Many thanks to David for the interest in ordinal fraction arithmetic. I have
not yet studied the implementation in full detail. Note however that when you
strip all the zeroes from 3$0 you get 0$0 rather than 1$0.
Allow me some explaining.
Consider the year number 2020. This year! It is an ordinal number: the
twothousandandtwentieth year.
2019 years have passed since the beginning of the calendar. 2019 is a cardinal
number: twothousandandnineteen years. Note that 2019=2000+0010+0009. Zero means
"nothing" and the plus sign means add. You may compute with cardinal numbers,
but not with ordinal numbers. That is why J'ers use cardinal numbers for
indexing.
Increasing each digit in 2019 by one gives 3 1 2 10 indicating that the year
2020 is the tenth year of the second decade of the first century of the third
millenium.
(4$10)&#:&.<:2020
3 1 2 10
But 10 is not a single digit. In order to avoid the ordinal number 10 we use a
pental-binary notation.
(7$5 2)&#:&.<:2020
3 1 1 1 2 2 5
Now separators are superfluous
10#.(7$5 2)&#:&.<:2020
3111225
meaning - reading right to left - that the year 2020 is the fifth year of the
second half of the second decade of the first half of the first century of the
first half of the third millenium.
Note that 3111225=3000000+0100000+0010000+0001000+0000200+0000020+0000005
This is ordinal fraction arithmetic. It looks like integer arithmetic, but the
zero means "everything" rather than "nothing", and the plus sign means "and"
rather than "add".
The cardinal number 2019 is the algebraic expression 2M+1X+9I =
2M+0D+0C+0L+1X+1V+4IThe ordinal fraction 3111225 is the logical expression
M=3, D=1, C=1, L=2, X=2, V=2, I=5The twentyfirst century is 3110000: the first
century of the first half of the third millenium. And so on.
Thanks! Bo.
Den mandag den 27. juli 2020 18.16.27 CEST skrev David Lambert
<[email protected]>:
"(Bo's) 1986-article on ordinal fractions may be of interest."
https://www.academia.edu/10031088/ORDINAL_FRACTIONS_-_the_algebra_of_data
Has anyone got an implementation?
Is `=' synonymous with `compatible'? If so, the following seems to work,
replacing for the literals with a vector.
NB. ordinal fractions, represented by vectors
Dyad=: [: :
Monad=: :[:
Commutes=: adverb def 'assert@:(u -: u~) Dyad' NB. assertion failure if u does
not commute
ANY=: , 0
ERROR=: , __
error=: ERROR"_
NB. removes trailing zeros
simplify=: ([: ANY"_^:(0=#) ({.~ (1 i.~([email protected]&=)\.))^:(0={:))"1 Monad
intersect=: dyad define"1
'X Y'=. x ,:&, y NB. fill extends the shorter vector
X=. X + Y * X = 0
Y=. Y + X * Y = 0
simplify X error^:(-.@:-:) Y
)
difference=: simplify@:(([ * =)/)@:(,:&,)"1 Dyad
true=: ANY = simplify
eq=: equal=: -:"1 Dyad
incompatible=: (ERROR -: intersect)"1
compatible=: -.@:incompatible
le=: ([ -: intersect)&:,"1
ge=: le~
lt=: subordinate=: le > eq NB. x subordinate y proper subset
gt=: superordinate=: ge > eq NB. x superordinate y proper superset
,.(#~ (1 -.@e.&> ])) <@:(". ::]);._2]0 :0 NB. tests evaluate to 1
iff pass
(,3) -: simplify 3 0 0 0
ANY -: simplify ANY
ANY -: ANY intersect ANY
(,1) -: 1 intersect&:,1
ERROR -: 1 intersect&:,2
ERROR -: 1 2 3 intersect 1 2 1
2 3 -: ANY intersect 2 3 0
0 2 3 -: ANY intersect 0 2 3 0
2 3 -: 2 3 intersect 2 3 0
2 3 8 -: 2 3 intersect 2 3 8
2 3 compatible 2 3 0
0 2 3 incompatible 2 3 0
1 [ 2 1 1 1 difference Commutes 2 1 2 1
1 [ 2 1 0 3 intersect Commutes 0 1 8 3
)
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
