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Synopsis_09
Data Structures

Synopsis_09 - Data Structures



NAME

Synopsis_09 - Data Structures


AUTHOR

Larry Wall <larry@wall.org>


VERSION


  Maintainer: Larry Wall <larry@wall.org>

  Date: 13 Sep 2004

  Last Modified: 22 Nov 2005

  Number: 9

  Version: 5


Overview

This synopsis summarizes the non-existent Apocalypse 9, which discussed in detail the design of Perl 6 data structures. It was primarily a discussion of how the existing features of Perl 6 combine to make it easier for the PDL folks to write numeric Perl.


Lazy lists

All list contexts are lazy by default. They still flatten eventually, but only when forced to. You have to use unary ** to get a non-lazy flattening list context (that is, to flatten immediately like Perl 5).


Sized types

Sized low-level types are named most generally by appending the number of bits to a generic low-level type name:


    int1

    int2

    int4

    int8

    int16

    int32       (aka int on 32-bit machines)

    int64       (aka int on 64-bit machines)

    uint1       (aka bit)

    uint2

    uint4

    uint8       (aka byte)

    uint16

    uint32

    uint64

    num32

    num64       (aka num on most architectures)

    num128

    complex32

    complex64   (aka complex on most architectures)

    complex128

Complex sizes indicate the size of each num component rather than the total. This would extend to tensor typenames as well if they're built-in types. Of course, the typical tensor structure is just reflected in the dimensions of the array--but the principle still holds that the name is based on the number of bits of the simple base type.

The unsized types int and num are based on the architecture's normal size for int and double in whatever version of C the run-time system (presumably Parrot) is compiled in. So int typically means int32 or int64, while num usually means num64, and complex means two of whatever num turns out to be.

You are, of course, free to use macros or type declarations to associate additional names, such as ``short'' or ``single''. These are not provided by default. An implementation of Perl is not required to support 64-bit integer types or 128-bit floating-point types unless the underlying architecture supports them.

And yes, an int1 can store only -1 or 0. I'm sure someone'll think of a use for it...


Compact structs

A class whose attributes are all low-level types can behave as a struct. (Access from outside the class is still only through accessors, though.) Whether such a class is actually stored compactly is up to the implementation, but it ought to behave that way, at least to the extent that it's trivially easy (from the user's perspective) to read and write to the equivalent C structure. That is, when byte-stringified, it should look like the C struct, even if that's not how it's actually represented inside the class. (This is to be construed as a substitute for at least some of the current uses of pack/unpack.)


Compact arrays

In declarations of the form:


    my bit @bits;

    my int @ints;

    my num @nums;

    my int4 @nybbles;

    my buf @buffers;

    my ref[Array] @ragged2d;

    my complex128 @longdoublecomplex;

the presence of a low-level type tells Perl that it is free to implement the array with ``compact storage'', that is, with a chunk of memory containing contiguous (or as contiguous as practical) elements of the specified type without any fancy object boxing that typically applies to undifferentiated scalars. (Perl tries really hard to make these elements look like objects when you treat them like objects--this is called autoboxing.)

The declarations above declare one-dimensional arrays of indeterminate length. Such arrays are autoextending just like ordinary Perl arrays (at the price of occasionally copying the block of data to another memory location). For many purposes, though, it's useful to define array types of a particular size and shape that, instead of autoextending, throw an exception if you try to access outside their declared dimensionality. Such arrays tend to be faster to allocate and access as well.

A multidimensional array is indexed by a semicolon list, which is really a list of pipes in disguise. Each sublist is a slice/pipe of one particular dimension. So


    @array[0..10; 42; @x]

is really short for


    @array.postcircumfix:<[ ]>( <== 0..10 <== 42 <== @x );

The compiler is free to optimize to something faster when it is known that lazy multidimensional subscripts are not necessary.

Note that


    @array[@x,@y]

is always interpreted as a one-dimensional slice in the outermost dimension, which is the same as:


    @array[@x,@y;]

or more verbosely:


    @array.postcircumfix:<[ ]>( <== @x,@y );

To interpolate an array at the semicolon level rather than the comma level, use the [;] reduce operator:


    @array[[;] @x]

which is equivalent to


    @array.postcircumfix:<[ ]>( <== @x[0] <== @x[1] <== @x[2]...);

Alternately, use a multislice array, indicated by a ; twigil:


    @array[@;x]

Multislice arrays have the advantage of being able to collect individual pipes and keep them distinct:


    my @;x;

    @;x <==  %hash.keys.grep: {/^X/};

    @;x <== =<>;

    @;x <== 1...;

    @;x <== gather { loop { take rand 100 } };

    %hash{@;x}

To declare a multidimensional array, you may declare it with a signature as if it were a function returning one of its entries:


    my num @nums (Int);   # one dimension, @nums[Int]

or alternately:


    my @nums (Int --> num);   # one dimension, @nums[Int]

You can use ranges as types:


    my @nums (0..2 --> num);   # one dimension, @nums[0..2]

    my @ints (0..3, 0..1 --> int);   # one dimension, @ints[0..3; 0..1]

That includes the ``upto'' range type:


    my @ints (^4, ^2 --> int);   # one dimension, @ints[0..3; 0..1]

You can pretend you're programming in Fortran, or awk:


    my int @ints (1..4, 1..2); # two dimensions, @ints[1..4; 1..2]

Note that this only influences your view of the array, not the actual shape of the array. If you pass this array to another module, it will see it as have a shape of (0..3, 0..1) unless it also declares a variable to view it differently.

Alternately, you may declare it using a prototype subscript, but then you must remember to use semicolons instead of commas to separate dimensions, because each slice represents an enumeration of the possible values, so the following are all equivalent:


    my @ints (0..3, 0..1 --> int);

    my int @ints (0..3, 0..1);

    my int @ints[^4;^2];

    my int @ints[0..3; 0..1];

    my int @ints[0,1,2,3; 0,1];

You can pass a multislice for the shape as well:


    my int @ints[[;]@fooshape];

    my int @ints[@;fooshape];   # same thing

Again, the [;] list operator interpolates a list into a semicolon list, which we do for consistency with subscript notation, not because it makes a great deal of sense to allow slices for dimensional specs (apart from ranges). So while the following is okay:


    my int @ints[0,1,2,3,4];    # same as 0..4

the following is a semantic error that the compiler should catch:


    my int @ints[^3,^3,^3];     # oops, comma instead of semicolon

The shape may be supplied entirely by the object at run-time:


    my num @nums = Array of num.new(:shape(^3;^3;^3));

    my num @nums .=new():shape(^3;^3;^3); # same thing

Any dimension of the array may be specified as ``Int'', in which case that dimension will autoextend. Typically this would be used in the final dimension to make a ragged array functionally equivalent to an array of arrays:


    my int @ints[^42; Int];

    push(@ints[41], getsomeints());

The shape may also be specified by types rather than sizes:


    my int @ints[Even; Odd];

or by both:


    my int @ints[0..100 where Even; 1..99 where Odd];

(presuming Even and Odd are types already constrained to be even or odd).


PDL support

An array @array can be tied to a piddle at declaration time:


    my num @array[@;mytensorshape] is Piddle;

    my @array is Piddle(:shape(^2;^2;^2;^2)) of int8;

Piddles are allowed to assume a type of num by default rather than the usual simple scalar. (And in general, the type info is merely made available to the ``tie'' implementation to do with what it will. Some data structures may ignore the ``of'' type and just store everything as general scalars. Too bad...)

Arrays by default are one dimensional, but may be declared to have any dimensionality supported by the implementation. You may use arrays just like scalar references--the main caveat is that you have to use binding rather than assignment to set one without copying:


    @b := @a[0...:by(2)]

With piddles in particular, this might alias each of the individual elements rather than the array as a whole. So modifications to @b are likely to be reflected back into @a. (But maybe the PDLers will prefer a different notation for that.)

The dimensionality of an array may be declared on the variable, but the actual dimensionality of the array depends on how it was created. Reconciling these views is a job for the particular array implementation. It's not necessarily the case that the declared dimensionality must match the actual dimensionality. It's quite possible that the array variable is deliberately declared with a different dimensionality to provide a different ``view'' on the actual value:


    my int @array[^2;^2] is Puddle .= new(:shape(^4) <== 0,1,2,3);

Again, reconciling those ideas is up to the implementation, Puddle in this case. The traits system is flexible enough to pass any metadata required, including ideas about sparseness, raggedness, and various forms of non-rectangleness such as triangleness. The implementation should probably carp about any metadata it doesn't recognize though. The implementation is certainly free to reject any object that doesn't conform to the variable's shape requirements.


Subscript and slice notation

A subscript indicates a ``slice'' of an array. Each dimension of an array is sliced separately, so we say a subscript is a semicolon-separated list of slice specifiers, also known as a multislice. A three-dimensional slice might look like this:


    @x[0..10; 1,0; 1...:by(2)]

It is up to the implementation of @x to decide how aggressively or lazily this subscript is evaluated, and whether the slice entails copying. (The PDL folks will generally want it to merely produce a virtual piddle where the new array aliases its values back into the old one.)

Of course, a single element can be selected merely by providing a single index value to each slice list:


    @x[0;1;42]


The semicolon operator

At the statement level, a semicolon terminates the current expression. Within any kind of bracketing construct, semicolon notionally separates slices, the interpretation of which depends on the context. Such a semicolon list always provides list context to each of its sublists. The storage of these sublists is hidden in the inner workings of the list. It does not produce a list of lists.

Single dimensional arrays expect simple slice subscripts, meaning they will treat a list subscript as a slice in the single dimension of the array. Multi-dimensional arrays, on the other hand, know how to handle multiple slices, one for each dimension. You need not specify all the dimensions; if you don't, the unspecified dimensions are ``wildcarded''. Supposing you have:


    my num @nums[^3;^3;^3];

Then


    @nums[0..2]

is the same as


    @nums[0..2;]

which is the same as


    @nums[0,1,2;*;*]

But you should maybe write the last form anyway just for good documentation, unless you don't actually know how many more dimensions there are.

If you wanted that 0..2 range to mean


    @nums[0;1;2]

instead, then you need to use the [;] reduction operator:


    @nums[[;] 0..2]

The zero-dimensional slice:


    @x[]

is assumed to want everything, not nothing. It's particularly handy because Perl 6 (unlike Perl 5) won't interpolate a bare array without brackets:


    @x = (1,2,3);

    say "@x = @x[]";    # prints @x = 1 2 3

Lists are lazy in Perl 6, and the slice lists are no exception. In particular, things like range objects are not flattened until they need to be, if ever. So a PDL implementation is free to steal the values from these ranges and ``piddle'' around with them:


    @nums[$min..$max:by(3)]

    @nums[$min..$max]

    @nums[$min...:by(3)]

    @nums[1...:by(2)]           # the odds

    @nums[0...:by(2)]           # the evens

That's all just the standard Perl 6 notation for ranges. Additional syntactic relief is always available as long as it's predeclared somehow. It's possible the range operator could be taught that :2 means :by(2), for instance. (But I rather dislike the RFC-proposed 0:10:2 notation that makes colon mean two different things so close together, plus it conflicts with Perl 6's general adverb notation if the next thing is alphabetic. On top of which, we're using :2 as a general radix notation.)

Another thing that's not going to fly easily is simply dropping out terms. Perl depends rather heavily on knowing when it's expecting a term or an operator, and simply leaving out terms before or after a binary operator really screws that up. For instance,


    0..:by(2)

parses as


    0 .. (by => 2)

rather than


    0 .. Inf :by(2)

That why we have postfix ... to mean ..Inf. But then if you leave out the first argument:


    ...:by(2)

you've written the yada-yada-yada operator, which is actually a term that will not produce an infinite range for you. Don't do that.

Maybe you should just find some nice Unicode characters for your operators...


PDL signatures

To rewrite a Perl 5 PDL definition like this:


       pp_def(

            'inner',

            Pars => 'a(n); b(n); [o]c(); ', # the signature, see above

            Code => 'double tmp = 0;

                     loop(n) %{ tmp += $a() * $b(); %}

                     $c() = tmp;' );

you might want to write a macro that parses something vaguely resembling this:


    role PDL_stuff[::TYPE] {

        PDLsub inner (@a[$n], @b[$n] --> @c[]) {

            my TYPE $tmp = 0;

            for ^$n {

                $tmp += @a[$_] * @b[$_];

            }

            @c[] = tmp;

        }

    }

where that turns into something like this:


    role PDL_stuff[::TYPE] {

        multi sub inner (TYPE @a, TYPE @b --> TYPE) {

            my $n = @a.shape[0];        # or maybe $n is just a parameter

            assert($n == @b.shape[0]);  #  and this is already checked by PDL

            my TYPE $tmp = 0;

            for ^$n {

                $tmp += @a[$_] * @b[$_];

            }

            return $tmp;

        }

    }

Then any class that does PDL_stuff[num] has an inner() function that can (hopefully) be compiled down to a form useful to the PDL threading engine. Presumably the macro also stores away the PDL signature somewhere safe, since the translated code hides that information down in procedural code. Possibly some of the [n] information can come back into the signature via where constraints on the types. This would presumably make multimethod dispatch possible on similarly typed arrays with differing constraints.

(The special destruction problems of Perl 5's PDL should go away with Perl 6's GC approach, as long as PDL's objects are registered with Parrot correctly.)


Junctions

A junction is a superposition of data values pretending to be a single data value. Junctions come in four varieties:


    list op     infix op

    =======     ========

    any()       |

    all()       &

    one()       ^

    none()      (no "nor" op defined)

Note that the infix ops are ``list-associative'', insofar as


    $a | $b | $c

    $a & $b & $c

    $a ^ $b ^ $c

mean


    any($a,$b,$c)

    all($a,$b,$c)

    one($a,$b,$c)

rather than


    any(any($a,$b),$c)

    all(all($a,$b),$c)

    one(one($a,$b),$c)

Some contexts, such as boolean contexts, have special rules for dealing with junctions. In any scalar context not expecting a junction of values, a junction produces automatic parallelization of the algorithm. In particular, if a junction is used as an argument to any routine (operator, closure, method, etc.), and the scalar parameter you are attempting to bind the argument to is inconsistent with the Junction type, that routine is ``autothreaded'', meaning the routine will be called automatically as many times as necessary to process the individual scalar elements of the junction in parallel.

The results of these separate calls are then recombined into a single junction of the same species as the junctive argument. If two or more arguments are junctive, then the argument that is chosen to be ``autothreaded'' is:

  • the left-most conjunction or injunction (if any), or else

  • the left-most abjunction or disjunction

with the tests applied in that order.

Each of the resulting set of calls is then recursively autothreaded until no more junctive arguments remain. That is:


       substr("camel", 0|1, 2&3)

    -> all( substr("camel", 0|1, 2),      # autothread the conjunctive arg

            substr("camel", 0|1, 3)

          )

    -> all( any( substr("camel", 0, 2),   # autothread the disjunctive arg

                 substr("camel", 1, 2),

               ),

            any( substr("camel", 0, 3),   # autothread the disjunctive arg

                 substr("camel", 1, 3),

               )

          )

    -> all( any( "ca",                    # evaluate

                 "am",

               ),

            any( "cam",

                 "ame",

               )

    -> ("ca"|"am") & ("cam"|"ame")        # recombine results in junctions

Junctions passed as part of a container do not cause autothreading unless individually pulled out and used as a scalar. It follows that junctions passed as members of a ``slurpy'' array or hash do not cause autothreading on that parameter. Only individually declared parameters may autothread. (Note that positional array and hash parameters are in fact scalar parameters, though, so you could pass a junction of array or hash references.)


Parallelized parameters and autothreading

Within the scope of a use autoindex pragma (or equivalent, such as use PDL (maybe)), any closure that uses parameters as subscripts is also a candidate for autothreading. For each such parameter, the compiler supplies a default value that is a range of all possible values that subscript can take on (where ``possible'' is taken to mean the declared shape of a shaped array, or the actual shape of an autoextending array). That is, if you have a closure of the form:


    -> $x, $y { @foo[$x;$y] }

then the compiler adds defaults for you, something like:


    -> $x = @foo.shape[0].range,

       $y = @foo.shape[1].range { @foo[$x;$y] }

where each such range is autoiterated for you.

In the abstract (and often in the concrete), this puts an implicit loop around the block of the closure that visits all the possible subscript values for that dimension (unless the parameter is actually supplied to the closure, in which case that is what is used as the slice subscript).

So to write a typical tensor multiplication:


    Cijkl = Aij * Bkl

you can just write this:


    use autoindex;

    do { @c[$^i, $^j, $^k, $^l] = @a[$^i, $^j] * @b[$^k, $^l] };

or equivalently:


    -> $i, $j, $k, $l { @c[$i, $j, $k, $l] = @a[$i, $j] * @b[$k, $l] }();

or even:


    do -> $i, $j, $k, $l {

        @c[$i, $j, $k, $l] = @a[$i, $j] * @b[$k, $l]

    }

That's almost pretty.

It is erroneous for an unbound parameter to match multiple existing array subscripts differently. (Arrays being created don't count.)

Note that you could pass any of $i, $j, $k or $l explicitly, or prebind them with a .assuming method, in which only the unbound parameters autothread.

If you use an unbound array parameter as a semicolon-list interpolator (via the [;] reduction operator), it functions as a wildcard list of subscripts that must match the same everywhere that parameter is used. For example,


    do -> @wild { @b[[;] reverse @wild] = @a[[;] @wild]; };

produces an array with the dimensions reversed regardless of the dimensionality of @a.

The optimizer is, of course, free to optimize away any implicit loops that it can figure out how to do more efficiently without changing the semantics.

See RFC 207 for more ideas on how to use autothreading (though the syntax proposed there is rather different).


Hashes

Everything we've said for arrays applies to hashes as well, except that if you're going to limit the keys of one dimension of a hash, you have to provide an explicit list of keys to that dimension of the shape, or an equivalent range:


    my num %hash{<a b c d e f>; Str};

    my num %hash{'a'..'f'; Str};                # same thing

To declare a hash that can take any object as a key rather than just a string, say something like:


    my %hash{Any};

Likewise, you can limit the keys to objects of particular types:


    my Fight %hash{Dog;Cat};

The standard Hash is just


    my Any %hash{Str};

Note that any type used as a key must be intrinsically immutable, or it has to be able to make a copy that functions as an immutable key, or it has to have copy-on-write semantics. It is erroneous to change a key object's value within the hash except by deleting it and reinserting it.


Autosorted hashes

The default hash iterator is a property called .iterator that can be user replaced. When the hash itself needs an iterator for .pairs, .keys, .values, or .kv, it calls %hash.iterator() to start one. In scalar context, .iterator returns an iterator object. In list context, it returns a lazy list fed by the iterator. It must be possible for a hash to be in more than one iterator at at time, as long as the iterator state is stored in a lazy list. However, there is only one implicit iterator (the each iterator) that works in scalar context to return the next pair. [Or maybe not.]

The downside to making a hash autosort via the iterator is that you'd have to store all the keys in sorted order, and resort it when the hash changes. Alternately, the entire hash could be tied to an ISAM implementation (not included (XXX or should it be?)).

For multidimensional hashes, the key returned by any hash iterator is a list of keys, the size of which is the number of declared dimensions of the hash. [XXX but this seems to imply another lookup to find the value. Perhaps the associated value can also be bundled in somehow.]


Autovivification

Autovivification will only happen if the vivifiable path is used as a container, by binding, assigning, or taking a reference. On the other hand, value extraction does not autovivify.

This is as opposed to Perl 5, where autovivification could happen unintentionally, even when the code looks like a non-destructive test:


    my %hash;

    exists $hash{foo}{bar}; # creates $hash{foo} as an empty hash reference

In Perl 6 these read-only operations are indeed non-destructive:


    my %hash;

    exists $hash{foo}{bar}; # %hash is still empty

But these ones do autovivify:


    my %hash;

    my $val := $hash{foo}{bar};

    my @array;

    my $ref = \$array[0][0];

    my %hash;

    $hash{foo}{bar} = "foo"; # duh

This rule applies to dereferencing arrays, hashes, and scalar references.

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