The C3 algorithm#

The C3 algorithm is used as method resolution order for new style classes in Python. The implementation here is used to order the list of super categories of a category.


  • Simon King (2011-11): initial version.

sage.misc.c3.C3_algorithm(start, bases, attribute, proper)#

An implementation of the C3 algorithm.

C3 is the algorithm used by Python to construct the method resolution order for new style classes involving multiple inheritance.

After github issue #11943 this implementation was used to compute the list of super categories of a category; see all_super_categories(). The purpose is to ensure that list of super categories matches with the method resolution order of the parent or element classes of a category.

Since github issue #13589, this implementation is superseded by that in sage.misc.c3_controlled, that puts the C3 algorithm under control of some total order on categories. This guarantees that C3 always finds a consistent Method Resolution Order. For background, see sage.misc.c3_controlled.


  • start – an object; the returned list is built upon data provided by certain attributes of start.

  • bases – a string; the name of an attribute of start providing a list of objects.

  • attribute – a string; the name of an attribute of the objects provided in getattr(start,bases). That attribute is supposed to provide a list.


Our implementation of the algorithm only works on lists of objects that compare equal if and only if they are identical.


A list, the result of the C3 algorithm applied to the list [getattr(X,attribute) for X in getattr(start,bases)].


We create a class for elements in a hierarchy that uses the C3 algorithm to compute, for each element, a linear extension of the elements above it:

.. TODO:: Move back the __init__ at the beginning

sage: from sage.misc.c3 import C3_algorithm sage: class HierarchyElement(UniqueRepresentation): ….: @lazy_attribute ….: def _all_bases(self): ….: return C3_algorithm(self, ‘_bases’, ‘_all_bases’, False) ….: def __repr__(self): ….: return self._name ….: def __init__(self, name, bases): ….: self._name = name ….: self._bases = list(bases)

We construct a little hierarchy:

sage: T = HierarchyElement("T", ())
sage: X = HierarchyElement("X", (T,))
sage: Y = HierarchyElement("Y", (T,))
sage: A = HierarchyElement("A", (X, Y))
sage: B = HierarchyElement("B", (Y, X))
sage: Foo = HierarchyElement("Foo", (A, B))

And inspect the linear extensions associated to each element:

sage: T._all_bases
sage: X._all_bases
[X, T]
sage: Y._all_bases
[Y, T]
sage: A._all_bases
[A, X, Y, T]
sage: B._all_bases
[B, Y, X, T]

So far so good. However:

sage: Foo._all_bases
Traceback (most recent call last):
ValueError: Cannot merge the items X, Y.

The C3 algorithm is not able to create a consistent linear extension. Indeed, its specifications impose that, if X and Y appear in a certain order in the linear extension for an element of the hierarchy, then they should appear in the same order for any lower element. This is clearly not possibly for Foo, since A and B impose incompatible orders. If the above was a hierarchy of classes, Python would complain that it cannot calculate a consistent Method Resolution Order.