# Vocabulary/lcapco

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`[x] u L: n y`Level At Conjunction

Rank Infinity *-- operates on [x and] y as a whole --*
WHY IS THIS IMPORTANT?

`(u L:0 y)` creates a copy of `y` in which each leaf has been replaced by the result of applying `u` to the leaf.

A leaf of `y` is a noun inside `y` that itself has no nested contents.

A leaf is either empty or unboxed.

NB. The leaves are 'abc', 'de', 4 5, and i. 6 ]a =. (<<'abc'),(<(<'de'),(<4 5)),(<<<i. 6) +-----+--------+---------------+ |+---+|+--+---+|+-------------+| ||abc|||de|4 5|||+-----------+|| |+---+|+--+---+|||0 1 2 3 4 5||| | | ||+-----------+|| | | |+-------------+| +-----+--------+---------------+ $L:0 a NB. Replace each leaf by its shape +---+-----+-----+ |+-+|+-+-+|+---+| ||3|||2|2|||+-+|| |+-+|+-+-+|||6||| | | ||+-+|| | | |+---+| +---+-----+-----+ |.L:0 a v NB. Reverse each leaf +-----+--------+---------------+ |+---+|+--+---+|+-------------+| ||cba|||ed|5 4|||+-----------+|| |+---+|+--+---+|||5 4 3 2 1 0||| | | ||+-----------+|| | | |+-------------+| +-----+--------+---------------+

More generally...

Every atom of a boxed noun can contain another boxed noun, each atom of which can contain another, and so on,
creating a *hierarchy of boxing* in which each box has a *position* with respect to higher levels of boxing,
and a boxing level of the contents telling how many levels of boxing hierarchy are within the box.

The boxing level of unboxed contents is 0.

`(u L: n)` forms a new verb which applies verb `u` to the contents of each box whose contents have boxing level `n`.

The result of `u` replaces the contents it was applied to, **but keeping the same position** in the boxing hierarchy.

This definition is accurate only when each box in `y` has contents with the same boxing level. See below for the general case.

] y=: (<0 1),(<(<'ab'),(<2 3)),(<(<0),(<1),<<4 5) +---+--------+-----------+ |0 1|+--+---+|+-+-+-----+| | ||ab|2 3|||0|1|+---+|| | |+--+---+|| | ||4 5||| | | || | |+---+|| | | |+-+-+-----+| +---+--------+-----------+ |.L:0 y +---+--------+-----------+ |1 0|+--+---+|+-+-+-----+| | ||ba|3 2|||0|1|+---+|| | |+--+---+|| | ||5 4||| | | || | |+---+|| | | |+-+-+-----+| +---+--------+-----------+ |.L:1 y +---+--------+-----------+ |1 0|+---+--+|+-+-+-----+| | ||2 3|ab|||0|1|+---+|| | |+---+--+|| | ||4 5||| | | || | |+---+|| | | |+-+-+-----+| +---+--------+-----------+

Contrast this with Spread (`S:`), which returns an array whose items are the results of applying `u` to the various contents.

### Common uses

1. Apply verb `u` to the leaves (innermost unboxed items) of a boxed noun `y`

] y=: 'alpha' ; 'bravo' ;'charlie' +-----+-----+-------+ |alpha|bravo|charlie| +-----+-----+-------+ toupper L:0 y +-----+-----+-------+ |ALPHA|BRAVO|CHARLIE| +-----+-----+-------+

**Tip:**
Use `leaf` as a mnemonic for ` L:0`

leaf L:0 toupper leaf y +-----+-----+-------+ |ALPHA|BRAVO|CHARLIE| +-----+-----+-------+

`leaf` is a

- Standard Library word(s) residing in the 'z'-locale.
- Defined in the factory script
`stdlib.ijs`. - View definition(s) by entering in the J session:
`open'stdlib'`

### Related Primitives

Level Of (`L. y`),
Spread (`u S: n`)

### More Information

1. The boxing level is specified with 3 numbers representing monadic and dyadic values, just as for rank.

2. The number `(n)` may be negative, in which case the value used for `n` is the given value
plus the boxing level of the argument, but never less than 0.

]a =: <,<2$<,'abc' NB. A noun +-----------+ |+---------+| ||+---+---+|| |||abc|abc||| ||+---+---+|| |+---------+| +-----------+ L. a NB. Boxing level 3 3 $L:0 a +-------+ |+-----+| ||+-+-+|| |||3|3||| ||+-+-+|| |+-----+| +-------+ $L:_3 a NB. same as L:0 +-------+ |+-----+| ||+-+-+|| |||3|3||| ||+-+-+|| |+-----+| +-------+ $L:1 a +---+ |+-+| ||2|| |+-+| +---+ $L:_2 a NB. same as L:1 +---+ |+-+| ||2|| |+-+| +---+

3. A positive value of `n` means "apply `u` on contents that are `n` levels above the leaves".

But a negative value of `n` **does not** mean "apply `u` at levels `-n` down from the top".
Even for negative `n`, the operation is based on distance from the leaves

]a =: (<<'abc') +-----+ |+---+| ||abc|| |+---+| +-----+ |.L:_2 a NB. L:_2 same as L:0 +-----+ |+---+| ||cba|| |+---+| +-----+ ]b =: (<<'abc'),(<<<0 1 2) +-----+---------+ |+---+|+-------+| ||abc|||+-----+|| |+---+|||0 1 2||| | ||+-----+|| | |+-------+| +-----+---------+ |.L:_2 b NB. Now L:_2 same as L:1. abc is at level 0 +-----+---------+ |+---+|+-------+| ||abc|||+-----+|| |+---+|||0 1 2||| | ||+-----+|| | |+-------+| +-----+---------+

### Details

Executing ` u L: n ` involves examining each box and (provided the contents of the box have the right boxing level) executing `u&.>` on the box.
This guarantees that the boxing hierarchy stays unchanged above the level at which `u` is applied.

If `n` is greater than or equal to the boxing level of the argument(s),
`u` is applied on the entire argument(s); otherwise the boxes are opened one by one and recursively processed.
This is described in detail below.

We start with the monadic case; the dyadic is slightly more elaborate.

#### Monadic L:

1. The *absolute level an* is calculated.
It is simple `n` unless `n` is negative, in which case it is ` 0 <. n + L. y ` (i.e. `n` plus the boxing level of `y`, but never less than 0).
*an* never changes during the remainder of the processing.

2. `Level y` is executed, performing the function.

The recursive verb `Level` is defined as follows. It takes one argument, `y`.

1. If the boxing level of `y` is less than or equal to *an*, the result of `Level y` is ` u y `.

2. Otherwise, recursion is performed by opening each atom of `y` and executing `Level` on the contents.

I.e. the result of `Level y` is ` Level&.> y `.

Every time a box of `y` is opened for recursion, the boxing level of the contents drops by at least 1.

This guarantees the recursion will terminate. But an atom of `y` may have a boxing level less that that of the entire `y`.
This means that opening a box may reduce the boxing level by more than 1

]c =: (<'abc';1 2 3),(<(<'def'),(<0 1 2;4 5)) +-----------+-----------------+ |+---+-----+|+---+-----------+| ||abc|1 2 3|||def|+-----+---+|| |+---+-----+|| ||0 1 2|4 5||| | || |+-----+---+|| | |+---+-----------+| +-----------+-----------------+ |.L:1 c +-----------+-----------------+ |+-----+---+|+---+-----------+| ||1 2 3|abc|||fed|+---+-----+|| |+-----+---+|| ||4 5|0 1 2||| | || |+---+-----+|| | |+---+-----------+| +-----------+-----------------+

The question to ponder is: *Why did applying |. at level 1 reverse def, which is obviously a leaf at level 0?*

The answer is that the larger sub-argument, of which `def` was a part, was this:

+---+-----------+ |def|+-----+---+| | ||0 1 2|4 5|| | |+-----+---+| +---+-----------+

This noun has level 2, so `u` cannot be applied to it.
Each box is opened and processed, and the boxing level of the first atom skips from 2 to 0, at which point `u` is applied.

Similarly:

|.L:2 c +-----------+-----------------+ |+-----+---+|+-----------+---+| ||1 2 3|abc|||+-----+---+|def|| |+-----+---+|||0 1 2|4 5|| || | ||+-----+---+| || | |+-----------+---+| +-----------+-----------------+

The first box, whose contents are at level 1, is reversed because `|.` is applied as soon as the box is opened.

The rule to remember is that **the boxes of y are opened from the top down, and u is applied when an opened contents is within an levels of the leaves.**

#### Dyadic L:

The processing of ` x u L: n y ` is like ` u L: n y ` with these added features

`n`can give different levels for`x`and`y``u`is executed only when**both**level requirements are met- if the arguments for recursion do not have the same shape, they follow the rules for cell replication with verb rank
`0 0`

1. The *absolute levels anx* and *any* are calculated, as for the monadic case. *anx* and *any* never change during the remainder of the processing.

2. `x Level y` is executed, performing the function.

The recursive verb `Level` is defined as follows. It takes two arguments, `x` and `y`.

1. If each argument is less than or equal to its absolute level, the result of `x Level y` is ` u y `.

2. Otherwise, recursion is performed. a. If only one argument is less than or equal to its absolute level, that argument is boxed.

This adds one to its boxing level, which has the effect of 'marking time' as it waits for the other operand to reach its absolute boxing level.

b. Whether or not step (a) was executed, the atoms of `x` and `y` are matched and `Level` is executed on the opened contents.

I.e. the result of ` x Level y ` is ` x Level&.> y `.

]y =: 'AB';('CD';'EFG');<(<2 2$'HIJ';'K') +--+--------+---------+ |AB|+--+---+|+-------+| | ||CD|EFG|||+---+-+|| | |+--+---+|||HIJ|K||| | | ||+---+-+|| | | |||HIJ|K||| | | ||+---+-+|| | | |+-------+| +--+--------+---------+ x ,L:0 y +-------------+--------------+-----------------------------+ |+-----+-----+|+-----+------+|+-------------+-------------+| ||abcAB|defAB|||abcCD|defEFG|||+------+----+|+------+----+|| |+-----+-----+|+-----+------+|||abcHIJ|abcK|||defHIJ|defK||| | | ||+------+----+|+------+----+|| | | |||abcHIJ|abcK|||defHIJ|defK||| | | ||+------+----+|+------+----+|| | | |+-------------+-------------+| +-------------+--------------+-----------------------------+

This follows the rules given above.