perm filename SLIDES.HPP[HPP,DBL] blob
sn#200748 filedate 1976-02-06 generic text, type C, neo UTF8
COMMENT ⊗ VALID 00016 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00003 00002 .DEVICE XGP
C00004 00003 .COMMENT Facets
C00006 00004 .COMMENT Facets: COMPOSE
C00008 00005 .COMMENT Planning
C00009 00006 .COMMENT Tree of Disc
C00010 00007 .COMMENT Complete the Square
C00011 00008 .COMMENT AM Conjec
C00013 00009
C00014 00010 .COMMENT Heur1: going to extremes
C00015 00011 .COMMENT Chain: plausible scenaria of discoveries
C00016 00012 .COMMENT Chain OVERLAY
C00017 00013 .COMMENT Factorings: exs of divisors
C00018 00014 .COMMENT Excerpt: Cardinality
C00021 00015 .COMMENT Defn of EQUAL
C00022 00016 .COMMENT OVERLAY Defn of EQUAL
C00023 ENDMK
C⊗;
.DEVICE XGP
.!XGPCOMMANDS←"/TMAR=50/PMAR=2100/BMAR=50"
.FONT 1 "BASB30"
.FONT 2 "BDR66"
.FONT 4 "BDI40"
.FONT 7 "BDR40"
.FONT 8 "BDR25"
.FONT 9 "GRFX35"
.TURN ON "↑α[]↓_π{"
.TURN ON "⊗" FOR "%"
.TABBREAK
.ODDLEFTBORDER ← EVENLEFTBORDER ← 1000
.PAGE FRAME 54 HIGH 91 WIDE
.AREA TEXT LINES 1 TO 53
.DOUBLE SPACE
.PREFACE 2
.NOFILL
.PREFACE 1
.!XGPLFTMAR←100
.MACRO B ⊂ BEGIN NOFILL SELECT 9 INDENT 0 GROUP PREFACE 0 MILLS TURN OFF "{↑↓}[]α" ⊃
.MACRO E ⊂ APART END ⊃
.NEXT PAGE
.INDENT 0
.SELECT 1
.COMMENT Facets;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Facets of a Concept_↓
.BEGIN 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;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Facets of "COMPOSE"_↓
.BEGIN SELECT 2 PREFACE 0
Characterizations
⊗7Name(s): Compose ⊗*
⊗7Definitions: recursive, opaque, wffs⊗*
⊗7Algorithms: opaque, transparent, destructive ⊗*
⊗7Domain/range: Relations x Relations → Relations⊗*
⊗7Intuitions: refiring arrows, time sequence⊗*
Ties to other concepts
⊗7Specializations: Compose f with itself⊗*
⊗7Generalizations: Relation⊗*
⊗7Examples: (Intersect, Complement) → Set-difference⊗*
⊗7Conjec: (AoB)oC ≡ Ao(BoC)⊗*
⊗7Analogies: multiplying two matrices⊗*
Heuristics
⊗7Worth: Primitive. Creates new active Concepts⊗*
⊗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 Planning;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Control Structure_↓
.BEGIN SELECT 7 TABS 15,55 TURN ON "\↑↓←→"
THINGS WORTH DOING
(Fill in exs of Primes)
(Improve algs for Compose)
(Generalize Defn of Equality)
.GROUP SKIP 3
.ONCE CENTER
Select 1 activity
Execute this plan
→Assemble relevant heuristics
.END
.SKIP TO COLUMN 1
.COMMENT Tree of Disc;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
⊗2↓_Graph of Development_↓⊗*
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "→∞\α" TABS 30,50,60,70
Bags\Equality\Cross-product
\Numbers\\Projection
\Multiplication
Exponentiation\Divisors
Hyper-exponentiation\Max-Divis\Primes
.END
.SKIP TO COLUMN 1
.COMMENT Complete the Square;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Complete the Square_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TABS 40,64,75 TURN ON "→∞α{}\" TURN OFF "↑↓"
.SELECT 2
\⊗7count⊗*
Pairs of Bags∞-→α→ Pairs of Numbers
|\\\|
|\\\
|\\\|
|\\\
|\\\|
|\\\
| ⊗7cross-product⊗*\\\| ⊗7(?)⊗*
|\\\
|\\\|
|\\\
|\\\|
|\\\
↓\⊗7count⊗*\\↓
Bags∞-\-∞-\α→ Numbers
.TURN ON "↑↓" SELECT 7
?(x,y) = Count ( Cross-product ( Count↑-↑1(x), Count↑-↑1(y) ) ).
.END
.SKIP TO COLUMN 1
.COMMENT AM Conjec;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Maximally Divisible Numbers_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑↓[]{}&" SELECT 2
⊗2Max-divis(N) iff (∀m<n) d(m) < d(n)
↓_CONJECTURE:_↓ if N = ↓2↑a⊗71⊗*↓3↑a⊗72⊗*↓5↑a⊗73⊗*...p⊗7↓k⊗*↑a⊗7k⊗*
⊗2where p↓i is the i↑t↑h prime,
and (a↓i + 1) / (a↓j + 1) "=" log(p↓j ) / log(p↓i)
then Max-divis(n).
**************************************************
For example: n could be
2⊗7↑8⊗*3⊗7↑5⊗*5⊗7↑3⊗*7⊗7↑2⊗*11⊗7↑2⊗*13⊗7↑1⊗*17⊗7↑1⊗*19⊗7↑1⊗*23⊗7↑1⊗*29⊗7↑1⊗*31⊗7↑1⊗*37⊗7↑1⊗*41⊗7↑1⊗*43⊗7↑1⊗*47⊗7↑1⊗*53⊗7↑1⊗*
.SELECT 2
(which equals 25,608,675,584).
(a↓i + 1)'s are (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)
n has 3,981,312 divisors.
AM Conjecture says that
n is the smallest integer with that many divisors.
.END
.SKIP TO COLUMN 1
.COMMENT Heur1: going to extremes;
.GROUP SKIP 4
.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
\\⊗7a\\b⊗*
\S⊗7↑-↑1⊗*\\\\S
.CENTER
If S is some interesting subset of B,
(e.g.: ↓_extremal_↓), then try to isolate
the elements of A that f maps into S.
.END
.SKIP TO COLUMN 1
.COMMENT Chain: plausible scenaria of discoveries;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Chain of Discoveries_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "→∞\α" TABS 30,50,60
⊗2SETS\NUMBERS→CROSS-PRODUCT⊗*
\\\complete the square
\⊗2MULTIPLICATION⊗*
\\look at the inverse
\⊗2DIVISORS⊗*
\\go to extremes: minimal
\\\⊗2PRIMES⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Chain OVERLAY;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
⊗2.⊗*
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "→∞\α" TABS 30,50,60,70
⊗2 ⊗*
→inverse
→⊗2PROJEC⊗*
\⊗2 ⊗*
maximal
⊗2MAX-DIVIS⊗*
.END
.SKIP TO COLUMN 1
.COMMENT 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 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), (4,2,2), (2,2,2,2) }
Factorings-of(58) =
\{ (58,1), (29,2) }
.END
.SKIP TO COLUMN 1
.COMMENT Excerpt: Cardinality;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_As the User Sees It_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0
(1) Looking for examples of things which are EQUAL.
5 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.
(2) 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".⊗*
(3) 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) ).
(4) 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 Defn of EQUAL;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Recursive Defn of EQUAL_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓[]{}&" SELECT 2
Definition of ↓_EQUAL_↓(X,Y):
(LAMBDA (X Y)
(IF X and Y are both atomic,
then X must be identically EQ to Y,
else
IF X and Y are both lists,
then both
(CAR(X) is ↓_EQUAL_↓ to CAR(Y))
and
(CDR(X) is ↓_EQUAL_↓ to CDR(Y))
]
.END
.SKIP TO COLUMN 1
.COMMENT OVERLAY Defn of EQUAL;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_._↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓[]{}&" SELECT 2
Definition of ↓_EQ1_↓XX
(
(
.
.
.
.
XXXXXXXXXXXXXXXXXX T
and
(CDR(X) is ↓_EQ1_↓XX to CDR(Y))
]
.END
.SKIP TO COLUMN 1