Left & right identity

We touched briefly that there are two identity functions in J. Left [ and right ] identity. And they performed identically for us when solving day 6 problem using sliding windows.

The difference comes when this function is invoked with both arguments, in this case left [ identity will return its left argument and right ] - right one respectively.

   3 [ 5
3
   3 ] 5
5

As with many things in J, this comes in handy when composing multiple functions. Let's consider a plus + primitive and write in a fork expression like following.

   plus =: [ + ]

Now this looks like an arbitrary set of symbols to anyone unfamiliar with J's special execution order for fork. This fork has its both arguments so it will evaluate slightly differently from the one we described previously.

Rules for this one would look like following.

   x (f h g) y

Will be evaluated to:

   (x f y) h (x g y)

In this case branches of the fork with get both x and y, then h will be applied to their results. When we apply fork evaluation rules for our plus example we would get behavior similar to following direct definition.

   plus_direct =: {{ (x [ y) + (x [ y) }}

Summing that with specifics of identity functions when calling with multiple argument we can see that this complex notation is actually equivalent so a single + primitive, since identity in first set of braces will evaluate to x, and identity in second - to y, leaving us with the same plus definition that we wrote a few chapters back.