This and other RFCs are available on the web at http://dev.perl.org/rfc/ =head1 TITLE Lazily evaluated list generation functions =head1 VERSION Maintainer: Jeremy Howard <[EMAIL PROTECTED]> Date: 10 August 2000 Last Modified: 8 September 2000 Mailing List: [EMAIL PROTECTED] Number: 81 Version: 3 Status: Developing =head1 ABSTRACT This RFC proposes that the existing C<..> operator produce a lazily evaluated list. In addition, a new operation C<:> is proposed that allows for the generation of lazily evaluated lists based on any Perl expression. This proposal only discusses these operators in a list context. The current meaning of '..' in a scalar context is not affected. =head1 CHANGES =head2 Since v2 =over 4 =item * Clarified the order of arguments passed to the list generation function =item * Made ':' an alias for '..' =back =head2 Since v1 =over 4 =item * Changed notation to generate lists using previous element value from I<(@start:&gen:$num_steps)> to I<(@start..&gen:$num_steps)> =item * Made I<(@start:&gen:$num_steps)> create a list that does not require intermediate values to be calculated =back =head1 DESCRIPTION This RFC proposes that Perl incorporate a broader tool box of list generation techniques: =over 8 =item * Lazy evaluation of generated lists =item * Generation of arbitary lists from a function =back These techniques would allow programs written in Perl to follow a structure familiar to programmers used to numerical programming environments. It would provide a more compact notation for many common mathematical algorithms, and give Perl important information to make key optimisations. =head2 Lazy evaluation of generated lists The C<..> of previous Perls is a I<list generation operator>, which creates a list based on its parameters: ($start..$stop); # ($start, $start+inc, $start+2*inc, ... $stop) where 'inc' is 1 if $start<$stop, or -1 otherwise. The list is generated as soon as it is declared. These makes some code rather inefficient: @a = (1..1000000); # One million element list generated here print $a[999999]; creates a one million element list despite only using one element of it. Under I<lazy evaluation>, elements of the list are only created when they are required, and saved for later use. In the previous example only $a[999999] would be calculated by interpolation (not sequentially) and stored when using lazy evaluation. Lists, whether generated lazily or not, are assumed to be I<stable>. That is, the value of $a[999999] will be the same everywhere in a program, unless @a itself is modified. This means that lazily evaluated lists provide a handy notation for memoization, as we will see later. It is proposed that once an element has been calculated in a list, that it is cached for use later rather than recalculated each time. Lazy list elements get calculated when they are output, or used in an expression that is output. If list elements are not output then they are never calculated. =head2 Introduction of C<:> to generate arbitary lists It is proposed that a new operator be added to Perl's list generation arsenal, C<:>. The ':' character is chosen because it reflects standard notation for array slicing, which is an important use of this operator. C<:> is only meaningful when called in a list context, generating a lazily evaluated list in one of 3 ways. =over 4 =item 1. I<($start..$end:$step)> Although earlier Perls could create ascending and descending lists incrementing by one, other increments required an unwieldy map: @threes = map {3*$_} (1..5); # (3,6,9,12,15) which was also less than intuitive to those used to the simple slicing notation of numerical programming languages such as Matlab and IDL. This proposed use of C<:> is identical to C<..> without C<:>, except that it increments by $step rather than 1. Specifically, returns a list ($start, $start+$step, $start+2*$step, ... $end). If $step does not go into ($end-$start) exactly, the last element of the list is the largest number in the series less than or equal to $end. $step is any non-zero number (not necessarily an integer). $step is optional--if it's missing then $step is the same as if the C<:> wasn't there at all. For example: (1..5:2); # (1,3,5) (1..5:); # Same as (1..5) (1..-5:-2); # (1,-1,-3,-5) (1..5:-2); # () Empty list This slicing notation is particularly useful for dealing with arrays, matrices, and tensors: @matrix = (1,3,4, 2,6,7); @column1of3 = (1..10000:3); # (1,4,7,...10000) print sum(@matrix[@column1of3]); # Prints 3 @matrix2 = readBig3ColumnMatrixFromSomewhere(); $column1Sum = sum(@matrix2[@column1of3]); # No need to redefine our slice! Note that more complex slicing, masking, and indirection across n-dimensional tensors would make the win of not having to respecify the indexes more substantial than in this simplified example. =item 2. I<($start:$end:$step)> An alias for ($start..$end:$step), to keep things familiar for folks moving over from over languages supporting similar notation. =item 3. I<(@start..&gen:$num_steps)> The most general form of this is to provide a syntax to create bounded or unbounded lists whose elements are generated by any arbitary function: ($start, &gen($start), &gen(&gen($start)), {$steps times}...); This is created using the notation: ($start..&gen:$steps); As before, $start is required (but need not be an integer). The first argument to &gen is the index of the element being calculated. For example: @first5PowersOf2 = (1..sub {$_[0]**2}:5); # (1,2,4,8,16) The second argument to &gen is the value of the previous element: @first5PowersOf2 = (1..sub {$_[1]*2}:5); # (1,2,4,8,16) Because lists are memoized automatically, when we later say: print $powersOf2[4]; The value of $powersOf2[4] is pulled from the memoization cache rather than recalculated. What's more: print $powersOf2[5]; is calculated from $powersOf2[4], rather than having to recalculate all the in-betweens again, because of that caching. This form of list generation can not generate the Fibonnaci sequence, because it is not defined by a single $start, and it has more than one parameter in its generator function. This requires a more generalised form: ((@start)..&gen:$num_steps) which allows us to say: @fib = ((1,1).. ^2 + ^1: 5); # (1,1,2,3,5) if we use the higher-order function notation proposed in RFC 23. As this example shows, the values of all previous elements are available to the list generation function as the second and later arguments to the function. As described earlier, the first argument is the index of the element being generated. =item 4. I<(@start:&gen:$num_steps)> The C<(@start..&gen:$num_steps)> notation makes it easy to generate lists that depend on the value of the previous element. For lists where the element is a direct function of the index, the arguments beyond the first can simply be ignored: @even_numbers = (1.. ^0 * 2: 5); # (2,4,6,8,10) However, because we want Perl to know how to generate one element of these lists without having to generate all previous elements, the following notation is proposed to achieve the same effect: @even_numbers = (1: ^0 * 2: 5); # (2,4,6,8,10) The values of previous elements are not passed to the list generation function with the ':' syntax. This means that Perl can calculate directly the value of any element without calculating the value of previous elements. This version of a lazily generated list is effectively a memoized function that restricts its domain to the natural numbers. However it can be used in some important additional roles--in particular, as a list index: @sub_list = @a[@even_numbers]; =back =head1 EXTENSIONS If infinite lists are available in Perl 6 (see RFC 24), the $num_steps and $end arguments to the list generation operators can be null. This indicates that there are infinite elements in the list: @column1of3 = (1.. :3); # (1,4,7,...) @all_even_numbers = (1: ^0 * 2:); # (2,4,6,8,...) =head1 JUSTIFICATION One particularly important use of generated lists is for slicing arrays. This is difficult in Perl 5, where it is tackled by Perl Data Language (PDL). Note that currently (perl5) we have to say $n1 = $n-1; # since we need to stringify $y = $x->slice("0:$n1:4"); This should be contrasted with the less cluttered syntax offered by numerical Python and commercial systems such as Matlab and IDL: y = x[0:n-1:4]; The syntax in this RFC would provide notation that is familiar to users of standard numerical programming tools, but is also a natural (and compatible) step from the existing Perl C<..> operator. =head1 IMPLEMENTATION It will be common in numerical programming to see multiple layers of indirection: @a = (1:5:100000); @b = getBigImage(); @c = @a(@b); print $c[5]; In these cases Perl should consolidate as much as possible at compile time to avoid too much overhead. When a lazy list is passed to a function it is not evaluated. The function can then access only the elements it needs, which are calculated as required. Furthermore, the arguments that generated the list are available as attributes of the list, and can therefore be used directly without actually accessing the list: @a = (1:5:100000); ($gen_type) = attributes::get(@a); # 'increment' ($start_val, $end_val, $increment) = attributes::get(@a)[1..3]; # (1, 5, 100000) @first5PowersOf2 = (1..sub {$_[1]*2}:5); # (1,2,4,8,16) ($gen_type) = attributes::get(@a); # 'recursive' ($start_val, &gen_func, $num_steps) = attributes::get(@a)[1..3]; @even_numbers = (1: ^0 * 2: 5); # (2,4,6,8,10) ($gen_type) = attributes::get(@a); # 'function' ($start_val, &gen_func, $num_steps) = attributes::get(@a)[1..3]; =back =head1 REFERENCES RFC 23: Higher-order functions RFC 24: Semi-finite (lazy) lists RFC 82: Apply operators component-wise in a list context RFC 205: New operator ';' for creating array slices Memoization in Perl: http://www.plover.com/~mjd/perl/MiniMemoize/ List comprehension in Haskell: http://www.numeric-quest.com/haskell/hcompanion/principles.html#List comprehension Fethi Rabhi and Guy Lapalme: I<Algorithms: A functional programming approach>, Addison-Wesley, 235 pages, paperback, 1999. ISBN 0-201-59604-0 =head1 ACKNOWLEDGEMENTS Damian Conway: Reviewed first draft Karl Glazebrook: Suggested splitting from infinite lists proposal Christian Soeller: Original 'slice' RFC; suggested argument reordering Dan Sugalski: Implementation ideas
