a useful, connected manner,
08000
08100 Giving judgements (short of formal proof or disproof) about the truth
08200 of conjectures,
08300
08400 Offering reasonable conjectures,
08500
08600 Having a sense of interestingness or worth-of-pursuing,
08700
08800 Ability to weigh evidence for or against claims,
08900
09000 To assess the difficulty of problems,
09100
09200 To extend and generalize from examples,
09300
09400 Giving plans for proof/solution to the extent that they are present in
09500 the intuition,
09600
09700 To be dynamic; readjusting old schemas, shifting, reorganizing,
09800
09900 To effectively mobilize facts and techniques by using analogy, relevant
10000 features, etc.
10100
10200 To exercise a notion of the relatedness of propositions APART from
10300 the logical notions (probable implication, co-dependence on something
10400 else, support for, interdependence...): to give a convincing argument,
10500 to explain meaningfully, to be convinced or explained to,
10600
10700 Ability to have several organizations over the same knowledge for
10800 different uses
10900
11000 Understanding of math as a logical whole with many interconnections;
11100 ability to take different starting points,
11200
11300 To enter a "reflective" node and consider its own content; to name things,
11400 to isolate things, to reorganize itself,
11500
11600 Inventing mini-theories on a topic, to do small-scale research by tying
11700 fragments and observations together into a coherent whole; generalizing
11800 from the results of working on a few problems.
11900
12000 To be taught by various techniques.
12100
12200
12300
12400 .GROUP SKIP 5
12500 Elementary number theory as a domain has the (possible) advantages of
12600 being non-visual to a large extent, fairly basic and, in parts,
12700 hierarchically organized.
12800
12900 ⊗5An analogy:⊗*
13000 Artificial Intelligence is unable to use most psychological studies
13100 on learning, because the latter deal typically with rote
13200 memorization, not with creative assimilation. The reason for this
13300 is that the simple type of learning is more accessable to quantitative
13400 investigation. The same unfortunate dodge may be occurring in
13500 automatic theorem proving. The difficult problems of creative invention
13600 in mathematics have been avoided, and work has been on the mechanical
13700 methods of deduction. The tradeoff of discovery for completeness is
13800 never made by successful research meathematicians.
13900
14000 Why choose mathematics as a domain? The effort expended already on
14100 concept-formation program writers has shown that even a seemingly
14200 clean task domain presents massive dialogue problems. In math, much
14300 of this goes away; communication is tolerated by both man and by
14400 machine in a formal language (mathematical notation.)
00100 .NEXT PAGE
00200 ⊗22. TASK: GLOBAL LEVEL⊗*
00300
00400 ⊗5FACETS: Design considerations⊗*
00500 .NOFILL
00600
00700 (1) The given, initally supplied facts and methods.
00800 The format
00900 The content
01000
01100 (2) The teacher's inputs
01200 Format
01300 Content
01400
01500 (3) What is "learned"
01600 Format of internal new knowledge
01700 Content
01800
01900 (4) Abilities of the system to demonstrate its understanding
02000 Tests
02100 How easily it learns new information
02200
02300 (5) Methods of understanding:
02400 assimilating teacher's inputs into internal formats
02500 meaningful interaction between relevant knowledge
02600 .FILL
02700
02800 ⊗5The given knowledge and abilities⊗*
02900
03000 The system will be provided with background information on set theory,
03100 the meaning of definition, truth, falsity, proof, deduction, induction,
03200 inference, aesthetics, and efficiency values. The internal representations
03300 used will be devised by the system; there is room here for instruction
03400 by the user, and (more exciting) the development of new ways of representing
03500 mathematical concepts. For example, our exponential notation for numbers
03600 lends itself to fast algorithmns for addition and multiplication; perhaps
03700 there is a better representation for number-theoretic operations.
03800
03900 ⊗5The role of the human in this system⊗*
04000
04100 There will be a human watching and helping the system. His roles will
04200 include that of guide, teacher, and experimenter. The former roles will
04300 be played when the system is off the track, or boged down; the latter
04400 when the system is truly advanced.
04500 There should be communication in a formal yet natural meta-mathematical
04600 language, and also some non-linguistic modes of communication.
00100 .NEXT PAGE
00200 ⊗23. TASK: ACTIONS LEVEL⊗*
00300
00400 ⊗5An example: a plausible chain of specific discoveries⊗*
00500
00600
00700 Example of interacting modules of knowledge, discovering addition,
00800 multiplication,exponentiation, primes, the unique factorization theorem,
00900 greatest common divisor. Bear in mind that this is just one early idea
01000 of one possible development, based on a tentative set of given concepts.
01100 The knowledge base is assumed to contain notions of set theory
01200 (including joining by both UNION and by APPEND),
01300 truth and falsity, proof, relations, and operators. Specific
01400 acquaintence is assumed with the relations of ordering and equality,
01500 the concept of counting, and the operator COUNT (which takes a set
01600 and returns the number of elements in it). Despite this ability,
01700 there is not necessarily any known way of representing numbers.
01800 There are
01900 assumed various heuristics known to the program: ways to prove things,
02000 regularities to look for, ways of deciding how interesting an activity
02100 is, how to know when to make a new definition, etc.
02200
02300 Many orderings of activities, and ways of interacting, must be
02400 possible. One typical sketch of its actions might be as follows; note
02500 the virtual absence of communication with the user.
02600 1. Consider the composition COUNT * APPEND.
02700 2. See that it is associative, commutative.
02800 3. Notice that the distinguished number zero and the distinguished
02900 set NIL are associated as identities of this composition and of
03000 APPEND, respectively.
03100 4. It is very desirable to have a function operate on entities in
03200 a given domain and return a value in that same domain. One way to do this
03300 (with function f:DxD → D already in this form, but function g:D→E) is to
03400 consider g(f(g↑-↑1(x),g↑-↑1(y)), where g↑-↑1:E→D is the inverse of g. This is
03500 relevant when g * f is interesting, as in the current case. So the function
03600 COUNT * APPEND is considered applied to two numbers,
03700 by applying it actually
03800 to two sets which have those numbers of elements. It is seen that
03900 COUNT * APPEND * COUNT↑-↑1xCOUNT↑-↑1 is associative and commutative, and
04000 is (by construction) from NxN → N.
04100 5. Try to relate this composition to the known function SUCCESSOR.
04200 Notice that SUCCESSOR is the COUNT of the APPEND of a set and a singleton,
04300 so SUCCESSOR(COUNT(x)) will
04400 equal COUNT(APPEND(x,y)) iff y is a singleton set,
04500 a set whose count is 1.
04600 So SUCCESSOR(x) ≡ COUNT(APPEND(COUNT↑-↑1(x), COUNT↑-↑1(1))).
04700 6. Decide it's worth defining. Give it a name, e.g., PLUS.
04800 So our last statement would read SUCCESSOR(x) ≡ PLUS(x,1).
04900 This is much simpler, and hence encouraging.
05000 7. Since SUCCESSOR(x) can range over all nonzero numbers, so can
05100 PLUS(x,1). Also, notice that PLUS(0,0) is 0. So PLUS can assume any
05200 value in N.
05300 8. A corollary of (7) is that any number can be viewed as
05400 repeated additions of 1 to itself. In particular, view PLUS as operating
05500 on a BAG. Then any number n is PLUS of a bag containing n 1's.
05600 9. PLUS is thus repeated SUCCESSOR-ing. What would it mean to
05700 do repeated PLUS-ing? That is, consider a function M:NxN → N, where
05800 M(x,y) is the same as doing PLUS on a bag conaining y numbers, each of
05900 them being x. So M(x,y) ≡ PLUS(x,x,...<y times>,...,x).
06000 10. The new function has good domain/range, is
06100 associative, and is commutative.
06200 M(x,1) = x = PLUS(x,0), so the distinguished elements 1,0 are the
06300 identities of M, PLUS, respectively. This is also a good sign.
06400 11. The function is worth naming, call it TIMES. Now try to see
06500 how TIMES can be directly interpreted in the language of sets. It seems
06600 that TIMES(x,y) means one has a set with x elements, each of which is a
06700 set with y elements, and one is removing the boundaries of the inner
06800 sets, then counting the whole collection. The commutativity is clearly
06900 seen by arranging the elements into a matrix of x rows and y columns;
07000 turning this sideways resluts in amatrix of y rows and x columns, but
07100 obviously the number of objects in toto has been conserved.
07200 12. Once TIMES is accepted, new compositions open up for study.
07300 Consider PLUS(x, TIMES(y,z)). Work a while, then give up;
07400 perhaps find: PLUS(x,TIMES(x,y)) =
07500 TIMES(SUCCESSOR(x),y) = TIMES(PLUS(x,1),y).
07600 13. Consider PLUS(TIMES(x,y), TIMES(u,v)).
07700 Again, the only result
07800 found will probably be when x=u. Then
07900 PLUS(TIMES(x,y),TIMES(x,v)) = TIMES(x, PLUS(y,v)). AHA! This was the
08000 very next composition to be investigated, TIMES * PLUS. This rule is
08100 probably worth naming, call it DISTRIB.
08200 14. The commutativity of PLUS and of TIMES indicate how to show
08300 that they are not
08400 1-1. PLUS↑-↑1(3) contains (2,1) and (1,2), so this is true.
08500 The associativity
08600 indicates that it probably isn't even unique
08700 to within rearrangements of
08800 the arguments, viewing PLUS as operating on a BAG. This is also true,
08900 since PLUS↑-↑1(3) contains (1,2) and (0,3). Similarly for TIMES↑-↑1(4)
09000 containing (1,4) and (2,2).
09100 15. Further properties may surface by considering statements
09200 and relations involving ordering, (<, ≤). For example,
09300 PLUS(x,y) ≥ x. TIMES(x,y) ≥ PLUS(x,y) iff x and y are > 0.
09400 TIMES(x,y) is usually bigger than PLUS(x,y). The
09500 execeptions are if x or y is
09600 0 or 1, or if x=y=2. Thus 2 is distinguished now.
09700 16. Higher order operations are considered. E(x,y) is defined
09800 as TIMES operating on a bag of x elements, each of them a y. But even
09900 at this point, associativity and commutativity fail, so there is no
10000 reason to probe too hard. Probably the relation to be discovered is
10100 TIMES(E(x,y), E(x,z)) = E(x, PLUS(y,z)), and perhaps also
10200 E(E(x,y),z) = E(x, TIMES(y,z)).
10300 There may be some interesting relations
10400 involving hyper-exponentiation and hyper..., but it is unlikely.
10500 17. Once a representation for numeration is chosen, work can be
10600 directed toward efficient algorithms for doing PLUS and TIMES.
10700 Perhaps the problem should be reversed.
10800 We don't want to have to revert to
10900 counting sets to actually compute 8↑4 = 4096 in unary!
11000 So how should a system
11100 of numeration be devised that would allow us to quickly compute PLUS
11200 and TIMES of any numbers, perhaps at the cost of memorizing a finite
11300 set of facts?
11400 18. Although the inverses of PLUS and TIMES were not singletons,
11500 the COUNT of these sets may follow some regular pattern.
11600 COUNT * TIMES↑-↑1 is the number of ways to express a number as a product
11700 of two numbers. It will always include the number and 1, as one pair.
11800 The system may decide to name the numbers whose value of
11900 COUNT * TIMES↑-↑1
12000 is 1: PRIMES. TIMES may also be viewed as operating on a BAG; in this
12100 sense, TIMES↑-↑1 would return all possible bags whose product is the
12200 number. The PRIMES stay the same! The sys. notices that for
12300 every number, its image under COUNT * TIMES↑-↑1
12400 contains a bag containing only primes, and only one such bag.
12500 This is the unique factorization theroem, and the sys. has proposed it
12600 intuitively!
12700 19. My definition
12800 of BAG implicitly assumes that if any number of α's can be added to the
12900 bag without altering its properties, then a minimal number of them are
13000 actually added. Thus PLUS never operates on zero unless there are less
13100 than two nonzero operands; similarly for TIMES and 1.
13200 TIMES↑-↑1(0) is quickly
13300 isolated as a very special beast,
13400 and not included in any statements involving
13500 TIMES↑-↑1.
13600 20. Consider applying PLUS * TIMES↑-↑1, that is, add up each
13700 bag of factors. Perhaps choose a name for those numbers whose
13800 images contain a bag whose sum is < the number, one for those
13900 containing a bag whose sum is = the number. Similarly, consider
14000 COUNT * UNION * TIMES↑-↑1, the total number of factors of a number.
14100 PLUS * UNION * TIMES↑-↑1 is the sum of all the divisors of the number.
14200 This will always range from SUCCESSOR of the number to below
14300 TIMES of the number and itself. Those at the low end are the PRIMES.
14400 Those at PLUS(x,x) = TIMES(x,2) may be named (PERFECTS).
14500 Those near the high end are
14600 not so well grouped, and may not be named.
14700 21. Consider comparing TIMES↑-↑1 collections from two numbers.
14800 Clearly UNION * TIMES↑-↑1(x) is a subset of UNION * TIMES↑-↑1 (TIMES(x,y))
14900 for any nonzero y. When does equality occur? Some cases are when
15000 y is 1, when y is a member of the first set. Are these all? Why?
15100 22. Compare as above. What is the
15200 INTERSECTION of these two sets? That is, what factors do the two
15300 numbers have in common? Study this for many cases. Eventually
15400 consider that this intersection set is
15500 UNION * TIMES↑-↑1 of the largest member of the intersection set.
15600 That is, the largest common factor determines all the others!
15700 23. This is worth a name; call it GCD. The GCD of a prime
15800 and any number is either 1 or the prime, and the second case occurs
15900 iff the prime is a factor of the second number. In general, two
16000 numbers may have GCD 1 with neither being prime. Explore this.
16100
00100 .NEXT PAGE
00200 ⊗24. IDEAS: INTERNAL⊗*
00300
00400 ⊗5Ideas about the representation of knowledge⊗*
00500
00600 There seems to be a unifying concept, embarrassingly simple, which
00700 subsumes BEINGS and RULES. This is merely the idea of pointer
00800 structures; each node of information would be part of several distinct
00900 trees.
01000
01100 Example: "The associative law over N" is a piece of knowledge. It is
01200 pointed to by ⊗4N⊗* and in turn it points to ⊗4BAG⊗*, in the tree of
01300 ⊗4property-of⊗* pointers. It points to the function ⊗4REVERSE⊗* in the
01400 structure dealing with ⊗4execution⊗*.
01500
01600 The PUP6 BEING community can be viewed as thirty pointer structures,
01700 connecting a hundred knowledge nodes. Each node could be expressed as
01800 a packet of knowledge. Perhaps a better way to look at it is to conceive
01900 of each part of each BEING as a node. Then the structures for each part
02000 are fairly disjoint. There is one new system of pointers, tying each
02100 set of parts of a BEING together.
02200
02300 One reason this was not obvious is that most of the links are virtual:
02400 they are computed at run time by pattern-matching, by saying "who should
02500 I link to?"
02600
02700 Perhaps a better way of looking at the
02800 knowledge is to keep it unstructured
02900 (unlike BEINGs, which make one pointer structure explicit) physically, and
03000 to have methods (some simple, some sophistocated) for ascertaining the
03100 corpus of information relevant in any given situation. That is, perhaps
03200 one should ⊗4not⊗* fix a set of BEING parts at all!
03300
03400 Predicate calculus typically is organized this way, but has no methods at
03500 all for deciding which knowledge is relevant and which isn't (except to
03600 try and use it!)
03700
03800 Some rule-based systems have explicit pointer structures; many translation
03900 systems use that idea.
04000
04100 Further thought should be given to any advantage
04200 of viewing a non-deterministic
04300 pattern-invoked transfer of attention as if it were following a virtual
04400 pointer in a system relevant to the particular match desired.
04500
04600 ⊗5One possible internal organization of knowledge⊗*
04700
04800 Each module of knowledge would be a BEING. The specific set of BEING
04900 parts would differ from those used in PUP6.
05000
05100 Conjectures are formed by analogy, definitions are made on the basis
05200 of aesthetics, and plausibility is determined by empirical testing,
05300 by analogy, and by desirability. Also, the user may direct any of
05400 these activities.
05500
05600 Each new concept would mean the creation of a new BEING. Part of the
05700 drive, the aestheitics, would be to ⊗4complete⊗* each BEING.
05800
05900 Taking an idea from CLASSES, we group the BEINGS into communities,
06000 say into OBJECT, OPERATOR, CONCEPT categories.
06100
06200
06300 ⊗5Possible parts of an OBJECT BEING:⊗*
06400
06500 .NOFILL
06600 DEFINITIONS. several equivalent definitions
06700 NAME. significant English variations
06800 IDEN. tests to see if this object is the one
06900 being referred to
07000 NOT-IDEN see if this object is irrelevant
07100 QUICK-IDEN cheap test to see if object is definitely irrelevant
07200 QUICK-NOT-IDEN cheap test to see if object is definitely relevant
07300 EXAMPLES includes trivial, typical, and advanced cases
07400 NOT-EXAMPLES sim. to above
07500 BOUNDARY-EXAMPLES cases which just barely fall into this concept
07600 NOT-BOUNDARY-EXAMPLES cases which just miss
07700 IMAGE analogic interpretations: ties to simpler objects,
07800 to reality. a sequence of representations
07900 of increasing formality.
08000 WHY Why is this worth naming as a separate BEING?
08100 USE Where is this used frequently, to advantage?
08200 GENERALIZATIONS What is this a special case of? The new features.
08300 SPECIALIZATIONS What are special cases of this? What new properties
08400 exist there but not here?
08500 REPRESENTATION How should this be represented internally?
08600 OPERATIONS What operators are associated with this object?
08700 What can one do to it, what happens then?
08800 NOT-OPERATIONS What can't be done. Is this desirable? Permanent?
08900 Why not? WHat if you try to do it anyway?
09000 BOUNDARY-OPERATIONS What operators are relevant to patching up entities
09100 which just barely miss, so that they don't miss.
09200 NO-BOUNDARY-OPERATIONS What operators pull an entity out of this category.
09300 VALUE Aesthetic worth, efficiency ratings, complexity,
09400 level of certainty, analogic utility, ubiquity
09500
09600 ⊗5Possible parts of an OPERATION BEING:⊗*
09700
09800 DEFINITIONS. several equivalent definitions
09900 NAME. significant English variations
10000 IDEN. tests to see if this operator is the one
10100 being referred to
10200 NOT-IDEN see if this operator is irrelevant
10300 QUICK-IDEN cheap test to see if operator
10400 is definitely irrelevant
10500 QUICK-NOT-IDEN cheap test to see if operator is definitely relevant
10600 EXAMPLES includes trivial, typical, and advanced cases
10700 NOT-EXAMPLES sim. to above. a ⊗4case⊗* includes a typical
10800 operator in this class being applied to some object.
10900 BOUNDARY-EXAMPLES cases which just barely fall into this concept
11000 NOT-BOUNDARY-EXAMPLES cases which just miss
11100 IMAGE analogic interpretations: ties to simpler
11200 operations, to real-world manipulations.
11300 WHY Why is this worth naming as a separate BEING?
11400 USE Where is this used frequently, to advantage?
11500 GENERALIZATIONS What is this a special case of? The new features.
11600 SPECIALIZATIONS What are special cases of this? What new properties
11700 exist there but not here?
11800 REPRESENTATION How should this be represented internally?
11900 OBJECTS What objects are associated with this operator?
12000 What can one do it do, what happens then?
12100 NOT-OBJECTS What can't it be done to? Is this desirable?
12200 Is this restriction temporary or permanent?
12300 Why not? What if you try to apply it anyway?
12400 BOUNDARY-OPERATORS What operators are relevant to patching up
12500 operator which just barely miss, so that
12600 they no longer fall outside this concept?
12700 NO-BOUNDARY-OPERATIONS What operators pull an operator out
12800 of this category?
12900 VALUE Aesthetic worth, efficiency ratings, complexity,
13000 level of certainty, analogic utility, ubiquity
13100 RANGE The OBJECTS parts describe the domain; this part
13200 describes the number and character of results
13300 obtained from an application of the operator.
13400 Perhaps redundancy, in allowing ordered pairs of
13500 desriptions: (inputs descs., outputs descs.)
13600 NOT-RANGE Why certain values will not be obtained.
13700 VIEWS How to view as an operator, a function, a relation,
13800 a property, a correspondence, a set of tuples.
13900 ALGORITHMS How to compute this function. Ties to
14000 representation. Need for better algorithms.
14100
14200 ⊗5Possible parts of a CONCEPT BEING:⊗*
14300 (note this refers to math notation, meta-comments, directions, etc.)
14400
14500 DEFINITIONS. several equivalent definitions
14600 NAME. significant English variations
14700 IDEN. tests to see if this term is the one
14800 being referred to
14900 NOT-IDEN see if this term is irrelevant
15000 QUICK-IDEN cheap test to see if term is definitely irrelevant
15100 QUICK-NOT-IDEN cheap test to see if term is definitely relevant
15200 EXAMPLES includes trivial, typical, and advanced cases
15300 NOT-EXAMPLES sim. to above. a "case" is an example of usage.
15400 BOUNDARY-EXAMPLES cases which just barely fall into this concept
15500 NOT-BOUNDARY-EXAMPLES cases which just miss
15600 IMAGE analogic interpretations: ties to simpler terms,
15700 to reality. a sequence of representations
15800 of increasing formality.
15900 WHY Why is this worth naming as a separate BEING?
16000 USE Where is this used frequently, to advantage?
16100 GENERALIZATIONS What is this a special case of? The new features.
16200 SPECIALIZATIONS What are special cases of this? What new properties
16300 exist there but not here?
16400 REPRESENTATION How should this be represented internally?
16500 OPERATIONS What operators are associated with this term?
16600 What can one do to it, what happens then?
16700 NOT-OPERATIONS What can't be done. Is this desirable? Permanent?
16800 Why not? WHat if you try to do it anyway?
16900 OBJECTS What objects are associated with this term?
17000 BOUNDARY-OPERATIONS What operators are relevant to patching up entities
17100 which just barely miss, so that they don't miss.
17200 NO-BOUNDARY-OPERATIONS What operators pull an entity out of this category?
17300 VALUE Aesthetic worth, efficiency ratings, complexity,
17400 level of certainty, analogic utility, ubiquity
17500 DOMAIN/RANGE If this term involves a pattern with variables,
17600 what can go where? What are the number and
17700 character of the outputs? How do they depend on
17800 the inputs' character?
17900
18000
18100 .FILL
18200 The contents of each part of each BEING should be simple; either another
18300 BEING, a simple program, a set of rules. Perhaps each BEING part should
18400 have a specific structure. The concept of plausibility should be handled
18500 more thoroughly.
18600
18700 Common knowledge should in some cases be factored out. Possibilities:
18800 (i) always ask a specific BEING, who sometimes queries a more general
18900 one if some knowledge is missing; (ii) always query the most general
19000 BEING relevant, who then asks some specific ones (This sounds bad);
19100 (iii) ask all the BEINGS pseudo-simultaneously, and examine the
19200 responders (this sounds too costly.)
00100 .NEXT PAGE
00200 ⊗25. PROTOCOL⊗*
00300
00400 These excerpts don't reflect what the system ⊗4should⊗* do, but merely
00500 are plausible paths it might follow. They give the flavor of the dialogue,
00600 and point out problems which must be attended to.
00700
00800 ⊗5First rough attempt: 11/4/74⊗*
00900
01000 S: I AM AWARE OF SEVERAL OPERATORS, WHO THINK THEY MAY BE RELATED.
01100 I AM GOING TO TRY TO FILL IN THE RELATIONSHIPS BETWEEN THEM.
01200 COMMENTS/QUESTIONS?
01300
01400 U: I wonder if you could tell me what the operators of interest are?
01500
01600 S: TRANSLATED AS: WHAT OPERATORS?
01700 ANSWER: APPEND, UNION, INTERSECT, SETDIFFERENCE, COMPLEMENT, COUNT, EQUAL
01800
01900 U: Tell me whether you know what "composition" of operators means.
02000
02100 S: TRANSLATED AS: DEFINED "COMPOSITION"?
02200 ANSWER: GIVEN TWO FUNCTIONS, IF THE DOMAIN OF THE FIRST INTERSECTS WITH
02300 THE RANGE OF THE SECOND, IN A SUBSET S, THEN THE COMPOSITION MAY BE
02400 DEFINED OVER THE PART OF THE DOMAIN OF THE SECOND FUNCTION WHICH IS THE
02500 PREIMAGE OF S, TO THE PART OF THE RANGE OF THE FIRST FUNCTION WHICH IS
02600 THE IMAGE OF S, AS FOLLOWS: TO COMPUTE THE COMPOSITION AT XεS,
02700 APPLY THE SECOND FUNCTION TO THE VALUE OF THE FIRST FUNCTION APPLIED
02800 TO X.
02900
03000 U: This is my notation for the composition of f and g (two functions):
03100
03200 g (f (x)). f is applied first, and I would like to call it the first function, or the
03300 innermost. Okay?
03400
03500 S: YOUR "FIRST FUNCTION" CORRESPONDS TO MY "SECOND FUNCTION."
03600 YOUR "SECOND FUNCTION" CORRESPONDS TO MY "FIRST FUNCTION". THIS HAS BEEN
03700 ASSIMILATED AND CHANGED. I CANNOT ACCEPT YOUR NOTATION. HOW WOULD
03800 IT BE POSSIBLE TO REPRESENT G APPLIED TO F APPLIED TO X?
03900
04000 U: As g (f (x)).
04100
04200 S: THEN HOW COULD I DISTINGUISH THIS FROM THE COMPOSITION OF F AND G?
04300
04400 U: Tell me what you think the difference between the composition of f and
04500 g and g (f (x)) is.
04600
04700 S: THE FORMER IS AN OPERATOR AND THE LATTER IS AN ELEMENT OF THE RANGE OF G
04800
04900 U: I would rather denote the former by g o f.
05000
05100 S: OK. WHAT DOES THIS HAVE TO DO WITH THE ORIGINAL QUESTION?
05200 POSSIBLE ANSWER: TRY TO RELATE THE EXISTING OPERATORS BY COMPOSITION.
05300
05400 U: That is right. Pay special attention to COUNT as the second function
05500 (in my sense).
05600
05700 S: COUNT o UNION WAS INVESTIGATED; NOTHING INTERESTING.
05800 COUNT o APPEND IS BEING INVESTIGATED. THIS FUNCTION IS ASSOCIATIVE.
05900 THIS IS COMMUTATIVE. THE FUNCTION SEEMS INTERESTING, BUT ITS DOMAIN AND RANGE ARE DISTINCT.
06000 INCT. SHOULD I WORK TO TRY TO CORRECT THIS, PERHAPS ALSO BY COMPOSITION?
06100
06200 U: What do you mean by the domain and range being distinct? (Perhaps an example?)
06300
06400 S: THE COMPOSITION COUNToAPPEND MAPS TWO ELEMENTS FROM THE DOMAIN "LISTS"
06500 TO ONE ELEMENT OF THE DOMAIN "NUMBERS". IF THESE WERE THE SAME, ONE COULD
06600 INVESTIGATE REPEATED APPLICATIONS OF THIS FUNCTION. ALL THE INTERESTING FUNCTIONS
06700 I KNOW ARE OF THIS FORM, FOR EXAMPLE APPEND (TWO LISTS → ONE LIST).
06800
06900 U: Continue to investigate that, and report your results to me.
07000
07100 S: I CAN CONSTRUCT A FUNCTION (LAMBDA (X,Y) (COUNT (APPEND (COUNT↑-↑1 X)
07200 (COUNT↑-↑1 Y))))
07300 THIS MAPS TWO NUMBERS INTO ONE NUMBER. I CAN CONSTRUCT
07400 THE ANALOGOUS FUNCTION WITH COUNT↑-↑1 AT THE FRONT, BUT
07500 THIS IS TOTALLY UNINTERESTING.
07600 I AAM INVESTIGATING THE FIRST POSSIBILITY. THIS IS ASSOCIATIVE.
07700
07800 THIS IS COMMUTATIVE.
07900 THIS TIES IN WITH SUCCESSOR: SUCCESSOR OF A NUMBER IS THE SAME AS
08000 APPLYING THIS COMPOSITION TO THAT NUMBER AND 1.
08100 THIS WARRANTS GIVING THE COMPOSITION A NAME. WHAT IS ITS NAME:
08200
08300 U: We will call it ADD. Can you tell me (i) What you mean by COUNT↑-↑1, and
08400 (ii) What is the explicit relationship between SUCCESSOR and ADD.
08500
08600 S: COUNT↑-↑1 MAPS A NUMBER ONTO A SET WHICH HAS A COUNT OF THAT NUMBER. THE
08700 LIST IS IN CANONICAL FORM, A HEAD OF THE FOLLOWING LIST: (X X X X X ...).
08800 (II). A RETRANSLATION OF THE PRECEDING CONCLUSION:
08900
09000 ∀XεNUMBERS, SUCCESSOR(X) = ADD(X,1)
09100
09200 ALSO: I NOW WANT TO SEE IF THERE IS ANYTHING ABOUT THE PROPERTIES SPECIAL
09300 TO SUCCESSOR WHICH WILL TELL ME SOMETHING SPECIAL ABOUT ADD. OK?
09400
09500 U: Yes. First tell me what properties of SUCCESSOR you are considering.
09600
09700 S: THE SUCCESSOR OF ANY NUMBER CAN BE EXPRESSED AS
09800 SUCCESSOR(SUCCESSOR(...(SUCCESSOR(0)...)).
09900 THE SUCCESSOR OF A NUMBER IS > THAT NUMBER.
10000 EVERY NUMBER HAS A SUCCESSOR.
10100 EVERY NUMBER EXECPT ZERO IS THE SUCCESSOR OF PRECISELY ONE NUMBER.
10200
10300 SUCCESSOR IS A FUNCTION.
10400
10500 U: In the first claim, the ellipsis notation doesn't tell me how many applications
10600 are intended. Do you know?
10700
10800 S: NO. SHOULD I WORK ON THAT OR ON THE EARLIER IDEA OF APPLYING SOME OF
10900 THESE FACTS TO ADD?
11000
11100 U: First, consider the ellipsis notation problem. Then the other one.
11200
11300 S: THERE SEEM EMPIRICALLY TO BE X APPLICATIONS OF
11400 SUCCESSOR IN THE IDENTITY X = SUCCESSOR(...(SUCCESSOR (0)).
11500 NOW I AM INVESTIGATING THE SPECIAL KNOWLEDGE.
11600
11700 THE FIRST PIECE OF KNOWLEDGE TRANSLATES READILY INTO:
11800 THE SUCCESSOR OF ANY NUMBER X CAN BE EXPRESSED AS
11900 (ADD 1 (ADD 1 (... (ADD 1 0) ...))), WHERE THERE ARE X APPLICATIONS OF ADD.
12000 SINCE THE SUCCESSOR OF ANY NUMBER X IS (ADD X 1), THIS SAYS THAT, FOR ANY X,
12100 SINCE ADD IS ASSOCIATIVE, (ADD X 1) = (ADD 1 1 ... 1 0) WHERE THERE ARE
12200 ADD(X 1) 1'S BEING ADDED TOGETHER IN THE ELLIPSIS.
12300 THIS SEEMS TO INDICATE THAT Y IS EQUAL TO Y 1'S ADDED TOGETHER. YES, THIS
12400 CHECKS OUT. THIS INCLUDES THE CASE OF 1 = ADD(1 0). FROM THIS WE
12500 CONCLUDE THAT 0 IS THE IDENTITY OF ADD.
12600
12700 ........
12800
12900 .GROUP SKIP 10
13000 ⊗5Skipping further ahead, toward discovering REMAINDER. 11/4/74.⊗*
13100
13200 U: Suppose we have two numbers a and b such that a does not divide b.
13300 Can you see anything interesting in that relation? As examples, consider
13400 the pairs 2 and 7, 3 and 7, 4 and 7, if you like.
13500
13600 S: I GENRALLY PREFER TO CONSIDER HUNDREDS OF EXAMPLES, SO I AM IGNORING
13700 YOUR LAST SUGGGESTION. NOW, AS TO THE FORMER...
13800
13900 IT SEEMS THAT, FOR EACH X, ALL NUMBERS > X DO NOT DIVIDE X. THUS IF
14000 A DOES NOT DIVIDE B, THERE IS STRONG EVIDENCE THAT A > B.
14100
14200 I FIND A COUNTEREXAMPLE FOR THAT; BUT IT WILL BE KEPT AS A PLAUSIBLE
14300 INFERENCE.
14400 THUS, GIVEN UNEQUAL A AND B, PRECISELY ONE OF THE FOLLOWING TWO CASES HOLDS:
14500 A DIVIDES B MAKES SENSE TO ASK
14600 B DIVIDES A MAKES SENSE TO ASK.
14700 I PROPOSE A NEW DEFINITION, THE CONCEPT (LAMBDA (X Y) (OR
14800 (X|Y)(Y|X))
14900 I AM INVESTIGATING THIS NEW IDEA. WHAT IS ITS NAME?
15000
15100 U: Let us call it MUTUAL DIVISIBILITY, abbreviated MD. However, it is more
15200 interesting to put x > y in the hypothesis. Can you restate your concept
15300 with this hypothesis?
15400
15500 S: IF X>Y THEN MD(X,Y) = Y|X. THAT IS, UNDER THIS HYPOTHESIS, MD= |.
15600
15700 SO?
15800
15900 U: MD as a concept is probably not interesting. Can you go on investigating
16000 the relation DOES NOT DIVIDE?
16100
16200 S: I AM CONSIDERING 810. THE NUMBERS WHICH DONOTDIVIDE IT ARE.......
16300 HMMM. THIS TIES IN TO THE UNIQUE FACTORIZATION THM:
16400 THE NONDIVISORS ARE THE PRIME NONDIVISORS AND ANY OF THEIR PRODUCTS.
16500 THE IDENTITY OF THE NONDIVISORS IS THUS SOLVED.
16600
16700 U: Consider the greatest number c such that c < b and a|c.
16800
16900 S: SO C = QxA FOR SOME Q. ALSO, B =(ADD C R) FOR SOME R.
17000 SO B = ADD(QxA, R). THIS MAKES SENSE IF SUCH A C EXISTS. THIS IN TURN
17100 IS TRUE ALWAYS BECAUSE A FINITE SET OF NUMBERS HAS A LARGEST NUMBER.
17200
17300 LET US SEE IF Q AND R ARE UNIQUE.
17400 THEY ARE.
17500
17600 SINCE WE WERE JUSST THINKING OF THE UFT, LET US CONSIDER THE EXPANSIONS
17700 OF EACH OF THESE NUMBERS: SAY
17800 A = πP↓i↑α↑(↑i↑)
17900 B = ... W...
18000 Q = ... V ...
18100 THEN C = AQ = ...P V.....
18200 AND R = ... U ...
18300
18400 THIS DOESN'T GET US ANYWHERE.
18500
18600 .......
18700
18800 The lessons from these two pieces are not completely expected:
18900 (i) The system must understand a much larger amount of input syntax than
19000 was anticipated, if the dialogue is to seem natural.
19100 This poses no ⊗4conceptual⊗* difficulty.
19200 (ii) The system seems like it could very easily get sidetracked into
19300 investigating the consequences of barren definitions. This might be a
19400 serious handicap to advanced discovery; at the lower levels, a teacher
19500 can guide the system out of such bogs.
19600 (iii) The problem of precisely what to print out to the user occurs here.
19700 Too much is bewildering and boring. Too little is dangerous. How exactly
19800 should the user be able to direct the system? How does the level of
19900 initiative get sensed (when the system should listen, should act, should
20000 direct.)
00100 .NEXT PAGE
00200 ⊗26. TASK: INTERNAL LEVEL⊗*
00300
00400
00500 ⊗5What modules will be needed for this initial task sequence?⊗*
00600
00700 The BEINGs may be roughly categorized along two axes.
00800 First, as OBJECT/OPERATOR/CONCEPTS, and second as
00900 to the level on which they operate: META/WORKING/BASIC.
01000 .NOFILL
01100
01200
01300 OBJECTS OPERATORS CONCEPTS
01400
01500 META Human Create-New-Being Definition
01600 Complete-a-Being Aesthetics
01700 Efficiency
01800 Plausibility
01900 Representation
02000
02100 WORKING Singleton Subset Proof
02200 Ordered-Pair Ordering
02300 Induction
02400 Deduction
02500
02600 BASIC Atom ∪ Truth
02700 Bag ∩ Falsity
02800 Tuple Append
02900 Set Set-difference
03000 Set-complement
03100
03200
03300 The scheme of development might be:
03400
03500 SUBSET
03600 SET-EQUALITY
03700 EQUIVALENCE RELATION PROPERTIES ARE SATISFIED
03800 PARTITION ALL SETS USING SET-EQUALITY
03900 SUCCESSOR-SET OF A SET
04000 AN ELEMENT OF ANY PARTITION CAN BE CONSTRUCTED BY REPEATED SUCCESSORING
04100 DECIDE ON A STANDARD REPRESENTATIVE FROM EACH PARTITION, USING THIS IDEA
04200 NUMBERS (UNARY REPRESENTATION): COMPOSE LIST OF N "SUCCESSOR"'s and 1 "0".
04300 COUNTING A SET: put it in its canonical representation
04400 COUNToAPPEND. Note this equals APPENDo(COUNTxCOUNT). Note COUNT↑2 = COUNT.
04500 PLUS. This takes two sets, and does COUNTING and UNIONING (in either order.)
04600 ZERO←→NULL SET, ONE←→SINGLETON, ASSOCIATIVITY AND COMMUTATIVITY OF PLUS.
04700 IDENTITIY OF ZERO: PLUS(x,ZERO) = COUNT(x).
04800 RELATION BETWEEN PLUS AND SUCCESSOR. PLUS(x,ONE) = COUNT(SUCCESSOR(x)).
04900 ANY NUMBER CAN BE CONSTRUCTED BY REPEATED PLUSSING OF 1.
05000 REPEATED PLUSSING OF n: PLUS(n,n,n,...,n)
05100 TIMES of two sets A, B. Let their counts be X and Y. Then
05200 the value of TIMES is PLUS(X,X,X,...,X), where the COUNT of that bag is Y.
05300 ASSOCIATIVITY, COMMUTATIVITY OF TIMES.
05400 IDENTITY ELEMENT. NIL FOR ∪, ⊗4U⊗* FOR ∩, zero FOR PLUS, one FOR TIMES.
05500 DISTRIBUTIVE LAW
05600 EXPONENTIATION. repeated TIMESing of n with itself.
05700 F(2,2) = 4, WHERE F IS PLUS, TIMES, EXPONENTIATE, HYPEREXPONENTIATE,...
05800 NON-ASSOCIAITIVITY. STOP THIS PARTICULAR EXTENSION SCHEME.
05900 INVERSE OPERATION
06000 PROPERTIES OF PLUS↑-↑1
06100 PROPERTIES OF TIMES↑-↑1, EXPONENTIATE↑-↑1
06200 COMPOSITIONS INVOLVING INVERSES: COUNToTIMES↑-↑1, PLUSoTIMES↑-↑1,
06300 PRIME NUMBERS
06400 UNIQUE FACTORIZATION THEOREM
06500 COMPARE UNIONoTIMES↑-↑1 OF TWO NUMBERS. ∩ OF THESE TWO SETS.
06600 THIS ∩ IS DETERMINED BY ITS LARGEST MEMBER: GCD.
06700 RELATIVE PRIMENESS
06800 from here, follow curriculum in standard number theory text.
06900 DIVISIBILITY THEORY (LCM, Division, Remainder, Euclidean Algorithm,
07000 congruence, TAU, sieving)
07100 NUMERICAL FUNCTIONS (multiplicative fns., perfect nos., MU)
07200 ARITHMETIC OF CONGRUENCE CALSSES (PHI, Residue Systems)
07300 SOLVING CONGRUENCES (Chinese Remainder Thm., Quadratic Residues)
07400 ADVANCED TOPICS (Distribution of primes into congruence classes,
07500 primitive roots, quadratic reciprocity)
00100 .NEXT PAGE
00200 ⊗27. FORMAL LANGUAGE⊗*
00300
00400 ⊗5Standard Math Notation⊗*
00500
00600 .NOFILL
00700
00800 IMPLICATION
00900 IF ... THEN ...
01000 IMPLIES
01100 IFF
01200 IF
01300 ONLY IF
01400 IS IMPLIED BY
01500 THEREFORE
01600 THUS
01700 SUPPOSE ... THEN
01800 LET ... THEN
01900 THEN
02000 SO
02100 HENCE
02200 IN ORDER TO...
02300 IT SUFFICES THAT
02400 NECESSITY
02500 SUFFICIENCE
02600 →
02700 ←
02800 ↔
02900 WHENEVER
03000 WHEN
03100
03200 SPECIFICATION
03300 SUCH THAT
03400 SATISFYING
03500 WITH
03600 WHERE
03700 SOME
03800 THE
03900 A/AN
04000 ALL
04100 EVERY
04200 NO ... IN
04300 ∀
04400 ∃
04500 FIXED
04600 VARIABLE
04700 ANY
04800 EACH
04900 MOST
05000 THERE EXISTS
05100 WHICH
05200 THAT
05300 THIS
05400 OTHER
05500
05600
05700 COMBINATION
05800 AND
05900 OR
06000 ∧
06100 ∨
06200 NOT
06300 ¬
06400 ALSO
06500 BUT
06600
06700
06800 OPERATION
06900 FUNCTION
07000 RELATION
07100 PREDICATE
07200 FN/FCN/FUNC
07300 f/g/h
07400 DO
07500 APPLY
07600 COMPUTE
07700 OPERATE
07800 PRODUCE
07900 ACCORDING
08000 CORRESPOND
08100 ALGORITHM
08200 <silent imperative>
08300 COMPOSITION
08400 o
08500 MAP
08600 TAKE
08700 SEND
08800 PULL
08900 IMAGE
09000 RANGE
09100 DOMAIN
09200 f:D→R
09300 PREIMAGE
09400 INVERSE
09500 UNDEFINED
09600 DEFINED
09700 f(a,b,c)
09800 f↑-↑1(x)
09900 f↑-↑1
10000 CLOSED
10100 PARTIAL
10200 TOTAL
10300
10400
10500 DEFINITION
10600 DEFINE
10700 CALL
10800 =df
10900 NOTATION FOR ...
11000 REFER TO...
11100 NAME
11200
11300
11400 KNOWN RELATIONS
11500 EQUALITY
11600 =
11700 IS/ARE
11800 INEQUALITY
11900 GREATER
12000 LESS
12100 SUBSET
12200 ⊂
12300 ⊃
12400 CONTAINS
12500 INCLUDES
12600 MORE
12700 INTERSECTS
12800 ∩
12900 UNION
13000 ∪
13100 APPEND
13200 BETWEEN
13300 INSIDE
13400 OUTSIDE
13500 INCLUSION
13600 EXACTLY
13700 COMPLEMENT
13800 SETDIFFERENCE
13900 +,-,x for sets
14000 CONS
14100 CAR
14200 CDR
14300 FIRST
14400 LAST
14500 ALL BUT
14600 JOIN
14700 PREVIOUS
14800 PRECEDE
14900 SUCCEED
15000 FOLLOWING
15100 NEXT
15200 NEAR
15300 FAR
15400 CLOSE
15500 ANALOGOUS
15600
15700
15800 ENTITIES
15900 ATOM
16000 ELEMENT
16100 CONSTANT
16200 VARIABLE
16300 SET
16400 TUPLE
16500 BAG
16600 MEMBER
16700 ε
16800 THING
16900 ENTITY
17000 OBJECT
17100 IDENTIFIER
17200 NAME
17300 LABEL
17400 VALUE
17500
17600
17700 ⊗5Fixed Formats for Quasi-English Meta-Comments, Questions, Hints⊗*
17800
17900 ACTIVITIES
18000 DO...
18100 CONSIDER...
18200 USE
18300 LOOP
18400 REPORT
18500 DISTINGUISH... AND/FROM ...
18600 EXPLAIN
18700 DISCUSS
18800 GET
18900
19000 RESTRICTED CONCEPTS
19100 ELLIPSIS
19200 ETC.
19300 AND SO ON
19400 ...
19500 SIMILARLY
19600 ANALOGY
19700 SIMPLIFY
19800 REDUCE
19900 FAILURE
20000 SUCCESS
20100 THINK
20200 CONCENTRATE
20300 CONSIDER
20400 ATTEND
20500 ASSUME
20600 SOLVE
20700 PROVE
20800 pronouns
20900 SEE
21000 HYPOTHESIS
21100 PROBLEM
21200 SOLUTION
21300 INVESTIGATE
21400 DISCOVER
21500 UNDERSTAND
21600
21700 TIME AND SPACE REFERENCES
21800 EARLIER
21900 LATER
22000 BEFORE
22100 AFTER
22200 THEN
22300 NOW
22400 NEVER
22500 ALWAYS
22600 HERE
22700 THERE
22800 UNDER
22900 ANYWHERE
23000 NOWHERE
23100
23200
23300 INDEFINITES
23400 SHOULD
23500 WOULD
23600 COULD
23700 MIGHT
23800 POSSIBLE
23900 PROBABLE
24000 PLAUSIBLE
24100 BEAUTY
24200 POTENTIAL
24300 CAN
24400 forms of TO BE
24500 OUGHT
24600 CONFUSION
24700 DEFINITE/INDEFINITE
24800 CERTAIN/UNCERTAIN
24900 TRANSLATE
25000 DIFFICULTY
25100 PLEASURE
25200 SO
25300 UNIQUE
25400 EXISTENCE
25500
25600
25700 QUESTIONS
25800 WHAT x
25900 WHY/WHY NOT x
26000 HOW
26100 WHEN
26200 .FILL
00100 .NEXT PAGE
00200 ⊗28. BIBLIOGRAPHY⊗*
00300
00400
00500 ⊗5BOOKS and MEMOS⊗*
00600
00700 Atkin, A. O. L., and Birch, B. J., eds., ⊗4Computers in Number Theory⊗*,
00800 Proceedings of the 1969 SRCA Oxford Symposium, 1971.
00900
01000 Badre, Nagib A., ⊗4Computer Learning From English Text⊗*, Memorandum
01100 No. ERL-M372, Electronics Research Laboratory, UCB, December 20, 1972.
01200 Also summarized in ⊗4CLET -- A COmputer Program that Learns Arithmetic
01300 from an Elementary Textbook⊗*, IBM Research Report RC 4235, February
01400 21, 1973.
01500
01600
01700 Banks, J. Houston, ⊗4Elementary-School Mathematics⊗*, 1966.
01800
01900 Beth, Evert W., and Piaget, Jean, ⊗4Mathematical Epistemology and
02000 Psychology⊗*, 1966.
02100
02200 Charosh, Mannis, ⊗4Mathematical Challenges⊗*, NCTM, 1965.
02300
02400 Copeland, Richard W., ⊗4How Children Learn Mathematics⊗*, 1970.
02500
02600 Courant, Richard, and Robins, Herbert, ⊗4What is Mathematics⊗*, 1941.
02700
02800 D'Augustine, Charles, ⊗4Multiple Methods of Teaching Mathematics in the
02900 Elementary School⊗*, 1968.
03000
03100 Dudley, Underwood, ⊗4Elementary Number Theory⊗*, 1969.
03200
03300 Eynden, Charles Vanden, ⊗4Number Theory: An Introduction to Proof⊗*, 1970.
03400
03500 Goldstein, Ira, ⊗4Elementary Geometry Theorem Proving⊗*, MIT AI Memo 280,
03600 April, 1973.
03700
03800 Hadamard, Jaques, ⊗4The Psychology of Invention in the Mathematical
03900 Field⊗*, 1945.
04000
04100 Halmos, Paul R., ⊗4Naive Set Theory⊗*, 1960.
04200
04300 GCMP, ⊗4Key TOpics in Mathematics⊗*, SRA, 1965.
04400
04500 Lamon, William E., ⊗4Learning and the Nature of Mathematics⊗*, UCSB, 1972.
04600
04700 Lefrancois, Guy R., ⊗4Psychological Theories and Human Learning⊗*, 1972.
04800
04900 Le Lionnais, F., ⊗4Great Currents of Mathematical Thought⊗*, 1971.
05000
05100 Meyer, Jerome S., ⊗4Fun With Mathematics⊗*, 1952.
05200
05300 Mirsky, L., ⊗4Studies in Pure Mathematics⊗*, 1971.
05400
05500 Moore, Robert C., ⊗4D-SCRIPT: A Computational Theory of Descriptions⊗*,
05600 MIT AI Memo 278, February, 1973.
05700
05800 National Council of Teachers of Mathematics, ⊗4The Growth of Mathematical
05900 Ideas⊗*, 24th yearbook, 1959.
06000
06100 National Council of Teachers of Mathematics, ⊗4Piagetian Cognitive-
06200 Development Research and Mathematical Education⊗*, 1971.
06300
06400 Nevins, Arthur J., ⊗4A Human Oriented Logic for Automatic Theorem
06500 Proving⊗*, MIT AI Memo 268, October, 1972.
06600
06700 Niven, Ivan, ⊗4An Introduction to the Theory of Numbers⊗*, 1960.
06800
06900 Ore, Oystein, ⊗4Number Theory and its History⊗*, Yale, 1948.
07000
07100 Polya, George, ⊗4Mathematical Discovery⊗*, Vol. 1, 1962; Vol. 2, 1965.
07200
07300 Polya, George, ⊗4Mathematics and Plausible Reasoning⊗*,
07400 Vols. 1 and 2, 1954.
07500
07600 Schminke, C. W., and Arnold, William R., eds., ⊗4Mathematics is a Verb⊗*,
07700 1971.
07800
07900 Skemp, Richard R., ⊗4The Psychology of Learning Mathematics⊗*, 1971.
08000
08100 Stewart, B. M., ⊗4Theory of Numbers⊗*, Michigan State, 1952.
08200
08300 Stokes, C. Newton, ⊗4Teaching the MEanings of Arithmetic⊗*, 1951.
08400
08500 Waismann, Friedrich, ⊗4Introduction to Mathematical Thinking⊗*, 1951.
08600
08700 ⊗5ARTICLES⊗*
08800
08900 Amarel, Saul, ⊗4On Representations of Problems of Reasoning about
09000 Actions⊗*, Machine Intelligence 3, 1968, pp. 131-171.
09100
09200 Bledsoe, W. W., ⊗4Splitting and Reduction Heuristics in Automatic
09300 Theorem Proving⊗*, Artificial Intelligence 2, 1971, pp. 55-77.
09400
09500 Bledsoe and Bruell, Peter, ⊗4A Man-Machine Theorem-Proving System⊗*,
09600 Artificial Intelligence 5, 1974, 51-72.
09700
09800 Buchanan, Feigenbaum, and Sridharan, ⊗4Heuristic Theory Formation⊗*,
09900 Machine Intelligence 7, 1972, pp. 267-...
10000
10100 Guard, J. R., et al., ⊗4Semi-Automated Mathematics⊗*, JACM 16,
10200 January, 1969, pp. 49-62.
10300
10400 Kling, Robert E., ⊗4A Paradigm for Reasoning by Analogy⊗*,
10500 Artificial Intelligence 2, 1971, pp. 147-178.
10600
10700 Knuth,Donald E., ⊗4Ancient Babylonian Algorithms⊗*,
10800 CACM 15, July, 1972, pp. 671-677.
10900
11000 Lee, RIchard C. T., ⊗4Fuzzy Logic and the Resolution Principle⊗*,
11100 JACM 19, January, 1972, pp. 109-119.
11200
11300 McCarthy, John, and Hayes, Patrick, ⊗4Some Philosophical Problems
11400 from the Standpoint of Artificial Intelligence⊗*, Machine Intelligence
11500 4, 1969, pp. 463-502.
11600
11700 Martin, W., and Fateman, R., ⊗4The MACSYMA System⊗*, Second
11800 Symposium on Symbolic and Algebraic Manipulation, 1971, pp. 59-75.
11900
12000 Pager, David, ⊗4A Proposal for a Computer-based Interactive Scientific
12100 Community⊗*, CACM 15, February, 1972, pp. 71-75.
12200
12300 Pager, David, ⊗4On the Problem of Communicating Complex Information⊗*,
12400 CACM 16, May, 1973, pp. 275-281.
12500
12600 Sloman, Aaron, ⊗4Interactions Between Philosophy and Artificial
12700 Intelligence: The Role of Intuition and Non-Logical Reasoning in
12800 Intelligence⊗*, Artificial Intelligence 2, 1971, pp. 209-225.
12900