DanBron/Snippets/AlternatingValences

Given the capped fork behavor reported by HenryRich, we can create a verb which alternates between a fork and a capped fork, on every invocation.

That is, we can have a fork whose middle tine is invoked monadically or dyadically, depending on the number of times the train's been invoked before (instead of on the number of arguments to the train) :

   NB.  See Henry Rich's msg at  
   NB.  http://www.jsoftware.com/pipermail/programming/2006-May/002349.html
   left    =:  [:
   middle  =:  3 : ('left =: [:`[@.A_z_';'y') @: mrd
   right   =:  3 : 'A_z_ =: -.{:0,".''A_z_'' ' :($:@:])

   NB.  Returns  'monad'  iff middle tine was called monadically
   NB.  and 'dyad'  iff middle tine was called dyadically.
   mrd     =:  'monad'"_ :('dyad'"_) 
   f       =:  left middle right
   
   
   f 0
monad
   f 0
dyad
   f 0
monad
   
   f~ 0
dyad
   f~ 0
monad
   f~ 0
dyad
   
   f"0 i.3
monad
dyad 
monad

   f~"0 i.3
dyad 
monad
dyad 

  (left middle right) 0
monad

  (left middle right) 0
dyad

This trick was inspired by BJonas on the J IRC channels .

Can you find other ways to acheive the same end? You are permitted to use side effects, like above, but you can't use cheap tricks like having the monad invoke the dyad (or not), based on some flag. The first and only valence invoked must change at every execution.

Attempting another line of attack, I found some surprising behavior:

   g =: 3 : ('g =: 0&g';' ''monad'' ')  : (4 : ('g =: g@:]';' ''dyad'' '))
   g
3 : 0 :(4 : 0)
g =: g@:]
 'dyad' 
)
g =: 0&g
 'monad' 
)
   g 0
monad

   g 0
0
   g 0
0

   g 3
3

   g~ 4
4

   3 g 17
17
   3 g 'x'
x
   'x' g 3
|domain error: g
|   'x'    g 3
   NB.  Aha!  Powering function of  &  
   
   
   NB.  Since  g y  is  0&g y  is  0 g y  is  0 (0&g)  y
   NB.  and we have  x m&u y  <-->  m&u^:x y  then 
   NB.  g y  <-->  0&g^:0  y  .  And, and by definition
   NB.  u^:0 y  is simply  y  .

   'g' f.  
0&(0&g)

Is the above (i.e. that g f. isn't the same as 'g' f.) a bug?

There has to be an way to present this as an interesting challenge on the Forums (like <"0 and g^:_1) (link to these).

For example:

• what is the definition of g such that g f.  2 : ' ''u'' -:&:(5!:2)@:;: ''v'' ' 'g' f.?

Or maybe find a way to phrase:

• find an adverb which turns any verb into an identity function

But this one could be more than just (0&)~. For example, ([`) (]`) (`:6))).

Perhaps:

• find a verb which is the identity function when assigned to a name, but an error when not assigned to a name

Is there anything besides u=:0&u~ ?