The category graph should comply with Python's method resolution order

[simon-king-jena]

Let C be a category. C.all_super_categories() starts with C.super_categories() and finds all super categories inductively.

Unfortunately, the algorithm of C.all_super_categories() does not comply with the C3 algorithm, that is used by Python to determine the method resolution order for C.parent_class and C.element_class.

The aim of this ticket is to be more consistent. Eventually, for any category C (perhaps with modifications for hom categories), one should have

sage: C.parent_class.mro() == [X.parent_class for X in C.all_super_categories()] + [object]
True
sage: C.element_class.mro() == [X.element_class for X in C.all_super_categories()] + [object]
True

and that test should become part of the test suite.

At #11900, the implementation of all_super_categories() has been changed, so, work should rely on #11900.

Unfortunately, it seems that the C3 algorithm can not simply be imported from Python, even though Python uses it internally, namely in Objects/typeobject.c. Looking at the implementation, it requires that one gets a list of types, while we want to use it on a list of objects.

Therefore, we need to implement C3 from scratch. Implementing C3 is easy, but if it shall be fast, one needs to be careful.

In particular, one needs to be able to L.pop(0) quickly, where L is a list. Unfortunately, L.pop(0) is slow, even in Cython. I found that, if the lists are not too long, it is best to revert L and do L.pop() instead, which is optimized in Cython. See the discussion at sage-devel.

What my patch does:

• Provide the C3 algorithm in a new extension module sage.misc.c3
• Use it to compute all_super_categories()
• Add a test _test_category_graph, asserting that self.parent_class.mro() and self.all_super_categories() are compatible.

Apply:

attachment: trac11943_mro_for_all_super_categories_combined.patch

Depends on #11900
Depends on #7980

CC: @hivert @koffie

Component: categories

Keywords: category graph, method resolution order

Author: Simon King

Reviewer: Nicolas M. Thiéry

Merged: sage-5.1.beta0

Issue created by migration from https://trac.sagemath.org/ticket/11943

[simon-king-jena]

comment:1

With the new test, I found a bug in algebra_ideals:

sage: C = AlgebraIdeals(FreeAlgebra(QQ,2,'a,b'))
sage: C.parent_class
ERROR: An unexpected error occurred while tokenizing input
The following traceback may be corrupted or invalid
The error message is: ('EOF in multi-line statement', (396, 0))

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

/home/king/Seminar/<ipython console> in <module>()

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/lazy_attribute.pyc in __get__(self, a, cls)
    507         if a is None: # when doing cls.x for cls a class and x a lazy attribute
    508             return self
--> 509         result = self.f(a)
    510         if result is NotImplemented:
    511             # Workaround: we make sure that cls is the class

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/categories/category.pyc in parent_class(self)
    626         """
    627         return dynamic_class("%s.parent_class"%self.__class__.__name__,
--> 628                              tuple(cat.parent_class for cat in self.super_categories()),
    629                              self.ParentMethods,
    630                              reduction = (getattr, (self, "parent_class")))

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/cachefunc.pyc in __call__(self, *args, **kwds)
    553             return self.cache[k]
    554         except KeyError:
--> 555             w = self._cachedmethod._instance_call(self._instance, *args, **kwds)
    556             self.cache[k] = w
    557             return w

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/cachefunc.pyc in _instance_call(self, inst, *args, **kwds)
    776 
    777         """
--> 778         return self._cachedfunc.f(inst, *args, **kwds)
    779 
    780     def _get_instance_cache(self, inst):

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/categories/algebra_ideals.pyc in super_categories(self)
     67             [Category of algebra modules over Univariate Polynomial Ring in x over Rational Field]
     68         """
     69         from algebra_modules import AlgebraModules
     70         R = self.algebra()
---> 71         return [AlgebraModules(R)]

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/classcall_metaclass.pyc in __call__(cls, *args, **options)
    256             return cls.__classcall_private__(cls, *args, **options)
    257         elif hasattr(cls, "__classcall__"):
--> 258             return cls.__classcall__(cls, *args, **options)
    259         else:
    260             return type.__call__(cls, *args, **options)

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/cachefunc.pyc in __call__(self, *args, **kwds)
    176             return self.cache[k]
    177         except KeyError:
--> 178             w = self.f(*args, **kwds)
    179             self.cache[k] = w
    180             return w

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/structure/unique_representation.pyc in __classcall__(cls, *args, **options)
    447             True
    448         """
--> 449         instance = type.__call__(cls, *args, **options)
    450         assert isinstance( instance, cls )
    451         if instance.__class__.__reduce__ == UniqueRepresentation.__reduce__:

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/categories/algebra_modules.pyc in __init__(self, A)
     53         from sage.categories.commutative_algebras import CommutativeAlgebras
     54         if not hasattr(A, "base_ring") or not A in CommutativeAlgebras(A.base_ring()):
---> 55             raise TypeError, "A (=%s) must be a commutative algebra"%A
     56         Category_module.__init__(self, A)
     57 

TypeError: A (=Free Algebra on 2 generators (a, b) over Rational Field) must be a commutative algebra

Thus, apparently, that category has never been used.

The problem is that the super categories of the category of algebra ideals should be the category of algebra modules. However, while the former accepts non-commutative algebras, the latter wants to see commutative algebras.

The clean way to proceed would be: Make algebra modules work over non-commutative rings. If that is too much of work, C.super_categories() should be [] if C is the category of algebra ideals over a non-commutative rings.

[simon-king-jena]

Dependencies: #11900

[simon-king-jena]

comment:2

Patch's up!

To my surprise, the change in the order of super categories was relatively harmless. In few cases, a test involving all_super_categories had to change.

I added a test _test_category_graph to the Test Suite of categories. By that test, I found one bug in algebra_ideals, which I fixed. Hom categories can not use the new test yet, since it tests the mro of the parent class against the list of parent classes of all super categories - which is fairly inconsistent for hom categories. That is already a "to do", and should be part of the next category overhaule.

Apart from these small changes, all doc tests passed! In particular, I did not need to change super_categories (except for algebra ideals): The current category graph seems to be consistent!!

That was the good news.

The bad news is that we have a regression in the computation of all_super_categories. With #11900, we have

sage: L = [Algebras(GF(p)) for p in prime_range(10000)]
sage: %time X = [C.all_super_categories() for C in L]
CPU times: user 0.74 s, sys: 0.01 s, total: 0.75 s
Wall time: 0.75 s

With the patches from here added, we only have

sage: L = [Algebras(GF(p)) for p in prime_range(10000)]
sage: %time X = [C.all_super_categories() for C in L]
CPU times: user 1.06 s, sys: 0.02 s, total: 1.07 s
Wall time: 1.08 s

I tried hard to make my C3 implementation as fast as possible in the range we need: few lists (say, 4) of moderate length (not more than 60).

I think the performance should be improved. But a review can already be started.

[simon-king-jena]

Author: Simon King

[nthiery]

comment:3

Replying to @simon-king-jena:

With the new test, I found a bug in algebra_ideals:
The problem is that the super categories of the category of algebra ideals should be the category of algebra modules. However, while the former accepts non-commutative algebras, the latter wants to see commutative algebras.
The clean way to proceed would be: Make algebra modules work over non-commutative rings. If that is too much of work, C.super_categories() should be [] if C is the category of algebra ideals over a non-commutative rings.

+1. Actually my patch on #10963 is adding a TODO about it :-)

[nthiery]

comment:4

Replying to @simon-king-jena:

To my surprise, the change in the order of super categories was relatively harmless. In few cases, a test involving all_super_categories had to change.

I am not so surprised, since most categories are actually used by some
parent (which detects any incompatibility). But it's very nice to
detect such incompatibilities earlier on and to catch them
systematically in the TestSuite!

Cheers,
Nicolas

[simon-king-jena]

comment:5

I tried to track down the regression of all_super_categories. Recall that my patch from #11900 did improve the implementation of all_super_categories, but did not change the algorithm.

With #11900, the main work is done in _all_super_categories_raw. With the patch from here, the work is done in c3_algorithm - and according to %prun, less time is spent in c3_algorithm than it was in _all_super_categories_raw. However, more time is spent in the expression

C.all_super_categories() for C in self.super_categories()

I guess I have to improve the implementation of the recursion.

[simon-king-jena]

comment:6

PS: It also seems that much time is spent in super_categories(), which is a cached method. Probably it would be better to turn it into a lazy attribute (or have two lazy attributes, namely for the case proper=True and proper=False).

Namely, even though #11115 will almost totally reduce the overhead of calling a cached function, a lazy attribute will always be faster.

[simon-king-jena]

comment:7

Replying to @simon-king-jena:

PS: It also seems that much time is spent in super_categories(), which is a cached method. Probably it would be better to turn it into a lazy attribute (or have two lazy attributes, namely for the case proper=True and proper=False).

Sorry, I meant: One lazy attribute C.super_categories, and two lazy attributes C.all_super_categories and C.all_super_categories_proper (super_categories() has no arguments).