
TODO: introduce multidimensional array feature and approaches

The new \CFA standard library @array@ datatype supports multidimensional uses more richly than the C array.  The new array's multimentsional interface and implementation, follows an array-of-arrays setup, meaning, like C's @float[n][m]@ type, one contiguous object, with coarsely-strided dimensions directly wrapping finely-strided dimensions.  This setup is in contrast with the pattern of array of pointers to other allocations representing a sub-array.  Beyond what C's type offers, the new array brings direct support for working with a noncontiguous array slice, allowing a program to work with dimension subscripts given in a non-physical order.  C and C++ require a programmer with such a need to manage pointer/offset arithmetic manually.

Examples are shown using a $5 \times 7$ float array, @a@, loaded with increments of $0.1$ when stepping across the length-7 finely-strided dimension shown on columns, and with increments of $1.0$ when stepping across the length-5 corsely-strided dimension shown on rows.
\lstinputlisting[language=C, firstline=120, lastline=128]{hello-md.cfa}
The memory layout of @a@ has strictly increasing numbers along its 35 contiguous positions.

A trivial form of slicing extracts a contiguous inner array, within an array-of-arrays.  Like with the C array, a lesser-dimensional array reference can be bound to the result of subscripting a greater-dimensional array, by a prefix of its dimensions.  This action first subscripts away the most coaresly strided dimensions, leaving a result that expects to be be subscripted by the more finely strided dimensions.
\lstinputlisting[language=C, firstline=60, lastline=66]{hello-md.cfa}
\lstinputlisting[language=C, firstline=140, lastline=140]{hello-md.cfa}

This function declaration is asserting too much knowledge about its parameter @c@, for it to be usable for printing either a row slice or a column slice.  Specifically, declaring the parameter @c@ with type @array@ means that @c@ is contiguous.  However, the function does not use this fact.  For the function to do its job, @c@ need only be of a container type that offers a subscript operator (of type @ptrdiff_t@ $\rightarrow$ @float@), with governed length @N@.  The new-array library provides the trait @ix@, so-defined.  With it, the original declaration can be generalized, while still implemented with the same body, to the latter declaration:
\lstinputlisting[language=C, firstline=40, lastline=44]{hello-md.cfa}
\lstinputlisting[language=C, firstline=145, lastline=145]{hello-md.cfa}

Nontrivial slicing, in this example, means passing a noncontiguous slice to @print1d@.  The new-array library provides a ``subscript by all'' operation for this purpose.  In a multi-dimensional subscript operation, any dimension given as @all@ is left ``not yet subscripted by a value,'' implementing the @ix@ trait, waiting for such a value.
\lstinputlisting[language=C, firstline=150, lastline=151]{hello-md.cfa}

The example has shown that @a[2]@ and @a[[2, all]]@ both refer to the same, ``2.*'' slice.  Indeed, the various @print1d@ calls under discussion access the entry with value 2.3 as @a[2][3]@, @a[[2,all]][3]@, and @a[[all,3]][2]@.  This design preserves (and extends) C array semantics by defining @a[[i,j]]@ to be @a[i][j]@ for numeric subscripts, but also for ``subscripting by all''.  That is:

\begin{tabular}{cccccl}
    @a[[2,all]][3]@  &  $=$  &  @a[2][all][3]@  & $=$  &  @a[2][3]@  & (here, @all@ is redundant)  \\
    @a[[all,3]][2]@  &  $=$  &  @a[all][3][2]@  & $=$  &  @a[2][3]@  & (here, @all@ is effective)
\end{tabular}

Narrating progress through each of the @-[-][-][-]@ expressions gives, firstly, a definition of @-[all]@, and secondly, a generalization of C's @-[i]@.

\noindent Where @all@ is redundant:

\begin{tabular}{ll}
    @a@  & 2-dimensional, want subscripts for coarse then fine \\
    @a[2]@  & 1-dimensional, want subscript for fine; lock coarse = 2 \\
    @a[2][all]@  & 1-dimensional, want subscript for fine \\
    @a[2][all][3]@  & 0-dimensional; lock fine = 3
\end{tabular}

\noindent Where @all@ is effective:

\begin{tabular}{ll}
    @a@  & 2-dimensional, want subscripts for coarse then fine \\
    @a[all]@  & 2-dimensional, want subscripts for fine then coarse \\
    @a[all][3]@  & 1-dimensional, want subscript for coarse; lock fine = 3 \\
    @a[all][3][2]@  & 0-dimensional; lock coarse = 2
\end{tabular}

The semantics of @-[all]@ is to dequeue from the front of the ``want subscripts'' list and re-enqueue at its back.  The semantics of @-[i]@ is to dequeue from the front of the ``want subscripts'' list and lock its value to be @i@.

Contiguous arrays, and slices of them, are all realized by the same underlying parameterized type.  It includes stride information in its metatdata.  The @-[all]@ operation is a conversion from a reference to one instantiation, to a reference to another instantiation.  The running example's @all@-effective step, stated more concretely, is:

\begin{tabular}{ll}
    @a@       & : 5 of ( 7 of float each spaced 1 float apart ) each spaced 7 floats apart \\
    @a[all]@  & : 7 of ( 5 of float each spaced 7 floats apart ) each spaced 1 float apart
\end{tabular}

\begin{figure}
    \includegraphics{measuring-like-layout}
    \caption{Visualization of subscripting by value and by \lstinline[language=C,basicstyle=\ttfamily]{all}, for \lstinline[language=C,basicstyle=\ttfamily]{a} of type \lstinline[language=C,basicstyle=\ttfamily]{array( float, Z(5), Z(7) )}. The horizontal dimension represents memory addresses while vertical layout is conceptual.}
    \label{fig:subscr-all}
\end{figure}

\noindent While the latter description implies overlapping elements, Figure \ref{fig:subscr-all} shows that the overlaps only occur with unused spaces between elements.  Its depictions of @a[all][...]@ show the navigation of a memory layout with nontrivial strides, that is, with ``spaced \_ floats apart'' values that are greater or smaller than the true count of valid indeces times the size of a logically indexed element.  Reading from the bottom up, the expression @a[all][3][2]@ shows a float, that is masquerading as a @float[7]@, for the purpose of being arranged among its peers; five such occurrences form @a[all][3]@.  The tail of flatter boxes extending to the right of a poper element represents this stretching.  At the next level of containment, the structure @a[all][3]@ masquerades as a @float[1]@, for the purpose of being arranged among its peers; seven such occurrences form @a[all]@.  The verical staircase arrangement represents this compression, and resulting overlapping.

The new-array library defines types and operations that ensure proper elements are accessed soundly in spite of the overlapping.  The private @arpk@ structure (array with explicit packing) is generic over these two types (and more): the contained element, what it is masquerading as.  This structure's public interface is the @array(...)@ construction macro and the two subscript operators.  Construction by @array@ initializes the masquerading-as type information to be equal to the contained-element information.  Subscrpting by @all@ rearranges the order of masquerading-as types to achieve, in genernal, nontrivial striding.  Subscripting by a number consumes the masquerading-as size of the contained element type, does normal array stepping according to that size, and returns there element found there, in unmasked form.

The @arpk@ structure and its @-[i]@ operator are thus defined as:
\begin{lstlisting}
forall( ztype(N),               // length of current dimension
        dtype(S) | sized(S),    // masquerading-as
        dtype E_im,             // immediate element, often another array
        dtype E_base            // base element, e.g. float, never array
      ) {
    struct arpk {
        S strides[z(N)];        // so that sizeof(this) is N of S
    };

    // expose E_im, stride by S
    E_im & ?[?]( arpk(N, S, E_im, E_base) & a, ptrdiff_t i ) {
        return (E_im &) a.strides[i];
    }
}
\end{lstlisting}

An instantion of the @arpk@ generic is given by the @array(E_base, N0, N1, ...)@ exapnsion, which is @arpk( N0, Rec, Rec, E_base )@, where @Rec@ is @array(E_base, N1, ...)@.  In the base case, @array(E_base)@ is just @E_base@.  Because this construction uses the same value for the generic parameters @S@ and @E_im@, the resulting layout has trivial strides.

Subscripting by @all@, to operate on nontrivial strides, is a dequeue-enqueue operation on the @E_im@ chain, which carries @S@ instatiations, intact, to new positions.  Expressed as an operation on types, this rotation is:
\begin{eqnarray*}
suball( arpk(N, S, E_i, E_b) ) & = & enq( N, S, E_i, E_b ) \\
enq( N, S, E_b, E_b ) & = & arpk( N, S, E_b, E_b ) \\
enq( N, S, arpk(N', S', E_i', E_b), E_b ) & = & arpk( N', S', enq(N, S, E_i', E_b), E_b )
\end{eqnarray*}

