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In the APL [[array model]] and in particular [[leading axis theory]], a '''cell''' of an array is a [[subarray]] which is formed by selecting a single [[Index#Index along an axis|index]] along some number of leading [[Axis|axes]] and the whole of each trailing axis. Cells are classified by their [[rank]], which may be between 0 ([[scalar]]s) and the array's rank (in which case the cell must be the entire array). Cells with rank < | In the APL [[array model]] and in particular [[leading axis theory]], a '''cell''' of an array is a [[subarray]] which is formed by selecting a single [[Index#Index along an axis|index]] along some number of leading [[Axis|axes]] and the whole of each trailing axis. Cells are classified by their [[rank]], which may be between 0 ([[scalar]]s) and the array's rank (in which case the cell must be the entire array). Cells with rank <syntaxhighlight lang=apl inline>k</syntaxhighlight> are called '''k-cells''' of an array. A [[major cell]] is a cell whose rank is one less than the entire array, or a 0-cell of a [[scalar]]. | ||
== Characterization == | == Characterization == | ||
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</div> | </div> | ||
The k-cells of an array with [[rank]] < | The k-cells of an array with [[rank]] <syntaxhighlight lang=apl inline>r</syntaxhighlight> share the last <syntaxhighlight lang=apl inline>k</syntaxhighlight> [[Axis|axes]] with that array, and collapse the first <syntaxhighlight lang=apl inline>r-k</syntaxhighlight> axes by choosing a single [[Index#Index along an axis|index]] along each of them. Using [[bracket indexing]] this can be written | ||
< | <syntaxhighlight lang=apl> | ||
A[3;2; ; ; ] ⍝ 3-cell of a rank-5 array | A[3;2; ; ; ] ⍝ 3-cell of a rank-5 array | ||
</ | </syntaxhighlight> | ||
while [[Squad indexing]] allows a cell to be selected without knowing the argument's rank. | while [[Squad indexing]] allows a cell to be selected without knowing the argument's rank. | ||
< | <syntaxhighlight lang=apl> | ||
3 2⌷A ⍝ (r-2)-cell of a rank-r array | 3 2⌷A ⍝ (r-2)-cell of a rank-r array | ||
</ | </syntaxhighlight> | ||
</div> | </div> | ||
Squad indexing thus transforms [[vector]]s whose length is at most the [[rank]] of an array into cells of that array. This transformation is an exact correspondence or [[wikipedia:bijection|bijection]]: each cell has exactly one corresponding vector. For this reason a valid [[simple]] [[numeric]] left argument to Squad is called the [[Index#Index of a cell|index of a cell]]. | Squad indexing thus transforms [[vector]]s whose length is at most the [[rank]] of an array into cells of that array. This transformation is an exact correspondence or [[wikipedia:bijection|bijection]]: each cell has exactly one corresponding vector. For this reason a valid [[simple]] [[numeric]] left argument to Squad is called the [[Index#Index of a cell|index of a cell]]. | ||
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All cells of a given rank in a particular array have the same [[shape]]: | All cells of a given rank in a particular array have the same [[shape]]: | ||
< | <syntaxhighlight lang=apl> | ||
A ← 5 4 3 2⍴0 | A ← 5 4 3 2⍴0 | ||
(3 2)(1 4)(2 2)(4 4) {⍴⍺⌷⍵}¨⊂ A | (3 2)(1 4)(2 2)(4 4) {⍴⍺⌷⍵}¨⊂ A | ||
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│3 2│3 2│3 2│3 2│ | │3 2│3 2│3 2│3 2│ | ||
└───┴───┴───┴───┘ | └───┴───┴───┴───┘ | ||
</ | </syntaxhighlight> | ||
This is because the shape of a cell is always a [[suffix]] of the shape of the entire array. | This is because the shape of a cell is always a [[suffix]] of the shape of the entire array. | ||
< | <syntaxhighlight lang=apl> | ||
(4 3 2 1)(2 2)(5)(3 1)⍬ {⍴⍺⌷⍵}¨⊂ A | (4 3 2 1)(2 2)(5)(3 1)⍬ {⍴⍺⌷⍵}¨⊂ A | ||
┌┬───┬─────┬───┬───────┐ | ┌┬───┬─────┬───┬───────┐ | ||
││3 2│4 3 2│3 2│5 4 3 2│ | ││3 2│4 3 2│3 2│5 4 3 2│ | ||
└┴───┴─────┴───┴───────┘ | └┴───┴─────┴───┴───────┘ | ||
</ | </syntaxhighlight> | ||
Because there is only one possible shape for k-cells of an array < | Because there is only one possible shape for k-cells of an array <syntaxhighlight lang=apl inline>A</syntaxhighlight>, this shape is called the rank-k '''cell shape''' for <syntaxhighlight lang=apl inline>A</syntaxhighlight>. The cell shape is an important feature of the [[Rank operator]]: every invocation of Rank's left [[operand]] receives arrays of the same shape as [[argument]]s. | ||
== Leading and trailing axes == | == Leading and trailing axes == | ||
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In order to create a hierarchy of cells some ordering on the [[Axis|axes]] of an array must be used, with each smaller cell taking up a subset of the axes of a larger cell. For a particular cell rank, the axes will be split into two parts, with the axes used by the cells forming the ''cell shape'' and the other axes forming the [[frame]]. The choice to use trailing axes for a cell follows naturally from the [[ravel order]] of an array: because cells are handled as individual units but the frame is not, each cell should be contiguous in memory. The only choice which ensures this property is to use the trailing axes, those which are closer together in ravel order, for cells. | In order to create a hierarchy of cells some ordering on the [[Axis|axes]] of an array must be used, with each smaller cell taking up a subset of the axes of a larger cell. For a particular cell rank, the axes will be split into two parts, with the axes used by the cells forming the ''cell shape'' and the other axes forming the [[frame]]. The choice to use trailing axes for a cell follows naturally from the [[ravel order]] of an array: because cells are handled as individual units but the frame is not, each cell should be contiguous in memory. The only choice which ensures this property is to use the trailing axes, those which are closer together in ravel order, for cells. | ||
The interactive tool at the right displays both trailing-axis cells (those used by APL) and leading-axis cells (the opposite choice). While each trailing-axis cell can be denoted with a single rectangle, leading-axis cells may require multiple rectangles to represent. This fact mirrors the properties of cells in ravel order: each leading-axis cell is a single region in the array's ravel, but in a trailing-axis no two elements are adjacent (except in the degenerate cases of 0-cells, full-rank cells, and length-1 axes). | The interactive tool at the right displays both trailing-axis cells (those used by APL) and leading-axis cells (the opposite choice). While each trailing-axis cell can be denoted with a single rectangle, leading-axis cells may require multiple rectangles to represent. This fact mirrors the properties of cells in ravel order: each leading-axis cell is a single region in the array's ravel, but in a trailing-axis cell no two elements are adjacent (except in the degenerate cases of 0-cells, full-rank cells, and length-1 axes). | ||
== Hierarchy == | == Hierarchy == | ||
In an array of rank < | In an array of rank <syntaxhighlight lang=apl inline>r</syntaxhighlight>, every cell of rank <syntaxhighlight lang=apl inline>k</syntaxhighlight> is contained in exactly one cell of rank <syntaxhighlight lang=apl inline>j</syntaxhighlight> when <math>r \ge j \ge k \ge 0</math>. In particular, every 0-cell (and hence every [[element]]) is contained in exactly one cell of each rank. This is why the interactive tool at the right can display cells of three different ranks even though only a single element can be pointed at. | ||
Considering cells in terms of their [[Index#Index of a cell|indices]] makes the reason for this relationship clear. In an array of rank < | Considering cells in terms of their [[Index#Index of a cell|indices]] makes the reason for this relationship clear. In an array of rank <syntaxhighlight lang=apl inline>r</syntaxhighlight>, a cell of rank <syntaxhighlight lang=apl inline>k</syntaxhighlight> has an index vector <syntaxhighlight lang=apl inline>I</syntaxhighlight> of length <syntaxhighlight lang=apl inline>r-k</syntaxhighlight> (the index selects one index along each axis ''not'' in the cell). For <syntaxhighlight lang=apl inline>j</syntaxhighlight> such that <math>r \ge j \ge k</math>, we have <math>r-j \le r-k</math>, and the cell of rank <syntaxhighlight lang=apl inline>j</syntaxhighlight> which contains our chosen cell has index <syntaxhighlight lang=apl inline>(r-j)↑I</syntaxhighlight>, a [[prefix]] of the smaller cell's index. | ||
This construction fails for <math>j < k</math> because the index for a smaller (lower-rank) cell must have extra indices relative to the index for a larger one. There is no way to choose a single index: in fact, as long as the axes of our array all have length greater than 1, there are multiple choices of index and the larger cell contains multiple smaller cells. As an example, a matrix can contain many rows even though a row is only ever contained in exactly one matrix. This fact should not be surprising: it's the main reason why APL uses arrays rather than lists at all! | This construction fails for <math>j < k</math> because the index for a smaller (lower-rank) cell must have extra indices relative to the index for a larger one. There is no way to choose a single index: in fact, as long as the axes of our array all have length greater than 1, there are multiple choices of index and the larger cell contains multiple smaller cells. As an example, a matrix can contain many rows even though a row is only ever contained in exactly one matrix. This fact should not be surprising: it's the main reason why APL uses arrays rather than lists at all! | ||