# The C3 algorithm, under control of a total order.¶

The C3 algorithm, under control of a total order.

## Abstract¶

Python handles multiple inheritance by computing, for each class, a linear extension of the poset of all its super classes (the Method Resolution Order, MRO). The MRO is calculated recursively from local information (the ordered list of the direct super classes), with the so-called C3 algorithm. This algorithm can fail if the local information is not consistent; worst, there exist hierarchies of classes with provably no consistent local information.

For large hierarchy of classes, like those derived from categories in Sage, maintaining consistent local information by hand does not scale and leads to unpredictable C3 failures (the dreaded “could not find a consistent method resolution order”); a maintenance nightmare.

This module implements a final solution to this problem. Namely, it allows for building automatically the local information from the bare class hierarchy in such a way that guarantees that the C3 algorithm will never fail.

Err, but you said that this was provably impossible? Well, not if one relaxes a bit the hypotheses; but that’s not something one would want to do by hand :-)

## The problem¶

Consider the following hierarchy of classes:

sage: class A1(object): pass
sage: class A2(object):
....:     def foo(self): return 2
sage: class A3(object): pass
sage: class A4(object):
....:     def foo(self): return 4
sage: class A5(A2, A1):
....:     def foo(self): return 5
sage: class A6(A4, A3): pass
sage: class A7(A6, A5): pass


If a is an instance of A7, then Python needs to choose which implementation to use upon calling a.foo(): that of A4 or A5, but obviously not that of A2. In Python, like in many other dynamic object oriented languages, this is achieved by calculating once for all a specific linear extension of the hierarchy of the super classes of each class, called its Method Resolution Order (MRO):

sage: [cls.__name__ for cls in A7.mro()]
['A7', 'A6', 'A4', 'A3', 'A5', 'A2', 'A1', 'object']


Thus, in our example, the implementation in A4 is chosen:

sage: a = A7()
sage: a.foo()
4


Specifically, the MRO is calculated using the so-called C3 algorithm which guarantees that the MRO respects not only inheritance, but also the order in which the bases (direct super classes) are given for each class.

However, for large hierarchies of classes with lots of multiple inheritance, like those derived from categories in Sage, this algorithm easily fails if the order of the bases is not chosen consistently (here for A2 w.r.t. A1):

sage: class B6(A1,A2): pass
sage: class B7(B6,A5): pass
Traceback (most recent call last):
...
TypeError: Error when calling the metaclass bases
Cannot create a consistent method resolution
order (MRO) for bases ...


There actually exist hierarchies of classes for which C3 fails whatever order of the bases is chosen; the smallest such example, admittedly artificial, has ten classes (see below). Still, this highlights that this problem has to be tackled in a systematic way.

Fortunately, one can trick C3, without changing the inheritance semantic, by adding some super classes of A to the bases of A. In the following example, we completely force a given MRO by specifying all the super classes of A as bases:

sage: class A7(A6, A5, A4, A3, A2, A1): pass
sage: [cls.__name__ for cls in A7.mro()]
['A7', 'A6', 'A5', 'A4', 'A3', 'A2', 'A1', 'object']


Luckily this can be optimized; here it is sufficient to add a single base to enforce the same MRO:

sage: class A7(A6, A5, A4): pass
sage: [cls.__name__ for cls in A7.mro()]
['A7', 'A6', 'A5', 'A4', 'A3', 'A2', 'A1', 'object']


## A strategy to solve the problem¶

We should recall at this point a design decision that we took for the hierarchy of classes derived from categories: the semantic shall only depend on the inheritance order, not on the specific MRO, and in particular not on the order of the bases (see the section On the order of super categories in the category primer). If a choice needs to be made (for example for efficiency reasons), then this should be done explicitly, on a method-by-method basis. In practice this design goal is not yet met.

Note

When managing large hierarchies of classes in other contexts this may be too strong a design decision.

The strategy we use for hierarchies of classes derived from categories is then:

1. To choose a global total order on the whole hierarchy of classes.
2. To control C3 to get it to return MROs that follow this total order.

A basic approach for point 1., that will work for any hierarchy of classes, is to enumerate the classes while they are constructed (making sure that the bases of each class are enumerated before that class), and to order the classes according to that enumeration. A more conceptual ordering may be desirable, in particular to get deterministic and reproducible results. In the context of Sage, this is mostly relevant for those doctests displaying all the categories or classes that an object inherits from.

## Getting fine control on C3¶

This module is about point 2.

The natural approach would be to change the algorithm used by Python to compute the MRO. However, changing Python’s default algorithm just for our needs is obviously not an option, and there is currently no hook to customize specific classes to use a different algorithm. Pushing the addition of such a hook into stock Python would take too much time and effort.

Another approach would be to use the “adding bases” trick straightforwardly, putting the list of all the super classes of a class as its bases. However, this would have several drawbacks:

• It is not so elegant, in particular because it duplicates information: we already know through A5 that A7 is a subclass of A1. This duplication could be acceptable in our context because the hierarchy of classes is generated automatically from a conceptual hierarchy (the categories) which serves as single point of truth for calculating the bases of each class.

• It increases the complexity of the calculation of the MRO with C3. For example, for a linear hierachy of classes, the complexity goes from $$O(n^2)$$ to $$O(n^3)$$ which is not acceptable.

• It increases the complexity of inspecting the classes. For example, the current implementation of the dir command in Python has no cache, and its complexity is linear in the number of maximal paths in the class hierarchy graph as defined by the bases. For a linear hierarchy, this is of complexity $$O(p_n)$$ where $$p_n$$ is the number of integer partitions of $$n$$, which is exponential. And indeed, running dir for a typical class like GradedHopfAlgebrasWithBasis(QQ).parent_class with 37 super classes took $$18$$ seconds with this approach.

Granted: this mostly affects the dir command and could be blamed on its current implementation. With appropriate caching, it could be reimplemented to have a complexity roughly linear in the number of classes in the hierarchy. But this won’t happen any time soon in a stock Python.

This module refines this approach to make it acceptable, if not seamless. Given a hierarchy and a total order on this hierarchy, it calculates for each element of the hierarchy the smallest list of additional bases that forces C3 to return the desired MRO. This is achieved by implementing an instrumented variant of the C3 algorithm (which we call instrumented C3) that detects when C3 is about to take a wrong decision and adds one base to force the right decision. Then, running the standard C3 algorithm with the updated list of bases (which we call controlled C3) yields the desired MRO.

EXAMPLES:

As an experimentation and testing tool, we use a class HierarchyElement whose instances can be constructed from a hierarchy described by a poset, a digraph, or more generally a successor relation. By default, the desired MRO is sorted decreasingly. Another total order can be specified using a sorting key.

We consider the smallest poset describing a class hierarchy admitting no MRO whatsoever:

sage: P = Poset({10: [9,8,7], 9:[6,1], 8:[5,2], 7:[4,3], 6: [3,2], 5:[3,1], 4: [2,1] }, linear_extension=True, facade=True)


And build a $$HierarchyElement$$ from it:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: x = HierarchyElement(10, P)


Here are its bases:

sage: HierarchyElement(10, P)._bases
[9, 8, 7]


Using the standard C3 algorithm fails:

sage: x.mro_standard
Traceback (most recent call last):
...
ValueError: Can not merge the items 3, 3, 2.


We also get a failure when we relabel $$P$$ according to another linear extension. For easy relabelling, we first need to set an appropriate default linear extension for $$P$$:

sage: linear_extension = list(reversed(IntegerRange(1,11)))
sage: P = P.with_linear_extension(linear_extension)
sage: list(P)
[10, 9, 8, 7, 6, 5, 4, 3, 2, 1]


Now, we play with the fifth linear extension of $$P$$:

sage: L = P.linear_extensions()
sage: Q = L[5].to_poset()
sage: Q.cover_relations()
[[10, 9], [10, 8], [10, 7], [9, 6], [9, 3], [8, 5], [8, 2], [7, 4], [7, 1], [6, 2], [6, 1], [5, 3], [5, 1], [4, 3], [4, 2]]
sage: x = HierarchyElement(10, Q)
sage: x.mro_standard
Traceback (most recent call last):
...
ValueError: Can not merge the items 2, 3, 3.


On the other hand, both the instrumented C3 algorithm, and the controlled C3 algorithm give the desired MRO:

sage: x.mro
[10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
sage: x.mro_controlled
[10, 9, 8, 7, 6, 5, 4, 3, 2, 1]


The above checks, and more, can be run with:

sage: x._test_mro()


In practice, the control was achieved by adding the following bases:

sage: x._bases
[9, 8, 7]
sage: x._bases_controlled
[9, 8, 7, 6, 5]


Altogether, four bases were added for control:

sage: sum(len(HierarchyElement(q, Q)._bases) for q in Q)
15
sage: sum(len(HierarchyElement(q, Q)._bases_controlled) for q in Q)
19


This information can also be recovered with:

sage: x.all_bases_len()
15
sage: x.all_bases_controlled_len()
19


We now check that the C3 algorithm fails for all linear extensions $$l$$ of this poset, whereas both the instrumented and controlled C3 algorithms succeed; along the way, we collect some statistics:

sage: stats = []
sage: for l in L:
....:     x = HierarchyElement(10, l.to_poset())
....:     try: # Check that x.mro_standard always fails with a ValueError
....:         x.mro_standard
....:     except ValueError:
....:         pass
....:     else:
....:         assert False
....:     assert x.mro            == list(P)
....:     assert x.mro_controlled == list(P)
....:     assert x.all_bases_len() == 15
....:     stats.append(x.all_bases_controlled_len()-x.all_bases_len())


Depending on the linear extension $$l$$ it was necessary to add between one and five bases for control; for example, $$216$$ linear extensions required the addition of four bases:

sage: Word(stats).evaluation_sparse()
[(1, 36), (2, 108), (3, 180), (4, 216), (5, 180)]


We now consider a hierarchy of categories:

sage: from operator import attrgetter
sage: x = HierarchyElement(Groups(), attrcall("super_categories"), attrgetter("_cmp_key"))
sage: x.mro
[Category of groups, Category of monoids, Category of semigroups,
Category of inverse unital magmas, Category of unital magmas, Category of magmas,
Category of sets, Category of sets with partial maps, Category of objects]
sage: x.mro_standard
[Category of groups, Category of monoids, Category of semigroups,
Category of inverse unital magmas, Category of unital magmas, Category of magmas,
Category of sets, Category of sets with partial maps, Category of objects]


For a typical category, few bases, if any, need to be added to force C3 to give the desired order:

sage: C = FiniteFields()
sage: x = HierarchyElement(C, attrcall("super_categories"), attrgetter("_cmp_key"))
sage: x.mro == x.mro_standard
False
sage: x.all_bases_len()
66
sage: x.all_bases_controlled_len()
69

sage: C = GradedHopfAlgebrasWithBasis(QQ)
sage: x = HierarchyElement(C, attrcall("super_categories"), attrgetter("_cmp_key"))
sage: x._test_mro()
sage: x.mro == x.mro_standard
False
sage: x.all_bases_len()
82
sage: x.all_bases_controlled_len()
89


The following can be used to search through the Sage named categories for any that requires the addition of some bases:

sage: from sage.categories.category import category_sample
sage: sorted([C for C in category_sample()
....:         if len(C._super_categories_for_classes) != len(C.super_categories())],
....:        key=str)
[Category of affine weyl groups,
Category of fields,
Category of finite dimensional algebras with basis over Rational Field,
Category of finite dimensional hopf algebras with basis over Rational Field,
Category of graded hopf algebras with basis over Rational Field,
Category of hopf algebras with basis over Rational Field]


AUTHOR:

• Nicolas M. Thiery (2012-09): initial version.
sage.misc.c3_controlled.C3_merge(lists)

Return the input lists merged using the C3 algorithm.

EXAMPLES:

sage: from sage.misc.c3_controlled import C3_merge
sage: C3_merge([[3,2],[4,3,1]])
[4, 3, 2, 1]
sage: C3_merge([[3,2],[4,1]])
[3, 2, 4, 1]


This function is only used for testing and experimenting purposes, but exercised quite some by the other doctests in this file.

It is an extract of sage.misc.c3.C3_algorithm(); the latter could be possibly rewritten to use this one to avoid duplication.

sage.misc.c3_controlled.C3_sorted_merge(lists, key='identity')

Return the sorted input lists merged using the C3 algorithm, with a twist.

INPUT:

• lists – a non empty list (or iterable) of lists (or iterables), each sorted strictly decreasingly according to key
• key – a function

OUTPUT: a pair (result, suggestion)

result is the sorted list obtained by merging the lists in lists while removing duplicates, and suggestion is a list such that applying C3 on lists with its last list replaced by suggestion would return result.

EXAMPLES:

With the following input, C3_merge() returns right away a sorted list:

sage: from sage.misc.c3_controlled import C3_merge
sage: C3_merge([[2],[1]])
[2, 1]


In that case, C3_sorted_merge() returns the same result, with the last line unchanged:

sage: from sage.misc.c3_controlled import C3_sorted_merge
sage: C3_sorted_merge([[2],[1]])
([2, 1], [1])


On the other hand, with the following input, C3_merge() returns a non sorted list:

sage: C3_merge([[1],[2]])
[1, 2]


Then, C3_sorted_merge() returns a sorted list, and suggests to replace the last line by [2,1]:

sage: C3_sorted_merge([[1],[2]])
([2, 1], [2, 1])


And indeed C3_merge now returns the desired result:

sage: C3_merge([[1],[2,1]])
[2, 1]


From now on, we use this little wrapper that checks that C3_merge, with the suggestion of C3_sorted_merge, returns a sorted list:

sage: def C3_sorted_merge_check(lists):
....:     result, suggestion = C3_sorted_merge(lists)
....:     assert result == C3_merge(lists[:-1] + [suggestion])
....:     return result, suggestion


Base cases:

sage: C3_sorted_merge_check([])
Traceback (most recent call last):
...
ValueError: The input should be a non empty list of lists (or iterables)
sage: C3_sorted_merge_check([[]])
([], [])
sage: C3_sorted_merge_check([[1]])
([1], [1])
sage: C3_sorted_merge_check([[3,2,1]])
([3, 2, 1], [3, 2, 1])
sage: C3_sorted_merge_check([[1],[1]])
([1], [1])
sage: C3_sorted_merge_check([[3,2,1],[3,2,1]])
([3, 2, 1], [3, 2, 1])


Exercise different states for the last line:

sage: C3_sorted_merge_check([[1],[2],[]])
([2, 1], [2, 1])
sage: C3_sorted_merge_check([[1],[2], [1]])
([2, 1], [2, 1])


Explore (all?) the different execution branches:

sage: C3_sorted_merge_check([[3,1],[4,2]])
([4, 3, 2, 1], [4, 3, 2, 1])
sage: C3_sorted_merge_check([[4,1],[3,2]])
([4, 3, 2, 1], [3, 2, 1])
sage: C3_sorted_merge_check([[3,2],[4,1]])
([4, 3, 2, 1], [4, 3, 1])
sage: C3_sorted_merge_check([[1],[4,3,2]])
([4, 3, 2, 1], [4, 3, 2, 1])
sage: C3_sorted_merge_check([[1],[3,2], []])
([3, 2, 1], [2, 1])
sage: C3_sorted_merge_check([[1],[4,3,2], []])
([4, 3, 2, 1], [2, 1])
sage: C3_sorted_merge_check([[1],[4,3,2], [2]])
([4, 3, 2, 1], [2, 1])
sage: C3_sorted_merge_check([[2],[1],[4],[3]])
([4, 3, 2, 1], [3, 2, 1])
sage: C3_sorted_merge_check([[2],[1],[4],[]])
([4, 2, 1], [4, 2, 1])
sage: C3_sorted_merge_check([[2],[1],[3],[4]])
([4, 3, 2, 1], [4, 3, 2, 1])
sage: C3_sorted_merge_check([[2],[1],[3,2,1],[3]])
([3, 2, 1], [3])
sage: C3_sorted_merge_check([[2],[1],[2,1],[3]])
([3, 2, 1], [3, 2])


Exercises adding one item when the last list has a single element; the second example comes from an actual poset:

sage: C3_sorted_merge_check([[5,4,2],[4,3],[5,4,1]])
([5, 4, 3, 2, 1], [5, 4, 3, 2, 1])
sage: C3_sorted_merge_check([[6,4,2],[5,3],[6,5,1]])
([6, 5, 4, 3, 2, 1], [6, 5, 4, 3, 2, 1])

class sage.misc.c3_controlled.CmpKey

Bases: object

This class implements the lazy attribute Category._cmp_key.

The comparison key A._cmp_key of a category is used to define an (almost) total order on non-join categories by setting, for two categories $$A$$ and $$B$$, $$A<B$$ if A._cmp_key > B._cmp_key. This order in turn is used to give a normal form to join’s, and help toward having a consistent method resolution order for parent/element classes.

The comparison key should satisfy the following properties:

• If $$A$$ is a subcategory of $$B$$, then $$A < B$$ (so that A._cmp_key > B._cmp_key). In particular, Objects() is the largest category.
• If $$A != B$$ and taking the join of $$A$$ and $$B$$ makes sense (e.g. taking the join of Algebras(GF(5)) and Algebras(QQ) does not make sense), then $$A<B$$ or $$B<A$$.

The rationale for the inversion above between $$A<B$$ and A._cmp_key > B._cmp_key is that we want the order to be compatible with inclusion of categories, yet it’s easier in practice to create keys that get bigger and bigger while we go down the category hierarchy.

This implementation applies to join-irreducible categories (i.e. categories that are not join categories). It returns a pair of integers (flags, i), where flags is to be interpreted as a bit vector. The first bit is set if self is a facade set. The second bit is set if self is finite. And so on. The choice of the flags is adhoc and was primarily crafted so that the order between categories would not change too much upon integration of trac ticket #13589 and would be reasonably session independent. The number i is there to resolve ambiguities; it is session dependent, and is assigned increasingly when new categories are created.

Note

This is currently not implemented using a lazy_attribute for speed reasons only (the code is in Cython and takes advantage of the fact that Category objects always have a __dict__ dictionary)

Todo

• Handle nicely (covariant) functorial constructions and axioms

EXAMPLES:

sage: Objects()._cmp_key
(0, 0)
sage: SetsWithPartialMaps()._cmp_key
(0, 1)
sage: Sets()._cmp_key
(0, 2)
(1, ...)
sage: Sets().Finite()._cmp_key
(2, ...)
sage: Sets().Infinite()._cmp_key
(4, ...)
sage: EnumeratedSets()._cmp_key
(8, ...)
sage: FiniteEnumeratedSets()._cmp_key
(10, ...)
(16, ...)
sage: Posets()._cmp_key
(32, ...)
sage: LatticePosets()._cmp_key
(96, ...)
sage: Crystals()._cmp_key
(136, ...)
(256, ...)
sage: Magmas()._cmp_key
(4096, ...)
(256, ...)
sage: Rings()._cmp_key
(225536, ...)
sage: Algebras(QQ)._cmp_key
(225536, ...)
sage: AlgebrasWithBasis(QQ)._cmp_key
(227584, ...)
(226560, ...)
(228608, ...)


For backward compatibility we currently want the following comparisons:

sage: EnumeratedSets()._cmp_key > Sets().Facade()._cmp_key
True
sage: AdditiveMagmas()._cmp_key > EnumeratedSets()._cmp_key
True

'sage.categories.enumerated_sets'

sage: CommutativeAdditiveSemigroups()._cmp_key < Magmas()._cmp_key
True
sage: VectorSpaces(QQ)._cmp_key < Rings()._cmp_key
True
sage: VectorSpaces(QQ)._cmp_key < Magmas()._cmp_key
True

class sage.misc.c3_controlled.CmpKeyNamed

Bases: object

This class implements the lazy attribute CategoryWithParameters._cmp_key.

Note

• The value of the attribute depends only on the parameters of this category.
• This is currently not implemented using a lazy_attribute for speed reasons only.

EXAMPLES:

sage: Algebras(GF(3))._cmp_key == Algebras(GF(5))._cmp_key  # indirect doctest
True
sage: Algebras(ZZ)._cmp_key != Algebras(GF(5))._cmp_key
True

class sage.misc.c3_controlled.HierarchyElement(value, bases, key, from_value)

Bases: object

A class for elements in a hierarchy.

This class is for testing and experimenting with various variants of the C3 algorithm to compute a linear extension of the elements above an element in a hierarchy. Given the topic at hand, we use the following naming conventions. For $$x$$ an element of the hierarchy, we call the elements just above $$x$$ its bases, and the linear extension of all elements above $$x$$ its MRO.

By convention, the bases are given as lists of HierarchyElement s, and MROs are given a list of the corresponding values.

INPUT:

• value – an object
• succ – a successor function, poset or digraph from which one can recover the successors of value
• key – a function taking values as input (default: the identity) this function is used to compute comparison keys for sorting elements of the hierarchy.

Note

Constructing a HierarchyElement immediately constructs the whole hierarchy above it.

EXAMPLES:

See the introduction of this module sage.misc.c3_controlled for many examples. Here we consider a large example, originaly taken from the hierarchy of categories above HopfAlgebrasWithBasis:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: G = DiGraph({
....:     44 :  [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     43 :  [42, 41, 40, 36, 35, 39, 38, 37, 33, 32, 31, 30, 29, 28, 27, 26, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     42 :  [36, 35, 37, 30, 29, 28, 27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     41 :  [40, 36, 35, 33, 32, 31, 30, 29, 28, 27, 26, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     40 :  [36, 35, 32, 31, 30, 29, 28, 27, 26, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     39 :  [38, 37, 33, 32, 31, 30, 29, 28, 27, 26, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     38 :  [37, 33, 32, 31, 30, 29, 28, 27, 26, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     37 :  [30, 29, 28, 27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     36 :  [35, 30, 29, 28, 27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     35 :  [29, 28, 27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     34 :  [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     33 :  [32, 31, 30, 29, 28, 27, 26, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     32 :  [31, 30, 29, 28, 27, 26, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     31 :  [30, 29, 28, 27, 26, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     30 :  [29, 28, 27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     29 :  [28, 27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     28 :  [27, 26, 15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     27 :  [15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     26 :  [15, 14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     25 :  [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     24 :  [4, 2, 1, 0],
....:     23 :  [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     22 :  [21, 20, 18, 17, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     21 :  [20, 17, 4, 2, 1, 0],
....:     20 :  [4, 2, 1, 0],
....:     19 :  [18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     18 :  [17, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     17 :  [4, 2, 1, 0],
....:     16 :  [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     15 :  [14, 12, 11, 9, 8, 5, 3, 2, 1, 0],
....:     14 :  [11, 3, 2, 1, 0],
....:     13 :  [12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     12 :  [11, 9, 8, 5, 3, 2, 1, 0],
....:     11 :  [3, 2, 1, 0],
....:     10 :  [9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
....:     9 :  [8, 5, 3, 2, 1, 0],
....:     8 :  [3, 2, 1, 0],
....:     7 :  [6, 5, 4, 3, 2, 1, 0],
....:     6 :  [4, 3, 2, 1, 0],
....:     5 :  [3, 2, 1, 0],
....:     4 :  [2, 1, 0],
....:     3 :  [2, 1, 0],
....:     2 :  [1, 0],
....:     1 :  [0],
....:     0 :  [],
....:     })

sage: x = HierarchyElement(44, G)
sage: x.mro
[44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]
sage: x.cls
<class '44.cls'>
sage: x.cls.mro()
[<class '44.cls'>, <class '43.cls'>, <class '42.cls'>, <class '41.cls'>, <class '40.cls'>, <class '39.cls'>, <class '38.cls'>, <class '37.cls'>, <class '36.cls'>, <class '35.cls'>, <class '34.cls'>, <class '33.cls'>, <class '32.cls'>, <class '31.cls'>, <class '30.cls'>, <class '29.cls'>, <class '28.cls'>, <class '27.cls'>, <class '26.cls'>, <class '25.cls'>, <class '24.cls'>, <class '23.cls'>, <class '22.cls'>, <class '21.cls'>, <class '20.cls'>, <class '19.cls'>, <class '18.cls'>, <class '17.cls'>, <class '16.cls'>, <class '15.cls'>, <class '14.cls'>, <class '13.cls'>, <class '12.cls'>, <class '11.cls'>, <class '10.cls'>, <class '9.cls'>, <class '8.cls'>, <class '7.cls'>, <class '6.cls'>, <class '5.cls'>, <class '4.cls'>, <class '3.cls'>, <class '2.cls'>, <class '1.cls'>, <class '0.cls'>, <type 'object'>]

all_bases()

Return the set of all the HierarchyElements above self, self included.

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: P = Poset((divisors(30), lambda x,y: y.divides(x)), facade=True)
sage: HierarchyElement(1, P).all_bases()
{1}
sage: HierarchyElement(10, P).all_bases()  # random output
{10, 5, 2, 1}
sage: sorted([x.value for x in HierarchyElement(10, P).all_bases()])
[1, 2, 5, 10]

all_bases_controlled_len()

Return the cumulated size of the controlled bases of the elements above self in the hierarchy.

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: P = Poset((divisors(30), lambda x,y: y.divides(x)), facade=True)
sage: HierarchyElement(30, P).all_bases_controlled_len()
13

all_bases_len()

Return the cumulated size of the bases of the elements above self in the hierarchy.

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: P = Poset((divisors(30), lambda x,y: y.divides(x)), facade=True)
sage: HierarchyElement(30, P).all_bases_len()
12

bases()

The bases of self.

The bases are given as a list of HierarchyElements, sorted decreasingly accoding to the key function.

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: P = Poset((divisors(30), lambda x,y: y.divides(x)), facade=True)
sage: x = HierarchyElement(10, P)
sage: x.bases
[5, 2]
sage: type(x.bases[0])
<class 'sage.misc.c3_controlled.HierarchyElement'>
sage: x.mro
[10, 5, 2, 1]
sage: x._bases_controlled
[5, 2]

cls()

Return a Python class with inheritance graph parallel to the hierarchy above self.

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement
sage: P = Poset((divisors(30), lambda x,y: y.divides(x)), facade=True)
sage: x = HierarchyElement(1, P)
sage: x.cls
<class '1.cls'>
sage: x.cls.mro()
[<class '1.cls'>, <type 'object'>]
sage: x = HierarchyElement(30, P)
sage: x.cls
<class '30.cls'>
sage: x.cls.mro()
[<class '30.cls'>, <class '15.cls'>, <class '10.cls'>, <class '6.cls'>, <class '5.cls'>, <class '3.cls'>, <class '2.cls'>, <class '1.cls'>, <type 'object'>]

mro()

The MRO for this object, calculated with C3_sorted_merge().

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement, C3_sorted_merge, identity
sage: P = Poset({7: [5,6], 5:[1,2], 6: [3,4]}, facade = True)
sage: x = HierarchyElement(5, P)
sage: x.mro
[5, 2, 1]
sage: x = HierarchyElement(6, P)
sage: x.mro
[6, 4, 3]
sage: x = HierarchyElement(7, P)
sage: x.mro
[7, 6, 5, 4, 3, 2, 1]

sage: C3_sorted_merge([[6, 4, 3], [5, 2, 1], [6, 5]], identity)
([6, 5, 4, 3, 2, 1], [6, 5, 4])


TESTS:

sage: assert all(isinstance(v, Integer) for v in x.mro)

mro_controlled()

The MRO for this object, calculated with C3_merge(), under control of $$C3_sorted_merge$$

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement, C3_merge
sage: P = Poset({7: [5,6], 5:[1,2], 6: [3,4]}, facade=True)
sage: x = HierarchyElement(5, P)
sage: x.mro_controlled
[5, 2, 1]
sage: x = HierarchyElement(6, P)
sage: x.mro_controlled
[6, 4, 3]
sage: x = HierarchyElement(7, P)
sage: x.mro_controlled
[7, 6, 5, 4, 3, 2, 1]
sage: x._bases
[6, 5]
sage: x._bases_controlled
[6, 5, 4]
sage: C3_merge([[6, 4, 3], [5, 2, 1], [6, 5]])
[6, 4, 3, 5, 2, 1]
sage: C3_merge([[6, 4, 3], [5, 2, 1], [6, 5, 4]])
[6, 5, 4, 3, 2, 1]


TESTS:

sage: assert all(isinstance(v, Integer) for v in x.mro_controlled)

mro_standard()

The MRO for this object, calculated with C3_merge()

EXAMPLES:

sage: from sage.misc.c3_controlled import HierarchyElement, C3_merge
sage: P = Poset({7: [5,6], 5:[1,2], 6: [3,4]}, facade=True)
sage: x = HierarchyElement(5, P)
sage: x.mro_standard
[5, 2, 1]
sage: x = HierarchyElement(6, P)
sage: x.mro_standard
[6, 4, 3]
sage: x = HierarchyElement(7, P)
sage: x.mro_standard
[7, 6, 4, 3, 5, 2, 1]
sage: C3_merge([[6, 4, 3], [5, 2, 1], [6, 5]])
[6, 4, 3, 5, 2, 1]


TESTS:

sage: assert all(isinstance(v, Integer) for v in x.mro_standard)

sage.misc.c3_controlled.identity(x)

EXAMPLES:

sage: from sage.misc.c3_controlled import identity
sage: identity(10)
10