RFC 169 (v1) Proposed syntax for matrix element access and slicing.

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=head1 TITLE

Proposed syntax for matrix element access and slicing.

=head1 VERSION

   Maintainer: Buddha Buck <[email protected]>
   Date: 29 August 2000
   Mailing List: [email protected]
   Version: 1
   Number: 169
   Status: Developing

=head1 ABSTRACT

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.

=head1 DESCRIPTION

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.

=head2 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.

=head2 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.

=head2 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.

=head2 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.

=head2 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.

=head1 IMPLEMENTATION

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.

=head1 REFERENCES

RFC 24: Higher Order Functions