SYNOPSIS
use Set::Infinite;
$set = Set::Infinite->new(1,2); # [1..2]
print $set->union(5,6); # [1..2],[5..6]
DESCRIPTION
Set::Infinite is a Set Theory module for infinite sets.A set is a collection of objects. The objects that belong to a set are called its members, or ``elements''.
As objects we allow (almost) anything: reals, integers, and objects (such as dates).
We allow sets to be infinite.
There is no account for the order of elements. For example, {1,2} = {2,1}.
There is no account for repetition of elements. For example, {1,2,2} = {1,1,1,2} = {1,2}.
CONSTRUCTOR
new
Creates a new set object:
$set = Set::Infinite->new; # empty set
$set = Set::Infinite->new( 10 ); # single element
$set = Set::Infinite->new( 10, 20 ); # single range
$set = Set::Infinite->new(
[ 10, 20 ], [ 50, 70 ] ); # two ranges
- empty set
-
$set = Set::Infinite->new; - set with a single element
-
$set = Set::Infinite->new( 10 ); $set = Set::Infinite->new( [ 10 ] ); - set with a single span
-
$set = Set::Infinite->new( 10, 20 ); $set = Set::Infinite->new( [ 10, 20 ] ); # 10 <= x <= 20 - set with a single, open span
-
$set = Set::Infinite->new( { a => 10, open_begin => 0, b => 20, open_end => 1, } ); # 10 <= x < 20 - set with multiple spans
-
$set = Set::Infinite->new( 10, 20, 100, 200 ); $set = Set::Infinite->new( [ 10, 20 ], [ 100, 200 ] ); $set = Set::Infinite->new( { a => 10, open_begin => 0, b => 20, open_end => 0, }, { a => 100, open_begin => 0, b => 200, open_end => 0, } );
The "new()" method expects ordered parameters.
If you have unordered ranges, you can build the set using "union":
@ranges = ( [ 10, 20 ], [ -10, 1 ] );
$set = Set::Infinite->new;
$set = $set->union( @$_ ) for @ranges;
The data structures passed to "new" must be immutable. So this is not good practice:
$set = Set::Infinite->new( $object_a, $object_b );
$object_a->set_value( 10 );
This is the recommended way to do it:
$set = Set::Infinite->new( $object_a->clone, $object_b->clone );
$object_a->set_value( 10 );
clone / copy
Creates a new object, and copy the object data.empty_set
Creates an empty set.If called from an existing set, the empty set inherits the ``type'' and ``density'' characteristics.
universal_set
Creates a set containing ``all'' possible elements.If called from an existing set, the universal set inherits the ``type'' and ``density'' characteristics.
SET FUNCTIONS
union
$set = $set->union($b);
Returns the set of all elements from both sets.
This function behaves like an ``OR'' operation.
$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
$set2 = new Set::Infinite( [ 7, 20 ] );
print $set1->union( $set2 );
# output: [1..4],[7..20]
intersection
$set = $set->intersection($b);
Returns the set of elements common to both sets.
This function behaves like an ``AND'' operation.
$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
$set2 = new Set::Infinite( [ 7, 20 ] );
print $set1->intersection( $set2 );
# output: [8..12]
complement
minus
difference
$set = $set->complement;
Returns the set of all elements that don't belong to the set.
$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
print $set1->complement;
# output: (-inf..1),(4..8),(12..inf)
The complement function might take a parameter:
$set = $set->minus($b);
Returns the set-difference, that is, the elements that don't belong to the given set.
$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
$set2 = new Set::Infinite( [ 7, 20 ] );
print $set1->minus( $set2 );
# output: [1..4]
symmetric_difference
Returns a set containing elements that are in either set, but not in both. This is the ``set'' version of ``XOR''.DENSITY METHODS
real
$set1 = $set->real;
Returns a set with density ``0''.
integer
$set1 = $set->integer;
Returns a set with density ``1''.
LOGIC FUNCTIONS
intersects
$logic = $set->intersects($b);
contains
$logic = $set->contains($b);
is_empty
is_null
$logic = $set->is_null;
is_nonempty
This set that has at least 1 element.is_span
This set that has a single span or interval.is_singleton
This set that has a single element.is_subset( $set )
Every element of this set is a member of the given set.is_proper_subset( $set )
Every element of this set is a member of the given set. Some members of the given set are not elements of this set.is_disjoint( $set )
The given set has no elements in common with this set.is_too_complex
Sometimes a set might be too complex to enumerate or print.This happens with sets that represent infinite recurrences, such as when you ask for a quantization on a set bounded by -inf or inf.
See also: "count" method.
SCALAR FUNCTIONS
min
$i = $set->min;
max
$i = $set->max;
size
$i = $set->size;
count
$i = $set->count;
OVERLOADED OPERATORS
stringification
print $set;
$str = "$set";
See also: "as_string".
comparison
sort
> < == >= <= <=>
See also: "spaceship" method.
CLASS METHODS
Set::Infinite->separators(@i)
chooses the interval separators for stringification.
default are [ ] ( ) '..' ','.
inf
returns an 'Infinity' number.
minus_inf
returns '-Infinity' number.
type
type( "My::Class::Name" )
Chooses a default object data type.
Default is none (a normal Perl SCALAR).
SPECIAL SET FUNCTIONS
span
$set1 = $set->span;
Returns the set span.
until
Extends a set until another:
0,5,7 -> until 2,6,10
gives
[0..2), [5..6), [7..10)
start_set
end_set
These methods do the inverse of the ``until'' method.Given:
[0..2), [5..6), [7..10)
start_set is:
0,5,7
end_set is:
2,6,10
intersected_spans
$set = $set1->intersected_spans( $set2 );
The method returns a new set, containing all spans that are intersected by the given set.
Unlike the "intersection" method, the spans are not modified. See diagram below:
set1 [....] [....] [....] [....]
set2 [................]
intersection [.] [....] [.]
intersected_spans [....] [....] [....]
quantize
quantize( parameters )
Makes equal-sized subsets.
Returns an ordered set of equal-sized subsets.
Example:
$set = Set::Infinite->new([1,3]);
print join (" ", $set->quantize( quant => 1 ) );
Gives:
[1..2) [2..3) [3..4)
select
select( parameters )
Selects set spans based on their ordered positions
"select" has a behaviour similar to an array "slice".
by - default=All
count - default=Infinity
0 1 2 3 4 5 6 7 8 # original set
0 1 2 # count => 3
1 6 # by => [ -2, 1 ]
offset
offset ( parameters )
Offsets the subsets. Parameters:
value - default=[0,0]
mode - default='offset'. Possible values are: 'offset', 'begin', 'end'.
unit - type of value. Can be 'days', 'weeks', 'hours', 'minutes', 'seconds'.
iterate
iterate ( sub { } , @args )
Iterates on the set spans, over a callback subroutine. Returns the union of all partial results.
The callback argument $_[0] is a span. If there are additional arguments they are passed to the callback.
The callback can return a span, a hashref (see "Set::Infinite::Basic"), a scalar, an object, or "undef".
[EXPERIMENTAL] "iterate" accepts an optional "backtrack_callback" argument. The purpose of the "backtrack_callback" is to reverse the iterate() function, overcoming the limitations of the internal backtracking algorithm. The syntax is:
iterate ( sub { } , backtrack_callback => sub { }, @args )
The "backtrack_callback" can return a span, a hashref, a scalar, an object, or "undef".
For example, the following snippet adds a constant to each element of an unbounded set:
$set1 = $set->iterate(
sub { $_[0]->min + 54, $_[0]->max + 54 },
backtrack_callback =>
sub { $_[0]->min - 54, $_[0]->max - 54 },
);
first / last
first / last
In scalar context returns the first or last interval of a set.
In list context returns the first or last interval of a set, and the remaining set (the 'tail').
See also: "min", "max", "min_a", "max_a" methods.
type
type( "My::Class::Name" )
Chooses a default object data type.
default is none (a normal perl SCALAR).
INTERNAL FUNCTIONS
_backtrack
$set->_backtrack( 'intersection', $b );
Internal function to evaluate recurrences.
numeric
$set->numeric;
Internal function to ignore the set ``type''. It is used in some internal optimizations, when it is possible to use scalar values instead of objects.
fixtype
$set->fixtype;
Internal function to fix the result of operations that use the numeric() function.
tolerance
$set = $set->tolerance(0) # defaults to real sets (default)
$set = $set->tolerance(1) # defaults to integer sets
Internal function for changing the set ``density''.
min_a
($min, $min_is_open) = $set->min_a;
max_a
($max, $max_is_open) = $set->max_a;
as_string
Implements the ``stringification'' operator.Stringification of unbounded recurrences is not implemented.
Unbounded recurrences are stringified as ``function descriptions'', if the class variable $PRETTY_PRINT is set.
spaceship
Implements the ``comparison'' operator.Comparison of unbounded recurrences is not implemented.
CAVEATS
-
constructor ``span'' notation
$set = Set::Infinite->new(10,1);Will be interpreted as [1..10]
-
constructor ``multiple-span'' notation
$set = Set::Infinite->new(1,2,3,4);Will be interpreted as [1..2],[3..4] instead of [1,2,3,4]. You probably want ->new([1],[2],[3],[4]) instead, or maybe ->new(1,4)
-
``range operator''
$set = Set::Infinite->new(1..3);Will be interpreted as [1..2],3 instead of [1,2,3]. You probably want ->new(1,3) instead.
INTERNALS
The base set object, without recurrences, is a "Set::Infinite::Basic".A recurrence-set is represented by a method name, one or two parent objects, and extra arguments. The "list" key is set to an empty array, and the "too_complex" key is set to 1.
This is a structure that holds the union of two ``complex sets'':
{
too_complex => 1, # "this is a recurrence"
list => [ ], # not used
method => 'union', # function name
parent => [ $set1, $set2 ], # "leaves" in the syntax-tree
param => [ ] # optional arguments for the function
}
This is a structure that holds the complement of a ``complex set'':
{
too_complex => 1, # "this is a recurrence"
list => [ ], # not used
method => 'complement', # function name
parent => $set, # "leaf" in the syntax-tree
param => [ ] # optional arguments for the function
}
AUTHOR
Flavio S. Glock <[email protected]>COPYRIGHT
Copyright (c) 2003 Flavio Soibelmann Glock. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.The full text of the license can be found in the LICENSE file included with this module.