perm filename SL[TLK,DBL] blob
sn#216658 filedate 1976-05-22 generic text, type C, neo UTF8
COMMENT ⊗ VALID 00059 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00006 00002 .DEVICE XGP
C00008 00003 .COMMENT Primes
C00009 00004 .COMMENT Heur1: going to extremes
C00010 00005 .COMMENT Heur1 overlay
C00011 00006 .COMMENT Extreme sets of nos
C00012 00007 .COMMENT UFT
C00013 00008 .COMMENT UFT overlay
C00014 00009 .COMMENT 1 reduction
C00015 00010 .COMMENT Defn of Divisors-of
C00016 00011 .COMMENT 2nd reduction
C00017 00012 .COMMENT chain of discoveries
C00019 00013 .COMMENT s→a heuristics
C00020 00014 .COMMENT 3 tasks
C00021 00015 .COMMENT Math thy
C00022 00016 .COMMENT Math text order
C00023 00017 .COMMENT 3 tasks
C00024 00018 .COMMENT 3 tasks revisited
C00025 00019 .COMMENT Diagram of concepts
C00027 00020 .COMMENT Facets
C00030 00021 .COMMENT Facets: COMPOSE
C00032 00022 .COMMENT Merged List of facets and concepts
C00035 00023 .COMMENT 2 kinds of operators--ways to grow
C00037 00024 .COMMENT 2 kinds of operators--ways to grow 1a
C00038 00025 .COMMENT 2 kinds of operators--ways to grow 1b
C00039 00026 .COMMENT 2 kinds of operators--ways to grow 2b
C00041 00027 .COMMENT Control joblist
C00043 00028 .COMMENT 3 actions on right-hand-sides
C00044 00029 .COMMENT Fill-in-examples action 1
C00045 00030 .COMMENT Fill-in-conjecs action 1b
C00047 00031 .COMMENT Create new concepts action 2
C00049 00032 .COMMENT 3 actions on right-hand-sides -- copy
C00050 00033 .COMMENT Suggest new New job action 3
C00052 00034 .COMMENT Task1 -- suggested by action 3
C00053 00035 .COMMENT that last task1 was already on the agenda
C00055 00036 .ZZ(Where is that damn global formula??)
C00056 00037 .COMMENT Big example of an agenda
C00059 00038 .COMMENT where we are now
C00060 00039 .COMMENT Full genl branch
C00062 00040 .COMMENT Compose heuristics-- just compose
C00063 00041 .COMMENT Operation heuristics -- jfrom genl branch
C00064 00042 .COMMENT Remainder of genl branch
C00066 00043 .ZZ(Genl branch overlay: point to the int heurs which are true of ∪-o-')
C00067 00044 .COMMENT Excerpt: Cardinality
C00070 00045 .COMMENT Genls of Equality
C00071 00046 .COMMENT Defn of EQUAL
C00072 00047 .COMMENT OVERLAY Defn of EQUAL
C00073 00048 .COMMENT 2nd example: UFT
C00088 00049 .COMMENT Losers
C00089 00050 .COMMENT Winners
C00091 00051 .COMMENT open problem
C00092 00052 .COMMENT expts
C00093 00053 .ZZ(Should there be a slide for Geom experiment in general? p53)
C00094 00054 .COMMENT Prime angles
C00095 00055 .COMMENT AM Conjec
C00097 00056 .COMMENT closing slide
C00098 00057 .COMMENT SUPPLEMENTARY: Sys Names
C00099 00058 .COMMENT SUPPLEMENTARY: Factorings: exs of divisors
C00100 00059 .PORTION TODO
C00101 ENDMK
C⊗;
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.TURN ON "↑α[]↓_π{"
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.TABBREAK
.ODDLEFTBORDER ← EVENLEFTBORDER ← 1000
.PAGE FRAME 54 HIGH 98 WIDE
.AREA TEXT LINES 1 TO 53
.DOUBLE SPACE
.PREFACE 2
.NOFILL
.PREFACE 1
.!XGPLFTMAR←75
.MACRO B ⊂ B0 ⊃
.MACRO B0 ⊂ BEGIN NOFILL SELECT 9 INDENT 0 GROUP PREFACE 0 MILLS TURN OFF "{↑↓}[]α" ⊃
.MACRO E ⊂ APART END ⊃
.MACRO ZZ(A) ⊂ BEGIN TURN ON "{}"
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⊗1page {PAGE}: A⊗*
. ⊃
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.MACRO B0 ⊂ BEGIN SELECT 9 NOFILL PREFACE 0 MILLS TURN OFF "↑↓" GROUP ⊃
.MACRO WBOX (LEFTN,RIGHTN) ⊂ B0;
. NARROW LEFTN,RIGHTN;
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. TOPLIN ⊃
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⊂∞α→⊃
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%∞α→$
. ⊃
.AT "MBOX" L "$" ⊂ "~ ⊗7L⊗9∞ →~" ⊃
.NEXT PAGE
.INDENT 0
.SELECT 1
.COMMENT Primes;
.GROUP SKIP 20
.BEGIN CENTER SELECT 2
A natural number is ↓_prime_↓
iff
it has only two divisors
.END
.SKIP TO COLUMN 1
.COMMENT Heur1: going to extremes;
.ONCE CENTER SELECT 2
↓_Going to Extremes_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓→∞\α" TABS 10,20,50,67,77 SELECT 2
\⊗2A⊗*\\\\⊗2B⊗*
\\\f
\\\\b⊗*
\b⊗7↑-↑1⊗*\\\\
.CENTER
If b is some unusual subset of B,
Then isolate the subset f⊗7↑-↑1⊗*(b) of A.
.END
.SKIP TO COLUMN 1
.COMMENT Heur1 overlay;
.TURN OFF "{}"
.GROUP SKIP 6
.ONCE CENTER SELECT 7
Divisors(12) = {1,2,3,4,6,12}
.ZZ(Draw N and 2↑N on p.5; Heur1 overlay; div of 12)
.SKIP TO COLUMN 1
.COMMENT Extreme sets of nos;
.BEGIN CENTER SELECT 2 TURN OFF "{}"
Numbers having no divisors
{ }
Numbers having only 1 divisor
{ 1 }
Numbers having only 2 divisors
{ 2, 3, 5, 7, 11, 13, ... }
Numbers having only 3 divisors
{ 4, 9, 25, 49, 121, 169, ... }
Numbers having only 4 divisors
{ 6, 10, 14, 15, 21, 22, ... }
.END
.SKIP TO COLUMN 1
.COMMENT UFT;
.GROUP SKIP 5
.BEGIN CENTER SELECT 2
↓_UNIQUE FACTORIZATION CONJECTURE _↓
"Any number can be factored uniquely into
numbers each having only two divisors."
.END
.SKIP TO COLUMN 1
.COMMENT UFT overlay;
.GROUP SKIP 9
.BEGIN CENTER SELECT 2
Primes."
.END
.SKIP TO COLUMN 1
.COMMENT 1 reduction;
.BEGIN CENTER SELECT 2 PREFACE 0 TURN ON "\" TABS 55
How in the world could you
discover "Prime Numbers" ?
||
||
||
||
⊗4Investigate those items whose image,⊗*
⊗4under some operation, is extreme⊗*
||
||
||
||
How in the world could you
discover "Divisors-of" ?
.END
.SKIP TO COLUMN 1
.COMMENT Defn of Divisors-of;
.GROUP SKIP 14
.BEGIN CENTER SELECT 2 TURN OFF "{}" PREFACE 0
Divisors(x) ≡ { x | (∃y) Times(x,y)=n }
.END
.SKIP TO COLUMN 1
.COMMENT 2nd reduction;
.GROUP SKIP 12
.BEGIN CENTER SELECT 2 TURN OFF "↑↓" PREFACE 0 TURN ON "\" TABS 55
||
||
||
||
⊗4Investigate the inverse of⊗*
⊗4an interesting relation⊗*
||
||
||
||
How in the world could you
discover "Times" ?
.END
.SKIP TO COLUMN 1
.COMMENT chain of discoveries;
.B0 CENTER
⊗2Discovery of Prime Numbers⊗*
~
~
~
⊗4Look at extrema⊗*
~
~
~
⊗2Discovery of Divisors-of⊗*
~
~
~
⊗4Look at inverses⊗*
~
~
~
⊗2Discovery of Multiplication⊗*
/\
/ \
/ \
/ \
/ \
⊗4Repeat⊗* ⊗4Analog⊗*
/ \
/ \
/ \
/ \
/ \
⊗2Discovery of Addition⊗* ⊗2Discovery of Cross-prod⊗*
~ ~
~ ~
# #
# #
# #
.TURN ON "↑↓_"
# # # ⊗4(↓_BASE OF KNOWN CONCEPTS_↓)⊗* # # #
.END
.ZZ(Chain; p. 12; Overlay onto this a slide of spreading-out lines going upward)
.ZZ(Chain; p. 12; Overlay onto this a slide of diff-colored nodes/ops; to repr heurs)
.SKIP TO COLUMN 1
.COMMENT s→a heuristics;
.GROUP SKIP 0
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0 RETAIN
.ONCE CENTER
↓_FORMAT FOR THE HEURISTIC RULES_↓
.ONCE CENTER
↓_Too Explosive_↓:
"Conjoin any two concepts C1 and C2, thus creating a new one."
.ONCE CENTER
↓_Acceptable_↓:
"IF C1 and C2 have some common examples,
and they were not thought to be related,
THEN consider the new concept defined as their conjunction."
.END
.SKIP TO COLUMN 1
.COMMENT 3 tasks;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_MATHEMATICAL DISCOVERY_↓
.ONCE CENTER SELECT 2
↓_as HEURISTIC SEARCH_↓
.BEGIN SELECT 2 PREFACE 2 INDENT 20 SKIP 2
(1) Starting knowledge about math
(2) Legal operators
(3) Heuristic strategies
.END
.SKIP TO COLUMN 1
.COMMENT Math thy;
.ONCE CENTER SELECT 2
↓_Mathematical Theory_↓
.BEGIN SELECT 2 PREFACE 0 TURN ON "→"
.GROUP SKIP 6
.ONCE CENTER
DEFINITIONS and THEOREMS
Axioms and Postulates
Operators →FOUNDATION
Undefined objects
.END
.SKIP TO COLUMN 1
.COMMENT Math text order;
.ONCE CENTER SELECT 2
↓_Textbook Order_↓
.BEGIN CENTER SELECT 2 PREFACE 0 TURN ON "→"
State the Foundation
↓
↓
State some Definitions
↓
↓
State a Lemma
↓
↓
Prove the Lemma
↓
↓
State a Theorem
↓
↓
Prove the Theorem
↓
↓
Motivation for the
theorem (optional)
.END
.SKIP TO COLUMN 1
.COMMENT 3 tasks;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Mathematical Discovery_↓
.ONCE CENTER SELECT 2
↓_as Heuristic Search_↓
.BEGIN SELECT 2 PREFACE 2 INDENT 20 SKIP 2
(1) Starting knowledge about math
(2) Legal operators
(3) Heuristic strategies
.END
.SKIP TO COLUMN 1
.COMMENT 3 tasks revisited;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Mathematical Discovery_↓
.ONCE CENTER SELECT 2
↓_as Heuristic Search_↓
.BEGIN SELECT 2 PREFACE 2 INDENT 20 SKIP 2
(1) Starting knowledge about math ⊗9 ∂ ⊗*
(2) Legal operators
(3) Heuristic strategies
.END
.SKIP TO COLUMN 1
.COMMENT Diagram of concepts;
.ONCE CENTER SELECT 2
↓_Initial Concepts_↓
.BEGIN NOFILL SELECT 6 PREFACE 0 INDENT 0 TURN ON "\" TABS 30,65,75 RETAIN
.BEGIN CENTER
Anything
Any-concept
Activity Object
Predicate Operation Atom Conjec Structure
Constant-pred Equality-pred Truth-value Empty-struc
Always-T Always-F Obj-equality T F Mult-eles Non-mult Ordered Unord
.END
\\Osets
\Repeat-op\
\Substitute
\Compose-ops\
\Coalesce-op\Sets
\Invert-op
\Canonize-preds
\Identity\Lists
\Insert-struc
\Delete-struc
\Union-strucs\Bags Ord-pairs
\Intersect-strucs
\Difference-strucs
\Reverse-struc
.END
.SKIP TO COLUMN 1
.COMMENT Facets;
.ONCE CENTER SELECT 2
↓_Facets of a Concept_↓
.BEGIN NARROW 3,0 SELECT 2 PREFACE 0
Characterizations
⊗7Name(s)⊗*
⊗7Definitions ⊗*
⊗7Algorithms ⊗*
⊗7Domain/range⊗*
⊗7Intuitions: abstract representations⊗*
Ties to other concepts
⊗7Specializations⊗*
⊗7Generalizations⊗*
⊗7Examples⊗*
⊗7Operations one can do to this concept⊗*
⊗7Conjectures/theorems involving this concept⊗*
⊗7Analogies⊗*
Heuristics
⊗7Worth: Why this concept is worth naming⊗*
⊗7Interest: When an instance of it is (un)interesting⊗*
⊗7Fillin: Hints for filling in parts of instances⊗*
⊗7Suggest new activities for AM to consider⊗*
⊗7Check: things to watch out for⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Facets: COMPOSE;
.ONCE CENTER SELECT 2
↓_Facets of "COMPOSE"_↓
.BEGIN NARROW 3,0 SELECT 2 PREFACE 0
Characterizations
⊗7Name(s): Compose ⊗*
⊗7Definitions: λ (f1 f2 f3)...⊗8 ⊗7←recursive, opaque, wffs⊗2
⊗7Algorithms: λ (f1 f2)... ⊗8 ⊗7←opaque, transparent, destructive ⊗2
⊗7Domain/range: Activities x Activities → Activities⊗2
Ties to other concepts
⊗7Specializations: Compose f with itself⊗2
⊗7Generalizations: Operation⊗*
⊗7Examples: (Intersect, Complement) → Intersect⊗1o⊗7Complement⊗2
⊗7Conjec: (AoB)oC ≡ Ao(BoC)⊗*
⊗7Analogies: ⊗*
Heuristics
⊗7Worth: 500⊗*
⊗7Interest: Domain=Range; both args are interesting⊗*
⊗7Fillin: D/R are Domain(arg1) and Range(arg2)⊗*
⊗7Sugg: Check AoB for properties which A or B have⊗*
⊗7Check: Domain(arg2) should intersect Range(arg1)⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Merged List of facets and concepts;
.BEGIN NOFILL SELECT 6 PREFACE 0 RETAIN; TURN ON "\" TABS 48
.INDENT 3
⊗2 ↓_Initial Concepts_↓⊗*\⊗2↓_Facets of_↓⊗*
⊗2↓_Explicitly Provided_↓⊗*\⊗2↓_each concept_↓⊗*
Anything\Names
Any-concept\Definitions
Activity\Algorithms
Object\Domain/range
Predicate\Intu's
Operation\
Atom\Specializations
Conjec \⊗6Generalizations⊗*
Structure\⊗6Examples⊗*
Constant-pred \⊗6In-range-of
Equality-pred\In-domain-of
Truth-value\⊗6Conjectures
Empty-struc\⊗6Analogies⊗*
Identity-op
Always-T Always-F\⊗6Worth
Obj-equality\⊗6Interestingness
Multiple-elements-allowed \⊗6Fillin
No-multiple-elements-allowed\⊗6Suggest
Ordered \⊗6Check
Unordered
Repeat-op\
Substitute
Compose-ops\
Osets Sets
Lists Bags
Ordered-pairs
Coalesce-op
Invert-op
Canonize-preds
Insert-struc
Delete-struc
Union-strucs
Intersect-strucs
Difference-strucs
Reverse-struc
T F
.END
.SKIP TO COLUMN 1
.COMMENT 2 kinds of operators--ways to grow;
.GROUP SKIP 2
.ONCE CENTER SELECT 2
↓_Fill in a blank facet_↓
.BEGIN SELECT 7 PREFACE 0 TABS 50 TURN ON "\↑↓_"
⊗2 ↓_COMPOSE_↓⊗*\⊗2 ↓_COMPOSE_↓⊗*
Definition:\Definition:
Domain/Range:\Domain/Range:
Algorithms:\Algorithms:
Examples:\Examples:
Up isa:\Up isa:
Generalizations:\Generalizations:
...\ ...
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Blow up a facet into a whole new concept_↓
⊗2 ↓_COMPOSE_↓⊗*\⊗2↓_Union⊗7↑o⊗*Complement_↓⊗*
Definition:\Definition:
Domain/Range:\Domain/Range:
Algorithms:\Algorithms:
Examples:\Examples:
Up isa:\Up isa:
Generalizations:\Generalizations:
...\ ...
.END
.SKIP TO COLUMN 1
.COMMENT 2 kinds of operators--ways to grow 1a;
.GROUP SKIP 6
.BEGIN SELECT 7 PREFACE 0 TABS 50 TURN ON "\↑↓_"
⊗2 ↓_COMPOSE_↓⊗*
Definition:
Domain/Range:
Algorithms:
Examples:
Up isa:
Generalizations:
...
.END
.SKIP TO COLUMN 1
.COMMENT 2 kinds of operators--ways to grow 1b;
.GROUP SKIP 2
.ONCE CENTER SELECT 2
↓_Fill in a blank facet_↓
.BEGIN SELECT 7 PREFACE 0 TABS 60 TURN ON "\↑↓_"
\⊗2 ↓_COMPOSE_↓⊗*
\Definition:
\Domain/Range:
\Algorithms:
\Examples:
\Up isa:
\Generalizations:
\ ...
.END
.SKIP TO COLUMN 1
.COMMENT 2 kinds of operators--ways to grow 2b;
.GROUP SKIP 4
.BEGIN SELECT 7 PREFACE 0 TABS 60 TURN ON "\↑↓_"
⊗2 ⊗*
.GROUP SKIP 4
\⊗2↓_Union⊗7↑o⊗*Complement_↓⊗*
\Definition:
\Domain/Range:
\Algorithms:
\Examples:
\Up isa:
\Generalizations:
\ ...
.ONCE CENTER SELECT 2
↓_Blow up a facet into a whole new concept_↓
.END
.SKIP TO COLUMN 1
.COMMENT Control joblist;
.ONCE CENTER SELECT 2
↓_AGENDA of Tasks Worth Doing_↓
.BEGIN narrow 10,0 SELECT 7 TABS 15,25,40,50 TURN ON "\↑↓←\" PREFACE 0
⊗4(Generalize the Definitions facet of Equality concept)⊗*
\\Priority=700
\\Reasons=((Equality is rarely satisfied)
\\ (No known genls of Equality))
⊗4(Fill in Examples facet of Primes concept)⊗*
\\Priority=400
\\Reasons=((No known exs of primes yet))
\\ ("Divisors-of" asked for examples of primes))
⊗4(Fill in new Algorithms for Compose concept)⊗*
\\Priority=300
\\Reasons=((Empirical: taking too long))
.END
.SKIP TO COLUMN 1
.COMMENT 3 actions on right-hand-sides;
.BEGIN CENTER SELECT 2 PREFACE 0
↓_3 Kinds of Actions_↓
.FILL INDENT 5,0,0 PREFACE 2 SELECT 2
(1) Fill in entries on some facet of some concept
(2) Create a new concept
(3) Suggest new jobs for the agenda
.END
.SKIP TO COLUMN 1
.COMMENT Fill-in-examples action 1;
.GROUP SKIP 17
.BEGIN NOFILL INDENT 20 SELECT 7 PREFACE 0
If the current task is of the form (FILL-IN EXAMPLES of ←C),
and the In-range-of facet of C is not empty,
Then pick any operation F from that facet,
and find examples x→y of F,
and collect all such y's.
.END
.ZZ(Put a little hand pointing --overlaid-- to the fillinentries 1)
.SKIP TO COLUMN 1
.COMMENT Fill-in-conjecs action 1b;
.GROUP SKIP 18
.BEGIN NOFILL PREFACE 0 INDENT 20 SELECT 7
If the current task is to Check Examples of X,
and (Forsome Y) Y is a generalization of X,
and Y has at least 10 examples,
and all examples of Y are also examples of X,
(ignoring boundary cases)
Then print the following conjecture:
"X is really no more specialized than Y",
and add it to the Examples facet of Conjectures,
and add it to the Conjecs facet of X,
and add it to the Conjecs facet of Y,
and Check the truth of this conjecture on
each boundary example of Y,
and add "X" to the Generalizations facet of Y,
and add "Y" to the Specializations facet of X,
.END
.ZZ(Put a little hand pointing --overlaid-- to the fillinentries 1)
.SKIP TO COLUMN 1
.COMMENT Create new concepts action 2;
.group SKIP 10
.BEGIN NOFILL PREFACE 0 INDENT 0 SELECT 7
.COMMENT WARNING: Centering going on -- Don't JUSTIFY!!!;
If the current task is (Fill-in examples of ←F),
and F is an operation from domain A into range B,
and more than 100 examples of A are known,
and more than 10 examples of B are produced by
applying F to these 100 domain items,
and at least 1 of these range items is an
extremal distinguished kind of B,
Then (for each such distinguished member "b"εB)
create the following new concept:
.WBOX(8,6)
MBOX Name: F-Inverse-of-b $
MBOX Definition: λ (x) ( F(x) is b ) $
MBOX Generalization: A $
MBOX Worth: Average(Worth(A), Worth(F), Worth(B), Worth(b), ||Examples(B)||) $
MBOX Interest: Conjecs tying this concept to either F or Inverse(F) $
.EBOX
In case the user asks, the reason for doing this is:
"Worthwhile investigating those A's which have an
unusual F-value, namely, those whose F-value is b"
The total time to spend right now on all these new concepts is:
Half the remaining cpu time in the current task's quantum.
.END
.ZZ(Put a little hand pointing --overlaid-- to the createcon 2)
.SKIP TO COLUMN 1
.COMMENT 3 actions on right-hand-sides -- copy ;
.BEGIN CENTER SELECT 2 PREFACE 0
↓_3 Kinds of Actions_↓
.FILL INDENT 5,0,0 PREFACE 2 SELECT 2
(1) Fill in entries on some facet of some concept
(2) Create a new concept
(3) Suggest new jobs for the agenda
.END
.SKIP TO COLUMN 1
.COMMENT Suggest new New job action 3;
.GROUP SKIP 14
.BEGIN NOFILL PREFACE 0 INDENT 3 SELECT 7
If the current task was (Fill-in examples of ←X),
and X is a predicate,
and over 100 items are known in the domain of X,
and more than 10 cpu seconds were recently spent
trying to randomly run X on those arguments,
and the ratio of successes/failures is >0 and <.05
Then add the following task to the agenda:
(Fill-in generalizations of X),
for the following reason:
"X is rarely satisfied; a slightly less
restrictive concept might be more interesting".
This reason's rating is computed as:
3x(ratio of nonexamples/examples found)
.END
.ZZ(Put a little hand pointing --overlaid-- to the suggestnewjobs 3)
.SKIP TO COLUMN 1
.COMMENT Task1 -- suggested by action 3;
.GROUP SKIP 30
.BEGIN NOFILL PREFACE 0 INDENT 0 SELECT 4 CENTER
(Fillin Generalizations facet of "Equality" concept)
⊗7Reason=(Equality is rarely satisfied)⊗*
.END
.SKIP TO COLUMN 1
.COMMENT that last task1 was already on the agenda;
.ONCE CENTER SELECT 2
↓_AGENDA_↓ of tasks worth doing
.BEGIN SELECT 7 TABS 10,15,25,40 TURN ON "\↑↓←\" PREFACE 0
⊗4(Fill in Examples facet of "Primes" concept)⊗*
\\Priority=400
\\Reasons=((No known exs of primes yet)
\\\(Divisors-of asked for examples of primes))
⊗4(Check Domain/range facet of "Square-root" concept)⊗*
\\Priority=400
\\Reasons=((Square-root was unable to return a value for 8)
\\\(Square-root was unable to return a value for 3)
\\\(Should check the domain/range entries for any
\\\ operation created as an inverse)
⊗4(Fill in Generalizations facet of "Equality concept")⊗*
\\Priority=350
\\Reasons=((No known generalizations of Equality))
⊗4(Fill in new Algorithms for Compose concept)⊗*
\\Priority=300
\\Reasons=((Empirical: taking too long))
.END
.SKIP TO COLUMN 1
.ZZ(Where is that damn global formula??)
.SKIP TO COLUMN 1
.COMMENT Big example of an agenda;
.ONCE CENTER SELECT 2
↓_AGENDA_↓ of tasks worth doing
.BEGIN SELECT 7 TABS 10,15,25,40 TURN ON "\↑↓←\" PREFACE 0
⊗4(Fill in Generalizations facet of "Equality concept")⊗*
\\Priority=700
\\Reasons=((No known generalizations of Equality)
\\\(Equality is rarely satisfied)
\\\(Focus of attention: recently worked on Equality))
⊗4(Check Conjecs facet of "Equality" concept)⊗*
\\Priority=500
\\Reasons=((Worth of Equality has recently risen)
\\\(Conjecs involving Equality have just been found)
\\\(Focus of attention: recently worked on Equality))
⊗4(Fill in Examples facet of "Primes" concept)⊗*
\\Priority=400
\\Reasons=((No known exs of primes yet))
⊗4(Check Examples facet of "Set-union" concept)⊗*
\\Priority=350
\\Reasons=(Many examples of Set-union recently added)
\\\(Worth of Set-union has recently risen))
⊗4(Check Domain/range facet of "Square-root" concept)⊗*
\\Priority=300
\\Reasons=((Square-root was unable to return a value for 8)
\\\(Square-root was unable to return a value for 3)
\\\(Should check the domain/range entries for any
\\\ operation created as an inverse)
⊗4(Fill in new Algorithms for Compose concept)⊗*
\\Priority=200
\\Reasons=((Empirical: taking too long))
.END
.SKIP TO COLUMN 1
.COMMENT where we are now;
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 20
.GROUP SKIP 8
What individual heuristics can do
How AM locates relevant heuristics
What AM as a whole can do
.END
.SKIP TO COLUMN 1
.COMMENT Full genl branch;
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0 TURN ON "←→α↑↓"
.ONCE CENTER
ANYTHING
⊗1Interest: The User recently mentioned C.⊗*
⊗1Interest: C is the result of F(x) where F and x are very interesting⊗*
.ONCE CENTER
ANY-CONCEPT
⊗1Fillin: Find the results of examples of any operation whose range is C.⊗*
⊗1Fillin: Instantiate the definition of C.⊗*
⊗1Check: Run the definition of C, and ensure that it returns True.⊗*
.ONCE CENTER
ACTIVITY
⊗1Fillin: Run the algorithm for C on random members of its domain.⊗*
⊗1Interest: Al the domain components are identical.⊗*
.ONCE CENTER
OPERATION
⊗1Interest: The range is one component of the domain.⊗*
⊗1Fillin: For any j, include C(i,j)=j, where i is an identity of C.⊗*
.ONCE CENTER
COMPOSITION DOMAIN=RANGE-OP
⊗1Interest: CoC is nontrival. →Fillin: include any fixed-points of C.⊗*
⊗1Check: Dom(F) ∩ Ran(G) is nontrivial→Check: Ran(C) is not just a subset of Dom(C)⊗*
⊗1Interest: Conjecs about F also hold for FoG.→Interest: C(x,C(y,z))=C(C(x,y),z)⊗*
.ONCE CENTER
UNION⊗1↑o⊗*COMPLEMENT
.END
.SKIP TO COLUMN 1
.COMMENT Compose heuristics-- just compose;
.GROUP SKIP 16
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0 TURN ON "←→α↑↓"
.ONCE INDENT 5
COMPOSITION
⊗1Interest: CoC is nontrival⊗*
⊗1Check: Dom(F) ∩ Ran(G) is nontrivial.⊗*
⊗1Interest: Conjecs about F also hold for FoG.⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Operation heuristics -- jfrom genl branch;
.GROUP SKIP 11
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0 TURN ON "←→α↑↓"
.ONCE CENTER
OPERATION
⊗1Fillin: For any j, include C(i,j)=j, where i is an identity of C.⊗*
⊗1Interest: The range is one component of the domain.⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Remainder of genl branch;
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0 TURN ON "←→α↑↓"
.ONCE CENTER
ANYTHING
⊗1Interest: The User recently mentioned C.⊗*
⊗1Interest: C is the result of F(x) where F and x are very interesting⊗*
.ONCE CENTER
ANY-CONCEPT
⊗1Fillin: Find the results of examples of any operation whose range is C.⊗*
⊗1Fillin: Instantiate the definition of C.⊗*
⊗1Check: Run the definition of C, and ensure that it returns True.⊗*
.ONCE CENTER
ACTIVITY
⊗1Fillin: Run the algorithm for C on random members of its domain.⊗*
⊗1Interest: Al the domain components are identical.⊗*
.ONCE INDENT 45
DOMAIN=RANGE-OP
.BEGIN INDENT 50
⊗1Fillin: include any fixed-points of C.⊗*
⊗1Check: Ran(C) is not just a subset of Dom(C)⊗*
⊗1Interest: C(x,C(y,z))=C(C(x,y),z)⊗*
.END
.ONCE CENTER
UNION⊗1↑o⊗*COMPLEMENT
.END
.SKIP TO COLUMN 1
.ZZ(Genl branch overlay: point to the int heurs which are true of ∪-o-')
.SKIP TO COLUMN 1
.COMMENT Excerpt: Cardinality;
.ONCE CENTER SELECT 2
↓_As the User Sees It_↓
.BEGIN SELECT 6 INDENT 10 PREFACE 0
AM is looking for examples of things which are EQUAL.
4 examples were found, and 151 non-examples.
It is worth trying to generalize the predicate EQUAL.
⊗4USER: Why?⊗*
Because this heuristic is in the Fillin facet of the Examples concept:
"If the ratio of examples/nonexamples of any predicate P is
below .1, a generalization of P may be more interesting than P."
AM is trying to generalize the predicate EQUAL.
EQUAL has 2 definitions.
The second definition is recursive.
It involves the conjunction of 2 recursive calls.
I can generalize EQUAL by replacing either call by "T".
The first generalization is EQ1.
It is like EQUAL, but does not recurse on the CAR of its args.
The next generalization is EQ2.
It is like EQUAL, but does not recurse on the CDR of its args.
⊗4USER: Call EQ1 "SAME-LENGTH".⊗*
AM is looking for examples of things which are SAME-LENGTH.
25 examples were found, and 100 non-examples.
It is worth trying to find a canonical form for all objects, an
operation C(x) which converts them to canonical form,
so that SAME-LENGTH(x,y) iff EQUAL( C(x), C(y) ).
AM is trying to canonize objects wrt EQUAL and SAME-LENGTH.
Conclusion: the canonical args of SAME-LENGTH are
BAGS which contain only "T"'s.
It is worth restricting Bag-operations to these kinds of Bags.
⊗4USER: Call such Bags "NUMBERS".⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Genls of Equality;
.GROUP SKIP 2
.ONCE CENTER SELECT 2
↓_Rippling up from "Equality"_↓
.BEGIN SELECT 2 INDENT 0 PREFACE 0 TURN ON "\" TABS 10,20,30,40,50,60
\\\\ANYTHING
\\\ANY-CONCEPT
\\ACTIVITY
\PREDICATE
EQUALITY
.END
.SKIP TO COLUMN 1
.COMMENT Defn of EQUAL;
.ONCE CENTER SELECT 2
↓_Recursive Defn of EQUAL_↓
.GROUP SKIP 1
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓[]{}&" SELECT 2
Definition of ↓_EQUAL_↓(L,M):
(LAMBDA (L M)
(IF L and M are both atomic,
then L must be identically EQ to M,
else
IF L and M are both lists,
then both
(CAR(L) is ↓_EQUAL_↓ to CAR(M))
and
(CDR(L) is ↓_EQUAL_↓ to CDR(M))
]
.END
.SKIP TO COLUMN 1
.COMMENT OVERLAY Defn of EQUAL;
.ZZ(P.43; Overlay of Equal defn; make a diff. color)
.GROUP SKIP 2
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓[]{}&" SELECT 2
⊗2Do this by hand:⊗*
Definition of ↓_EQ1_↓XX
XXXXXXXXXXXXXXXXXX T
(CDR(L) is ↓_EQ1_↓XX to CDR(M))
.END
.SKIP TO COLUMN 1
.COMMENT 2nd example: UFT;
.ZZ(UFT example slide not done yet)
.BEGIN NOFILL PREFACE 0 TURN OFF "{}" TURN ON "↑↓\" TABS 18,21 SELECT 6
.AT "-1-" ⊂ "↑-↑1" ⊃
.AT "***" ENTRY "." ⊂
⊗2** ↓_ENTRY:_↓ **⊗6
. ⊃
***Task 65.
Filling in examples of the following concept: "Divisors-of".
3 Reasons:\(1) No known examples for Divisors-of so far.
\(2) TIMES, which is related to Divisors-of, has just risen in interest.
\(3) Focus of attention: AM recently defined Divisors-of.
26 examples found, in 9.2 seconds. e.g.: Divisors-of(6)={1 2 3 6}.
.GROUP SKIP 3
***Task 66.
Considering numbers which have unusual sets of Divisors-of.
2 Reasons:\(1) Worthwhile to look for extreme cases.
\(2) Focus of attention: AM recently worked on Divisors-of.
Filling in examples of numbers with 0 divisors.
0 examples found, in 4.0 seconds.
Conjecture: no numbers have 0 divisors.
Filling in examples of numbers with 1 divisors.
1 examples found, in 4.0 seconds. e.g.: Divisors-of(1)={1}.
Conjecture: 1 is the only number with 1 divisor.
Filling in examples of numbers with 2 divisors.
24 examples found, in 4.0 seconds. e.g.: Divisors-of(13)={1 13}.
No obvious conjecture. This kind of number merits more study.
Creating a new concept: "Numbers-with-2-divisors".
Filling in examples of numbers with 3 divisors.
11 examples found, in 4.0 seconds. e.g.: Divisors-of(49)={1 7 49}.
All numbers with 3 divisors are also Squares.
This kind of number merits more study.
Creating a new concept: "Numbers-with-3-divisors".
.SKIP TO COLUMN 1
***Task 67.
Considering the square-roots of Numbers-with-3-divisors.
2 Reasons:\(1) Numbers-with-3-divisors are unexpectedly also perfect Squares.
\(2) Focus of attention: AM just worked on Numbers-with-3-divisors.
All square-roots of Numbers-with-3-divisors seem to be Numbers-with-2-divisors.
e.g., Divisors-of( Square-root(169) ) = Divisors-of(13) = {1 13}.
Formulating the converse to this statement. Empirically, it seems to be true.
Squaring a Number-with-2-divisors seems to be a Number-with-3-divisors.
This is very unusual. It is not plausibly a coincidence.
Boosting interestingness factor of the concepts involved:
Interestingness factor of "Divisors-of" raised from 300 to 400.
Interestingness factor of "Numbers-with-2-divisors" raised from 100 to 600.
Interestingness factor of "Numbers-with-3-divisors" raised from 200 to 600.
.GROUP SKIP 3
⊗4USER: Call the set of numbers with 2 divisors "Primes".⊗*
.SKIP TO COLUMN 1
***Task 79.
Examining TIMES-1-(x), looking for patterns involving its values.
2 Reasons:\(1) TIMES-1- is related to the newly-interesting concept "Divisors-of".
\(2) Many examples of TIMES-1- are known, to induce from.
Looking specifically at TIMES-1-(12), which is { (12) (2 6) (2 2 3) (3 4) }.
13 conjectures proposed, after 2.0 seconds.
Testing them on other examples of TIMES-1-.
Only 2 of these 13 conjectures are verified for all examples of TIMES-1-:
Conjecture 1: TIMES-1-(x) always contains a singleton bag.
e.g., TIMES-1-(12), which is { (12) (2 6) (2 2 3) (3 4) }, contains (12).
e.g., TIMES-1-(13), which is { (13) }, contains (13).
Creating a new concept, "Single-times".
Single-times is a relation from Numbers to Bags-of-numbers.
Single-times(x) is all bags in TIMES-1-(x) which are singletons.
e.g, Single-times(12)={ (12) }.
e.g, Single-times(13)={ (13) }.
Conjecture 2: TIMES-1-(x) always contains a bag containing only primes.
e.g., TIMES-1-(12), which is { (12) (2 6) (2 2 3) (3 4) }, contains (2 2 3).
e.g., TIMES-1-(13), which is { (13) }, contains (13).
Creating a new concept, "Prime-times".
Prime-times is a relation from Numbers to Bags-of-numbers.
Prime-times(x) is all bags in TIMES-1-(x) which contain only primes.
e.g, Prime-times(12)={ (2 3 3) }.
e.g, Prime-times(13)={ (13) }.
.SKIP TO COLUMN 1
***Task 80.
Considering the concept "Prime-times".
2 Reasons:\(1) Conjecs about Prime-times will tell more about Primes and TIMES-1-
\(2) Focus of attention: AM recently defined Prime-times.
Looking specifically at Prime-times(48), which is { (2 2 2 2 3) }.
4 conjectures proposed, after .6 seconds.
Testing them on other examples of Prime-times.
Only 1 of these 4 conjectures are verified for all examples of Prime-times:
Conjecture 1: Prime-times(x) is always a singleton set.
That is, Prime-times is a function, not just a relation.
e.g., Prime-times(48), which is { (2 2 2 2 3) }, is a singleton set.
e.g., Prime-times(47), which is { (47) }, is a singleton set.
This holds for all 17 known examples of Prime-times.
This fails for 2 of the boundary cases (extreme numbers): 0 and 1.
Conjecture is amended: Each number >1 is the product of a unique bag of primes.
I suspect that this conjecture may be very useful.
.GROUP SKIP 3
⊗4USER: Call this conjecture "Unique factorization conjecture".⊗*
.SKIP TO COLUMN 1
***Task 84.
Examining ADD-1-(x), looking for patterns involving its values.
2 Reasons:\(1) Recent success with the analogous concept: "TIMES-1-".
\(2) Many examples of ADD-1- are known, to induce from.
Looking specifically at ADD-1-(6), which is { (1 1 1 1 1 1) (1 1 1 1 2) (1 1 1 3)
\(1 1 2 2) (1 1 4) (1 2 3) (1 5) (2 2 2) (2 4) (3 3) }.
17 conjectures proposed, after 3.9 seconds.
Testing them on other examples of ADD-1-.
Only 11 of these 17 conjectures are verified for all examples of ADD-1-:
Conjecture 1: ADD-1-(x) always contains a doubleton bag (i.e., size 2).
e.g., ADD-1-(6) contains (1 5), (2 4), and (3 3).
e.g., ADD-1-(4) contains (1 3), and (2 2).
Creating a new concept, "Pair-add".
Pair-add is a relation from Numbers to Pairs-of-numbers.
Pair-add(x) is all bags in ADD-1-(x) which are doubletons (i.e., of size 2).
e.g, Pair-add(12)={ (1 11) (2 10) (3 9) (4 8) (5 7) (6 6) }.
e.g, Pair-add(4)={ (1 3) (2 2) }.
Conjecture 2: ADD-1-(x) always contains a bag containing only 1's.
⊗9#⊗*
⊗9#⊗*
Conjecture 10: ADD-1-(x) usually (but not always) contains a pair of primes.
e.g., ADD-1-(10) contains (3 7), and (5 5).
e.g., ADD-1-(4) contains (2 2).
Creating a new concept, "Prime-add".
Prime-add is a relation from Numbers to Pairs-of-numbers.
Prime-add(x) is all bags in ADD-1-(x) which are pairs of primes.
e.g, Prime-add(12)={ (5 7) }.
e.g, Prime-add(10)={ (3 7) (5 5) }.
e.g., Prime-add(11) = { }
⊗9#⊗*
⊗9#⊗*
.SKIP TO COLUMN 1
***Task 106.
Considering the set of numbers for which Prime-add is has non-empty value.
1 Reason:\(1) Prime-add often has non-empty value. Worth isolating that case.
Warning: no task on the agenda has an interestingness value above 200!!!
Creating a new concept "Prime-addable".
Prime-addable is a kind of Number.
A number x is Prime-addable if Prime-add(x) is non-empty.
Will spend 5.0 seconds filling in examples of Prime-addable.
18 examples found. Here are some of them: 4 5 6 7 8 9 10 12 13 14 17 16 18.
Empirically, all even numbers are in this set (ignoring boundary cases: 0 2 4 6).
So conjectured. Danger: must examine boundary cases: the numbers 0, 2, 4, 6.
Two exceptions noticed, the smallest boundary cases: 0, 2.
Conjecture is amended: All even numbers >2 are the sum of two primes.
Warning: I expect this conjecture will be cute but useless.
⊗4USER: Call this conjecture "Goldbach's conjecture".⊗*
***Task 107.
Considering the set of numbers for which the relation Prime-add is single-valued.
3 Reasons:\(1) Prime-add often has singleton value. Worth isolating that case.
\(2) Restricted to this set, Prime-add would be a function.
\(3) Focus of attention: AM recently worked on Prime-add.
Creating a new concept "Uniquely-prime-addable".
"Uniquely-prime-addable" is a kind of Number.
A number x is Uniquely-prime-addable if Prime-add(x) is a singleton.
Will spend 10.0 seconds filling in examples of Uniquely-prime-addable.
11 examples found. Here are some of them: 4 5 7 8 9 12 13.
No obvious conjecture derived empirically.
Will forget "Uniquely-prime-addable numbers", if no Ties found in near future.
.END
.SKIP TO COLUMN 1
.COMMENT Losers;
.BEGIN SELECT 2 CENTER; TURN ON "↑↓" PREFACE 0
↓_SOME LOSERS_↓
Triple-valued-functions
(Compose⊗7↑o⊗*Compose)⊗7↑o⊗*(Compose⊗7↑o⊗*Compose)
Insert⊗7↑o⊗*Equality
Hyper-hyper-exponentiation
Same-first-element
(?) Maximally-divisibles
(?) Goldbach conjecture
(?) Uniquely-prime-addables
(?) Mixed-exponential notation
.END
.SKIP TO COLUMN 1
.COMMENT Winners;
.BEGIN SELECT 2 CENTER; TURN ON "↑↓" PREFACE 0
↓_SOME WINNERS_↓
Singletons
Doubletons
Tripletons
Single-valued-functions
Same-length
Numbers
Add
Multiply
Greater-than
Exponentiate
Divisors
Square
Square-root
Primes
Unique factorization
(?) Maximally-divisibles
(?) Goldbach conjecture
(?) Uniquely-prime-addables
(?) Mixed-exponential notation
.END
.SKIP TO COLUMN 1
.COMMENT open problem;
.GROUP SKIP 10
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0 RETAIN
↓_DEFINITION_↓
.CENTER
.ONCE CENTER
"Open Research Problem"
means
"a limitation of this system."
.END
.SKIP TO COLUMN 1
.COMMENT expts;
.GROUP SKIP 5
.BEGIN NOFILL SELECT 2 PREFACE 0 INDENT 0
.ONCE CENTER
↓_Experiments on AM_↓
Add concepts from another domain
Modify the global task-rating formula
Modify the initial estimates of Worth
Remove certain concepts and heuristics
.END
.SKIP TO COLUMN 1
.ZZ(Should there be a slide for Geom experiment in general? p53)
.SKIP TO COLUMN 1
.COMMENT Prime angles;
.BEGIN SELECT 2 NOFILL PREFACE 0
.BEGIN CENTER
A cute geometric interpretation
of Goldbach's conjecture
⊗4Any angle (under 180↑o) can be approximated (to within 1↑o)
as the sum of two angles each of prime degree.⊗*
.END
... 7↑o 11↑o 13↑o 17↑o ...
115↑o is approximately 101↑o plus 13↑o
.END
.SKIP TO COLUMN 1
.COMMENT AM Conjec;
.ONCE CENTER SELECT 2
↓_Maximally Divisible Numbers_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "_∞→\↑↓[]{}&" SELECT 2 TABS 60,64 RETAIN
⊗2Max-div-n(d) =⊗1↓d↓f⊗6 the smallest integer n with at least d divisors.⊗2
↓_CONJECTURE:_↓ ⊗6For all n, a crisp lower bound for Max-div-n(d) is⊗*
n = ↓2↑a⊗71⊗*↓3↑a⊗72⊗*↓5↑a⊗73⊗*...p⊗7↓k⊗*↑a⊗7k⊗*,
⊗4where ⊗2 p⊗7↓i ⊗6 is the i↑t↑h prime,⊗2
⊗4and⊗2 a⊗7↓i⊗2 = ⊗6k⊗7↑t↑h⊗6ROOT⊗2α{⊗6d⊗9#⊗*log2⊗9#⊗*log3⊗9#⊗*...⊗9#⊗*log(p↓k)⊗2α} ↓/ ↓[p⊗7↓i⊗*]⊗2 - 1.
↓_COROLLARY:_↓
Max-div-n(d) ≥ ↓[⊗2e⊗*]⊗6k⊗9#⊗*k⊗7↑t↑h⊗6ROOTα{d⊗9#⊗*log2⊗9#⊗*log3⊗9#⊗*...⊗9#⊗*log(p↓k)α} ⊗2↓[/ 2⊗9#⊗*3⊗9#⊗*...⊗9#⊗*p⊗7↓k]⊗2
↓_COROLLARY:_↓ ⊗6For all i,⊗* (a⊗7↓i⊗2+1)⊗9↑#⊗2log(p⊗7↓i⊗2) ⊗2is constant.
↓_COROLLARY:_↓
⊗2If k is approximated as log(d)/log(β),
Then Max-div-n(d) ≥ ⊗2↓d↑[⊗6α{loglog(d)⊗9#⊗*(β-1)/log(β)α}]⊗2
↓_∞ →_↓
.SELECT 7
For example: Max-div-n(5,971,968) is about →8,455,557,000,000,000,000,000,000,000.
AM Conjec predicted: →n ≥ 4,683,635,000,000,000,000,000,000,000.
The best previous formula predicted:→n ≥ 722,082,100,000,000,000,000,000,000.
.END
.SKIP TO COLUMN 1
.COMMENT closing slide;
.GROUP SKIP 14
.BEGIN CENTER SELECT 2 PREFACE 0
Machines can perform creative
scientific theory formation
.END
.SKIP TO COLUMN 1
.COMMENT SUPPLEMENTARY: Sys Names;
.GROUP SKIP 6
.BEGIN SELECT 2 CENTER; TURN ON "→↑↓←\" PREFACE 0
S.A.M.
A.M.
A.C.E
G00036
.END
.SKIP TO COLUMN 1
.COMMENT SUPPLEMENTARY: Factorings: exs of divisors;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Factorings of Numbers_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN OFF "{}" SELECT 2 TABS 7,20 TURN ON "\"
Factorings-of(7) =
\{ (7,1) }
Factorings-of(18) =
\{ (18,1), (9,2), (6,3), (3,3,2) }
Factorings-of(32) =
\{ (32,1), (16,2), (8,4), (8,2,2), (4,4,2),
\\(4,2,2,2), (2,2,2,2,2) }
Factorings-of(58) =
\{ (58,1), (29,2) }
.END
.SKIP TO COLUMN 1
.PORTION TODO
.BEGIN NOFILL PREFACE 0 TURN ON "α[]↑↓_{}∞→" SELECT 1
.ONCE CENTER
⊗2↓_Things still to do_↓⊗1
.RECEIVE
.END
.PAGE←0
.SELECT 1