Families

A Family is an associative container which models a family \((f_i)_{i \in I}\). Then, f[i] returns the element of the family indexed by i. Whenever available, set and combinatorial class operations (counting, iteration, listing) on the family are induced from those of the index set. Families should be created through the Family() function.

AUTHORS:

• Nicolas Thiery (2008-02): initial release
• Florent Hivert (2008-04): various fixes, cleanups and improvements.

TESTS:

Check for workaround trac ticket #12482 (shall be run in a fresh session):

sage: P = Partitions(3)
sage: Family(P, lambda x: x).category() # used to return ``enumerated sets``
Category of finite enumerated sets
sage: Family(P, lambda x: x).category()
Category of finite enumerated sets

class sage.sets.family.AbstractFamily

Bases: sage.structure.parent.Parent

The abstract class for family

Any family belongs to a class which inherits from AbstractFamily.

hidden_keys()

Returns the hidden keys of the family, if any.

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: f.hidden_keys()
[]

inverse_family()

Returns the inverse family, with keys and values exchanged. This presumes that there are no duplicate values in self.

This default implementation is not lazy and therefore will only work with not too big finite families. It is also cached for the same reason:

sage: Family({3: 'a', 4: 'b', 7: 'd'}).inverse_family()
Finite family {'a': 3, 'b': 4, 'd': 7}

sage: Family((3,4,7)).inverse_family()
Finite family {3: 0, 4: 1, 7: 2}

map(f, name=None)

Returns the family \(( f(\mathtt{self}[i]) )_{i \in I}\), where \(I\) is the index set of self.

Todo

good name?

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: g = f.map(lambda x: x+'1')
sage: list(g)
['a1', 'b1', 'd1']

zip(f, other, name=None)

Given two families with same index set \(I\) (and same hidden keys if relevant), returns the family \(( f(self[i], other[i]) )_{i \in I}\)

Todo

generalize to any number of families and merge with map?

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: g = Family({3: '1', 4: '2', 7: '3'})
sage: h = f.zip(lambda x,y: x+y, g)
sage: list(h)
['a1', 'b2', 'd3']

class sage.sets.family.EnumeratedFamily(enumset)

Bases: sage.sets.family.LazyFamily

EnumeratedFamily turns an enumerated set c into a family indexed by the set \(\{0,\dots, |c|-1\}\).

Instances should be created via the Family() factory. See its documentation for examples and tests.

cardinality()

Return the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import EnumeratedFamily
sage: f = EnumeratedFamily(Permutations(3))
sage: f.cardinality()
6

sage: from sage.categories.examples.infinite_enumerated_sets import NonNegativeIntegers
sage: f = Family(NonNegativeIntegers())
sage: f.cardinality()
+Infinity

keys()

Returns self’s keys.

EXAMPLES:

sage: from sage.sets.family import EnumeratedFamily
sage: f = EnumeratedFamily(Permutations(3))
sage: f.keys()
Standard permutations of 3

sage: from sage.categories.examples.infinite_enumerated_sets import NonNegativeIntegers
sage: f = Family(NonNegativeIntegers())
sage: f.keys()
An example of an infinite enumerated set: the non negative integers

sage.sets.family.Family(indices, function=None, hidden_keys=, []hidden_function=None, lazy=False, name=None)

A Family is an associative container which models a family \((f_i)_{i \in I}\). Then, f[i] returns the element of the family indexed by \(i\). Whenever available, set and combinatorial class operations (counting, iteration, listing) on the family are induced from those of the index set.

There are several available implementations (classes) for different usages; Family serves as a factory, and will create instances of the appropriate classes depending on its arguments.

INPUT:

• indices – the indices for the family
• function – (optional) the function \(f\) applied to all visible indices; the default is the identity function
• hidden_keys – (optional) a list of hidden indices that can be accessed through my_family[i]
• hidden_function – (optional) a function for the hidden indices
• lazy – boolean (default: False); whether the family is lazily created or not; if the indices are infinite, then this is automatically made True
• name – (optional) the name of the function; only used when the family is lazily created via a function

EXAMPLES:

In its simplest form, a list \(l = [l_0, l_1, \ldots, l_{\ell}]\) or a tuple by itself is considered as the family \((l_i)_{i \in I}\) where \(I\) is the set \(\{0, \ldots, \ell\}\) where \(\ell\) is len(l) - 1. So Family(l) returns the corresponding family:

sage: f = Family([1,2,3])
sage: f
Family (1, 2, 3)
sage: f = Family((1,2,3))
sage: f
Family (1, 2, 3)

Instead of a list you can as well pass any iterable object:

sage: f = Family(2*i+1 for i in [1,2,3]);
sage: f
Family (3, 5, 7)

A family can also be constructed from a dictionary t. The resulting family is very close to t, except that the elements of the family are the values of t. Here, we define the family \((f_i)_{i \in \{3,4,7\}}\) with \(f_3 = a\), \(f_4 = b\), and \(f_7 = d\):

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: f
Finite family {3: 'a', 4: 'b', 7: 'd'}
sage: f[7]
'd'
sage: len(f)
3
sage: list(f)
['a', 'b', 'd']
sage: [ x for x in f ]
['a', 'b', 'd']
sage: f.keys()
[3, 4, 7]
sage: 'b' in f
True
sage: 'e' in f
False

A family can also be constructed by its index set \(I\) and a function \(f\), as in \((f(i))_{i \in I}\):

sage: f = Family([3,4,7], lambda i: 2*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f[7]
14
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

By default, all images are computed right away, and stored in an internal dictionary:

sage: f = Family((3,4,7), lambda i: 2*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}

Note that this requires all the elements of the list to be hashable. One can ask instead for the images \(f(i)\) to be computed lazily, when needed:

sage: f = Family([3,4,7], lambda i: 2*i, lazy=True)
sage: f
Lazy family (<lambda>(i))_{i in [3, 4, 7]}
sage: f[7]
14
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]

This allows in particular for modeling infinite families:

sage: f = Family(ZZ, lambda i: 2*i, lazy=True)
sage: f
Lazy family (<lambda>(i))_{i in Integer Ring}
sage: f.keys()
Integer Ring
sage: f[1]
2
sage: f[-5]
-10
sage: i = iter(f)
sage: next(i), next(i), next(i), next(i), next(i)
(0, 2, -2, 4, -4)

Note that the lazy keyword parameter is only needed to force laziness. Usually it is automatically set to a correct default value (ie: False for finite data structures and True for enumerated sets:

sage: f == Family(ZZ, lambda i: 2*i)
True

Beware that for those kind of families len(f) is not supposed to work. As a replacement, use the .cardinality() method:

sage: f = Family(Permutations(3), attrcall("to_lehmer_code"))
sage: list(f)
[[0, 0, 0], [0, 1, 0], [1, 0, 0], [1, 1, 0], [2, 0, 0], [2, 1, 0]]
sage: f.cardinality()
6

Caveat: Only certain families with lazy behavior can be pickled. In particular, only functions that work with Sage’s pickle_function and unpickle_function (in sage.misc.fpickle) will correctly unpickle. The following two work:

sage: f = Family(Permutations(3), lambda p: p.to_lehmer_code()); f
Lazy family (<lambda>(i))_{i in Standard permutations of 3}
sage: f == loads(dumps(f))
True

sage: f = Family(Permutations(3), attrcall("to_lehmer_code")); f
Lazy family (i.to_lehmer_code())_{i in Standard permutations of 3}
sage: f == loads(dumps(f))
True

But this one does not:

sage: def plus_n(n): return lambda x: x+n
sage: f = Family([1,2,3], plus_n(3), lazy=True); f
Lazy family (<lambda>(i))_{i in [1, 2, 3]}
sage: f == loads(dumps(f))
Traceback (most recent call last):
...
ValueError: Cannot pickle code objects from closures

Finally, it can occasionally be useful to add some hidden elements in a family, which are accessible as f[i], but do not appear in the keys or the container operations:

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2])
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f.hidden_keys()
[2]
sage: f[7]
14
sage: f[2]
4
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

The following example illustrates when the function is actually called:

sage: def compute_value(i):
...       print('computing 2*'+str(i))
...       return 2*i
sage: f = Family([3,4,7], compute_value, hidden_keys=[2])
computing 2*3
computing 2*4
computing 2*7
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f.hidden_keys()
[2]
sage: f[7]
14
sage: f[2]
computing 2*2
4
sage: f[2]
4
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

Here is a close variant where the function for the hidden keys is different from that for the other keys:

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2], hidden_function = lambda i: 3*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f.hidden_keys()
[2]
sage: f[7]
14
sage: f[2]
6
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

Family accept finite and infinite EnumeratedSets as input:

sage: f = Family(FiniteEnumeratedSet([1,2,3]))
sage: f
Family (1, 2, 3)
sage: from sage.categories.examples.infinite_enumerated_sets import NonNegativeIntegers
sage: f = Family(NonNegativeIntegers())
sage: f
Family (An example of an infinite enumerated set: the non negative integers)

sage: f = Family(FiniteEnumeratedSet([3,4,7]), lambda i: 2*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
{3, 4, 7}
sage: f[7]
14
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

TESTS:

sage: f = Family({1:'a', 2:'b', 3:'c'})
sage: f
Finite family {1: 'a', 2: 'b', 3: 'c'}
sage: f[2]
'b'
sage: loads(dumps(f)) == f
True

sage: f = Family({1:'a', 2:'b', 3:'c'}, lazy=True)
Traceback (most recent call last):
ValueError: lazy keyword only makes sense together with function keyword !

sage: f = Family(range(1,27), lambda i: chr(i+96))
sage: f
    Finite family {1: 'a', 2: 'b', 3: 'c', 4: 'd', 5: 'e', 6: 'f', 7: 'g', 8: 'h', 9: 'i', 10: 'j', 11: 'k', 12: 'l', 13: 'm', 14: 'n', 15: 'o', 16: 'p', 17: 'q', 18: 'r', 19: 's', 20: 't', 21: 'u', 22: 'v', 23: 'w', 24: 'x', 25: 'y', 26: 'z'}
sage: f[2]
'b'

The factory Family is supposed to be idempotent. We test this feature here:

sage: from sage.sets.family import FiniteFamily, LazyFamily, TrivialFamily
sage: f = FiniteFamily({3: 'a', 4: 'b', 7: 'd'})
sage: g = Family(f)
sage: f == g
True

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2])
sage: g = Family(f)
sage: f == g
True

sage: f = LazyFamily([3,4,7], lambda i: 2*i)
sage: g = Family(f)
sage: f == g
True

sage: f = TrivialFamily([3,4,7])
sage: g = Family(f)
sage: f == g
True

The family should keep the order of the keys:

sage: f = Family(["c", "a", "b"], lambda x: x+x)
sage: list(f)
['cc', 'aa', 'bb']

TESTS:

Only the hidden function is applied to the hidden keys:

sage: f = lambda x : 2*x
sage: h_f = lambda x:x%2
sage: F = Family([1,2,3,4],function = f, hidden_keys=[5],hidden_function=h_f)
sage: F[5]
1

class sage.sets.family.FiniteFamily(dictionary, keys=None)

Bases: sage.sets.family.AbstractFamily

A FiniteFamily is an associative container which models a finite family \((f_i)_{i \in I}\). Its elements \(f_i\) are therefore its values. Instances should be created via the Family() factory. See its documentation for examples and tests.

EXAMPLES:

We define the family \((f_i)_{i \in \{3,4,7\}}\) with \(f_3=a\), \(f_4=b\), and \(f_7=d\):

sage: from sage.sets.family import FiniteFamily
sage: f = FiniteFamily({3: 'a', 4: 'b', 7: 'd'})

Individual elements are accessible as in a usual dictionary:

sage: f[7]
'd'

And the other usual dictionary operations are also available:

sage: len(f)
3
sage: f.keys()
[3, 4, 7]

However f behaves as a container for the \(f_i\)‘s:

sage: list(f)
['a', 'b', 'd']
sage: [ x for x in f ]
['a', 'b', 'd']

The order of the elements can be specified using the keys optional argument:

sage: f = FiniteFamily({"a": "aa", "b": "bb", "c" : "cc" }, keys = ["c", "a", "b"])
sage: list(f)
['cc', 'aa', 'bb']

cardinality()

Returns the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import FiniteFamily
sage: f = FiniteFamily({3: 'a', 4: 'b', 7: 'd'})
sage: f.cardinality()
3

has_key(k)

Returns whether k is a key of self

EXAMPLES:

sage: Family({"a":1, "b":2, "c":3}).has_key("a")
True
sage: Family({"a":1, "b":2, "c":3}).has_key("d")
False

keys()

Returns the index set of this family

EXAMPLES:

sage: f = Family(["c", "a", "b"], lambda x: x+x)
sage: f.keys()
['c', 'a', 'b']

values()

Returns the elements of this family

EXAMPLES:

sage: f = Family(["c", "a", "b"], lambda x: x+x)
sage: f.values()
['cc', 'aa', 'bb']

class sage.sets.family.FiniteFamilyWithHiddenKeys(dictionary, hidden_keys, hidden_function)

Bases: sage.sets.family.FiniteFamily

A close variant of FiniteFamily where the family contains some hidden keys whose corresponding values are computed lazily (and remembered). Instances should be created via the Family() factory. See its documentation for examples and tests.

Caveat: Only instances of this class whose functions are compatible with sage.misc.fpickle can be pickled.

hidden_keys()

Returns self’s hidden keys.

EXAMPLES:

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2])
sage: f.hidden_keys()
[2]

class sage.sets.family.LazyFamily(set, function, name=None)

Bases: sage.sets.family.AbstractFamily

A LazyFamily(I, f) is an associative container which models the (possibly infinite) family \((f(i))_{i \in I}\).

Instances should be created via the Family() factory. See its documentation for examples and tests.

cardinality()

Return the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import LazyFamily
sage: f = LazyFamily([3,4,7], lambda i: 2*i)
sage: f.cardinality()
3
sage: from sage.categories.examples.infinite_enumerated_sets import NonNegativeIntegers
sage: l = LazyFamily(NonNegativeIntegers(), lambda i: 2*i)
sage: l.cardinality()
+Infinity

TESTS:

Check that trac ticket #15195 is fixed:

sage: C = CartesianProduct(PositiveIntegers(), [1,2,3])
sage: C.cardinality()
+Infinity
sage: F = Family(C, lambda x: x)
sage: F.cardinality()
+Infinity

keys()

Returns self’s keys.

EXAMPLES:

sage: from sage.sets.family import LazyFamily
sage: f = LazyFamily([3,4,7], lambda i: 2*i)
sage: f.keys()
[3, 4, 7]

class sage.sets.family.TrivialFamily(enumeration)

Bases: sage.sets.family.AbstractFamily

TrivialFamily turns a list/tuple \(c\) into a family indexed by the set \(\{0, \dots, |c|-1\}\).

Instances should be created via the Family() factory. See its documentation for examples and tests.

cardinality()

Return the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import TrivialFamily
sage: f = TrivialFamily([3,4,7])
sage: f.cardinality()
3

keys()

Returns self’s keys.

EXAMPLES:

sage: from sage.sets.family import TrivialFamily
sage: f = TrivialFamily([3,4,7])
sage: f.keys()
[0, 1, 2]