[% setvar title Proposed syntax for matrix element access and slicing. %]

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Proposed syntax for matrix element access and slicing.


  Maintainer: Buddha Buck <bmbuck@14850.com>
  Date: 29 Aug 2000
  Last Modified: 14 Sep 2000
  Mailing List: perl6-language-data@perl.org
  Number: 169
  Version: 2
  Status: Withdrawn
  Superceded By: RFC 204, RFC 205, RFC 206, RFC 207


This RFC is withdrawn in favor of RFCs 202-207. Most of the ideas presented in this RFC have been incorporated in RFC 205 (New operator ; for creating array slices), RFC 206 (@#arr for getting the dimensions of an array), and RFC 207 (Efficient Array Loops). The combination of RFC 205 and RFC 204 create a more flexible array index technique than presented in this RFC.

The main body of this RFC is left here as a historical record.


I propose the use of ';' as a separator for index terms when accessing multi-dimensional arrays (matrices). The syntax "$matrix[$w;$x;$y;$z];" is clearer and more flexible than other proposed alternatives. The flexibility is mainly in the possibility of a robust slicing syntax.

Although I know of no RFC's that recommend the implementation of matrices, I expect such a proposal to be made, and this RFC does not make such a suggestion.


Several suggestions have been made for matrix element accessors, including:

    $matrix[$x][$y][$z]   # aka "(Lo)*L" notation ( (lists of)* lists)
    $matrix[$x,$y,$z]     # aka "[,,]" notation
    $matrix{"$x,$y,$z"}   # aka hash notation

All three of these have problems. The (Lo)*L notation either requires matrices be (lists of)* lists, or uses the same syntax for both matrices and (lists of)* lists. The hash notation has the same confusion, but with hashes. The [,,] notation is too similar to the existing syntax for array slices.

None of them allow for convenient matrix slicing.

Using $matrix[$i;$j;$k;$l] has the advantage that it is dissimilar from exising syntax, thus minimizing confusion with pre-existing data structures, and it provides a clean separation of indices, allowing for a flexible slice notation.

Matrix Element Access.

Matrix elements should be accessed as normal arrays, but using ";" to separate the indices:

   my @matrix;
   $matrix[1;2;3;4] = 5;

   sub tensor2mul {
     my @tensor1 = @{shift @_};
     my @tensor2 = @{shift @_};
     my @tensor3;

     for $i (0..3) {
       for $j (0..3) {
         for $k (0..3) {
           for $l (0..3) {
             $tensor3[$i;$j;$k;$l] = $tensor1[$i;$j] * $tensor2[$k;$l];
     } } } }
     return @tensor3;

Currently, ';' is only used for statement separators and as term separators in for(;;) loops. This new proposed use should not be a problem for the parser or programmers to understand in this context.

Matrix Slices

Because the indices are separate, array slice notation can easily be used for the individual indices.

   #exlude row $j and column $k, as if taking a determinant.
   @submatrix = @matrix[0..$j-1,$j+1..$n;0..$k-1,$k+1..$n];

While that should work obviously enough, a more flexible syntax would be nice:

   # leave index blank for "all"  # or use unterminated ".."
   @row = @matrix[1;];            # @row = @matrix[1;..];
   @col = @matrix[;1];            # @col = @matrix[..;1];
   @swaprow = @matrix[1,0,2..$n;]

   # Use ^var to get more complicated slices
   @diagonal    = @matrix[^i;^i];
   @rdiagonal   = @matrix[^i;$n-$i];
   @uptriangle  = @matrix[^i;0..$i];
   @lowtriangle = @matrix[^i;$i..$#];

   # These would be nice, but probably not feasable

   @trans[^i;^j] = @matrix[^j;^i];
   @tensor3[^i;^j;^k;^l] = @tensor1[^i;^j] * @tensor2[^k;^l];
   $dotproduct = reduce ^1+^2 , 0, a[^_] * b[^_];

Those above would work if the ^vars would scope across all indices in the statement, not just one.

   # use $# to get maximum val for index
   sub lowtriangle {
      @matrix = @{ shift @_ };
      return @matrix[^i;^i..$#];  # works for all 2-d matrices.

Matrices of more than two dimensions should be handled similarly:

   # Slice through middle of cube perpendicular to main diagonal
   @slice = @matrix[^i;^j;^i-^j];

The use of the ^var syntax is analogous to the use of ^placeholder syntax in Damian Conway's HOF RFC.

Computing the size of matrices

I'd propose that, analogous to the $#array notation for revealing the upper index of @array, @#matrix return the analogous list:

   my @matrix
   $matrix[2;2;2] = 1;
   @sizes  = @#matrix;   # @sizes == (2,2,2)
   $numdim = @#matrix;   # array in scaler context.

Unresolved issue -- variable dimensionality

Right now, the syntax above hardcodes the dimension of a matrix in the index list. This does not allow you to write programs which manipulate $n-dimensional matrices, where $n is computed at run-time. This is unfortunate, and I'd like to hear suggestions on how to deal with that issue.

One possible mechanism would be to allow @k = @matrix[$n;]; to always assign to @k a matrix of one less dimension that @matrix, regardless of the original dimensionality, with the exception that @array[$n;] will return a singleton list rather than a 0-dimension matrix (which should by rights be a scalar). How that should generalize is not clear.

Unresolved issue -- ^var slices and HOF

Damian Conway suggested a HOF syntax which used ^var as a placeholder. The "reduce" example above shows how confusing using both HOF and ^var slices in the same expression can be. Originally, I thought this wouldn't be a problem because closures are not allowed as array indices, so @a[^i;^j] would be unambiguous. But this precludes being able to say:

   my &closure = $matrix[^_;^_]; # closure(1,2) == $matrix[1;2]
   my &c2      = closure(^_,5) ;   c2(4)        == $matrix[4;5]

I think both functionalities are powerful and useful. I would appreciate how to get both of them unambiguously and cleanly.


Using ; as a indices separator should cause no serious problems of implementation.

The use of ^var indices could, in a reference implementation, be syntactic sugar for something like:

@matrix[^i;^j;^k;^k] ==> do { my @result; my ($i,$j,$k); my @indices = @#matrix; for $i (0..$indices[0]) { for $j (0..$indices[1]) { for $k (0..$indices[2] { $result[$i;$j;$k] = $matrix[$i,$j,$k,$k]; } } } @result; }

It should be noted that the matrix slice operations are likely to be inherently O(k), where k is the number of dimensions indexed over.


RFC 24: "Higher Order Functions"

RFC 204: "Arrays: Use list references for multidimensional array access"

RFC 205: "Arrays: New operator ';' for creating array slices"

RFC 206: "Arrays: @#arr for getting the dimensions of an array"