On Wednesday 18 May 2005 17:57, Matt Fowles wrote:
> All~
>
> What does the reduce metaoperator do with an empty list?
Here is the last answer from Ken Iverson, who invented reduce in
the 1950s, and died recently.
file:///usr/share/j504/system/extras/help/dictionary/intro28.htm
Identity Functions and Neutral
The monads 0&+ and 1&* are identity functions, and 0 and 1 are
said to be identity elements or neutrals of the dyads + and *
respectively. Insertion on an empty list yields the neutral of
the dyad inserted. For example:
+/ i.0 +/'' +/0{. 2 3 5
0 0 0
*/i.0 */'' */0{. 2 3 5
1 1 1
> my @a;
> [+] @a; # 0? exception?
> [*] @a; # 1? exception?
> [<] @a; # false?
> [||] @a; # false?
> [&&] @a; # true?
>
> Also if it magically supplies some correct like the above, how
> does it know what that value is?
>
> Thanks,
> Matt
The page
file:///usr/share/j504/system/extras/help/dictionary/d420.htm
gives examples, unfortunately not easily readable ones.
"If y has no items (that is, 0=#y), the result of u/y is the
neutral or identity element of the function u. A neutral of a
function u is a value e such that x u e â x or e u x â x, for
every x in the domain (or some significant sub-domain such as
boolean) of u . This definition of insertion over an argument
having zero items extends partitioning identities of the form
u/y â (u/k{.y) u (u/k}.y) to the cases k e. 0,#y .
"The identity function of u is a function ifu such that ifu y â
u/y if 0=#y ."
[The following table is greatly simplified by listing identity
elements rather than identity functions. Some are only left
identities, and some only right identities.]
Identity element For
0 < > + - +. ~: | (2 4 5 6 b.)
1 = <: >: * % *. %: ^ ! (1 9 11 13 b.)
_ <.
__ >.
'' ,
[and a few more that I will not explain here]
Glossary
J Description
+. or
~: objects are identical
| remainder, defined so that 0|N is N
b. Boolean functions from table
<: less than or equal, restricted to Booleans here
(1<:0 is 0, 1<:1 is 1)
>: greater than or equal, restricted to Booleans here
* times, restricted to Booleans here
% divide
*. and
%: root
^ exponential
! combinations
<. minimum
>. maximum
'' empty vector, list of length 0
, catenate, join lists
_ infinity
__ negative infinity
So (_ <. N) is N, as is (__ >. N).
All of these functions are defined in detail but quite tersely in
the J Dictionary, indexed on the page
file:///usr/share/j504/system/extras/help/dictionary/vocabul.htm
For examuple, the Boolean function b. is defined on the page
file:///usr/share/j504/system/extras/help/dictionary/dbdotn.htm
--
Edward Cherlin
Generalist & activist--Linux, languages, literacy and more
"A knot! Oh, do let me help to undo it!"
--Alice in Wonderland
http://cherlin.blogspot.com
