# NAME

Set::Infinite - Sets of intervals

# 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]``````

## 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_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;``

## 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)``

## 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 `\$_` 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 { \$_->min + 54, \$_->max + 54 },
backtrack_callback =>
sub { \$_->min - 54, \$_->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(,,,) 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
}``````