perm filename SLIDES.CMU[TLK,DBL] blob
sn#201940 filedate 1976-02-14 generic text, type C, neo UTF8
COMMENT ⊗ VALID 00040 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00004 00002 .DEVICE XGP
C00005 00003 .COMMENT Primes
C00006 00004 .COMMENT UFT
C00007 00005 .COMMENT UFT overlay
C00008 00006 .COMMENT Heur1: going to extremes
C00009 00007 .COMMENT Extreme sets of nos
C00010 00008 .COMMENT 1 reduction
C00011 00009 .COMMENT Modus ponens
C00012 00010 .COMMENT 2nd reduction
C00013 00011 .COMMENT Complete the Square
C00014 00012 .COMMENT OVERLAY Complete the Square
C00015 00013 .COMMENT chain of discoveries
C00016 00014 .COMMENT s→a heuristics
C00017 00015 .COMMENT 3 tasks
C00018 00016 .COMMENT Math thy
C00019 00017 .COMMENT Math text order
C00020 00018 .COMMENT Math research order
C00021 00019 .COMMENT 3 tasks
C00022 00020 USE the 100 concepts slide from before
C00023 00021 .COMMENT Types of heurs
C00025 00022 USE the 100 concepts + facets slide from before
C00026 00023 .COMMENT Knowledge bits
C00027 00024 .COMMENT genl. tree
C00028 00025 .COMMENT 2 kinds of operators
C00030 00026 .COMMENT Facets
C00032 00027 .COMMENT Facets: COMPOSE
C00034 00028 .COMMENT optimizing the defns
C00035 00029 .COMMENT Control
C00037 00030 .COMMENT Control.1
C00039 00031 .COMMENT Control OVERLAY
C00041 00032 .COMMENT Factorings: exs of divisors
C00050 00033 .COMMENT Excerpt: Cardinality
C00053 00034 .COMMENT Defn of EQUAL
C00054 00035 .COMMENT OVERLAY Defn of EQUAL
C00055 00036 .COMMENT Sys Results
C00056 00037 .COMMENT Sys Results
C00057 00038 .COMMENT Sys Names
C00058 00039 .COMMENT AM Conjec
C00060 00040 .COMMENT Specific Heur
C00061 ENDMK
C⊗;
.DEVICE XGP
.COMMENT !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 Primes;
.GROUP SKIP 20
.BEGIN CENTER SELECT 2
A natural number p is ↓_prime_↓
iff
its only divisors are 1 and p.
.END
.SKIP TO COLUMN 1
.COMMENT UFT;
.GROUP SKIP 10
.BEGIN CENTER SELECT 2
↓_UNIQUE FACTORIZATION CONJECTURE _↓
"Any number can be factored uniquely into
numbers which all have only 2 divisors."
.END
.SKIP TO COLUMN 1
.COMMENT UFT overlay;
.GROUP SKIP 10
.BEGIN CENTER SELECT 2
Primes."
.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 member of B, Then
isolate the members of A that f maps into b.
.END
.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 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 Modus ponens;
.BEGIN CENTER SELECT 2 TURN OFF "↑↓" PREFACE 0
Q
||
||
||
||
⊗4Modus⊗*
⊗4Ponens⊗*
||
||
||
||
P, P→Q
.END
.SKIP TO COLUMN 1
.COMMENT 2nd reduction;
.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 "Multiply"
.END
.SKIP TO COLUMN 1
.COMMENT Complete the Square;
.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
\⊗4count⊗*
Pairs of Sets∞-→α→ Pairs of Numbers
|\\\|
|\\\
|\\\|
|\\\
|\\\|
|\\\
| ⊗4cross-product⊗*\\\| ⊗4(?)⊗*
|\\\
|\\\|
|\\\
|\\\|
|\\\
↓\⊗4count⊗*\\↓
Sets∞-\-∞-\α→ Numbers
.TURN ON "↑↓" SELECT 4
?(x,y) = Count ( Cross-product ( Count↑-↑1(x), Count↑-↑1(y) ) ).
.END
.SKIP TO COLUMN 1
.COMMENT OVERLAY Complete the Square;
.ONCE CENTER SELECT 2
↓__↓
.BEGIN NOFILL SELECT 7
.INDENT 0 PREFACE 0 TABS 40,64,75 TURN ON "→∞α\" TURN OFF "↑↓{}"
.SELECT 2
⊗4{A,B}, {U,V,W}⊗*\\\2, 3
{⊗4 {A,U}, {A,V}, {A,W},⊗*\\\ 6
⊗4 {B,U}, {B,V}, {B,W}⊗*⊗2 }⊗*
.END
.SKIP TO COLUMN 1
.COMMENT chain of discoveries;
.BEGIN CENTER SELECT 2 PREFACE 0
Discovery of Prime Numbers
|
|
⊗4Look at extrema⊗*
|
|
Discovery of Divisors-of
|
|
⊗4Look at inverses⊗*
|
|
Discovery of Multiplication
|
|
⊗4Complete the square diagram⊗*
|
|
Discoveries of Counting, Cross-product
|
.........
|
⊗4(↓_BASE OF KNOWN CONCEPTS_↓)⊗*
.END
.COMMENT Overlay onto this a slide of spreading-out lines going upward;
.COMMENT Overlay onto this a slide of diff-colored nodes/ops, to repr heurs.;
.SKIP TO COLUMN 1
.COMMENT s→a heuristics;
.BEGIN CENTER SELECT 2 PREFACE 0
"Generalize a predicate to produce new ones"
"IF a predicate is rarely satisfied,
THEN try to generalize it"
↓_General form of a Meta-heuristic:_↓
"In situation S, Apply heuristic H"
.END
.SKIP TO COLUMN 1
.COMMENT 3 tasks;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_DISCOVERY AS_↓
.ONCE CENTER SELECT 2
↓_HEURISTIC SEARCH_↓
.BEGIN SELECT 2 PREFACE 2 INDENT 20 SKIP 2
(1) Starting knowledge
(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 Math research order;
.ONCE CENTER SELECT 2
↓_Research Order_↓
.BEGIN CENTER SELECT 2 PREFACE 0 TURN ON "→"
Motivation for some
of the future theorems
↓
↓
State some theorems
↓
↓
Make useful definitions
↓
↓
Decide which defns are to be primitive
Decide which theorems are to be axioms
.END
.SKIP TO COLUMN 1
.COMMENT check out with big SYS4 file on [tlk,dbl];
.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
USE the 100 concepts slide from before
.SKIP TO COLUMN 1
.COMMENT Types of heurs;
.ONCE CENTER SELECT 2
↓_Kinds of Heuristics_↓
.BEGIN SELECT 2 PREFACE 0 TURN ON "→"
"Look at the inverse of any relation"
"IF few examples satisfy the predicate P,
THEN try to generalize the definition of P"
"IF you want to fill in examples of A,
FIRST find an operation whose range is A
THEN apply the operation to anything"
"IF the '⊗4↓_look for extremals_↓⊗*' heuristic has been applied,
WITH f, A, B, b, and b↑-↑1 as before,
THEN tag b↑-↑1 with the following heuristic:
"b↑-↑1X is a good subset of A to test
conjectures involving f or f↑-↑1"
Tag PRIMES with the following heuristic:
"Primes are a good kind of number to use
when inventing or testing conjectures
that involve divisors or multiplication"
.END
.SKIP TO COLUMN 1
USE the 100 concepts + facets slide from before
.SKIP TO COLUMN 1
.COMMENT Knowledge bits;
.GROUP SKIP 4
.BEGIN SELECT 2 PREFACE 0 TURN ON "→" CENTER
↓_KN_↓⊗4owledge b⊗*↓_IT_↓
KNIT
.END
.SKIP TO COLUMN 1
.COMMENT genl. tree;
.BEGIN SELECT 2 PREFACE 0 TURN ON "→" INDENT 0
.TURN ON "\" TABS 9, 18, 27, 36, 45
.ONCE CENTER
↓_One Branch of Generalization Links_↓
ANYTHING
\ANY-CONCEPT
\\ACTIVITY
\\\OPERATION
\\\\COMPOSITION
\\\\\UNION⊗7↑o⊗*COMPLEMENT
.END
.SKIP TO COLUMN 1
.COMMENT 2 kinds of operators;
.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 Facets;
.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;
.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 optimizing the defns;
.GROUP SKIP 2
.ONCE CENTER SELECT 2
↓_Optimize Definitions_↓
.BEGIN SELECT 2 PREFACE 0 TABS 50 TURN ON "\↑↓_" CENTER
(Multiply A B)
≡
(Count (Cross-Product \(Count↑-↑1 A)
\(Count↑-↑1 B)))
For i from 1 to A, join B.
.END
.SKIP TO COLUMN 1
.COMMENT Control;
.ONCE CENTER SELECT 2
↓_Control Structure_↓
.BEGIN SELECT 7 TABS 15,25,40,50 TURN ON "\↑↓←\" PREFACE 0
↓_⊗4AGENDA⊗*_↓ OF THINGS WORTH DOING
⊗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))
⊗4(Fill in new Algorithms for Compose concept)⊗*
\\Priority=300
\\Reasons=((Empirical: taking too long))
\\⊗4(Fill in Examples facet of Primes concept)⊗*
\\\Priority=500
\\\Reasons=(("Divisors-of" asked for some
\\\\examples of primes))
REPEATEDLY:
1) Select 1 job from the agenda
2) Assemble relevant heuristics
3) Execute them
.END
.SKIP TO COLUMN 1
.COMMENT Control.1;
.ONCE CENTER SELECT 2
↓_Control Structure_↓
.BEGIN SELECT 7 TABS 15,25,40,50 TURN ON "\↑↓←\" PREFACE 0
↓_⊗4AGENDA⊗*_↓ OF THINGS WORTH DOING
⊗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))
⊗4(Fill in new Algorithms for Compose concept)⊗*
\\Priority=300
\\Reasons=((Empirical: taking too long))
REPEATEDLY:
1) Select 1 job from the agenda
2) Assemble relevant heuristics
3) Execute them
.END
.SKIP TO COLUMN 1
.COMMENT Control OVERLAY;
.ONCE CENTER SELECT 2
.BEGIN SELECT 7 TABS 15,25,40,50 TURN ON "\↑↓←\" PREFACE 0
\\⊗4(Fill in Examples facet of Primes concept)⊗*
\\\Priority=500
\\\Reasons=(("Divisors-of" asked for some
\\\\examples of primes))
.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 7,20 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
.COMMENT Excerpt: Cardinality;
.ONCE CENTER SELECT 2
↓_As the User Sees It_↓
.BEGIN SELECT 7 INDENT 0 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 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_↓(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;
.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(L) is ↓_EQ1_↓XX to CDR(M))
]
.END
.SKIP TO COLUMN 1
.COMMENT Sys Results;
.GROUP SKIP 7
.BEGIN SELECT 2 CENTER TURN OFF "↑↓←\" PREFACE 0
Set-equality ⊗4(given concept)⊗*
↓
Same-length
↓
Numbers
↓
Multiply
↓
Divisors
↓
Primes
↓
Unique factorization
.END
.SKIP TO COLUMN 1
.COMMENT Sys Results;
.GROUP SKIP 6
.BEGIN SELECT 2 NOFILL TURN ON "→↑↓←\" PREFACE 0
→Maximally-divisible
→numbers
.END
.SKIP TO COLUMN 1
.COMMENT Sys Names;
.GROUP SKIP 6
.BEGIN SELECT 2 NOFILL TURN ON "→↑↓←\" PREFACE 0
S.A.M.
A.C.E
A.M.
.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
⊗2Max-divis(n) =⊗4↓d↓f⊗* (∀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
(a↓i + 1)'s are (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)
AM Conjecture says that n is the smallest
integer with 3,981,312 divisors.
.END
.SKIP TO COLUMN 1
.COMMENT Specific Heur;
.ONCE CENTER SELECT 2
↓_A Typical Heuristic Rule_↓
.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "_∞→\↑↓[]{}&" SELECT 2 TABS 60
↓_IF_↓
X is a predicate
AND
Effort has been spent looking for examples of X
AND
The ratio of examples to non-examples is < .01
↓_THEN_↓
Add this new job to the agenda:
(Generalize Definitions of X)
for reason: "X is too rarely satisfied"
whose numeric value is
the ratio of no-examples to examples of X
(or 1000; whichever is smaller)
.END