Unique Representation¶
Abstract classes for cached and unique representation behavior.
See also
AUTHORS:
 Nicolas M. Thiery (2008): Original version.
 Simon A. King (201302): Separate cached and unique representation.
 Simon A. King (201308): Extended documentation.
What is a cached representation?¶
Instances of a class have a cached representation behavior when several instances constructed with the same arguments share the same memory representation. For example, calling twice:
sage: G = SymmetricGroup(6)
sage: H = SymmetricGroup(6)
to create the symmetric group on six elements gives back the same object:
sage: G is H
True
This is a standard design pattern. Besides saving memory, it allows for sharing cached data (say representation theoretical information about a group). And of course a lookup in the cache is faster than the creation of a new object.
Implementing a cached representation¶
Sage provides two standard ways to create a cached representation:
CachedRepresentation
and
UniqueFactory
. Note that, in spite of its
name, UniqueFactory
does not ensure unique
representation behaviour, which will be explained below.
Using CachedRepresentation
¶
It is often very easy to use CachedRepresentation
: One simply writes
a Python class and adds CachedRepresentation
to the list of base
classes. If one does so, then the arguments used to create an instance of this
class will by default also be used as keys for the cache:
sage: from sage.structure.unique_representation import CachedRepresentation
sage: class C(CachedRepresentation):
....: def __init__(self, a, b=0):
....: self.a = a
....: self.b = b
....: def __repr__(self):
....: return "C(%s, %s)"%(self.a, self.b)
....:
sage: a = C(1)
sage: a is C(1)
True
In addition, pickling just works, provided that Python is able to look up the
class. Hence, in the following two lines, we explicitly put the class into the
__main__
module. This is needed in doctests, but not in an interactive
session:
sage: import __main__
sage: __main__.C = C
sage: loads(dumps(a)) is a
True
Often, this very easy approach is sufficient for applications. However, there are some pitfalls. Since the arguments are used for caching, all arguments must be hashable, i.e., must be valid as dictionary keys:
sage: C((1,2))
C((1, 2), 0)
sage: C([1,2])
Traceback (most recent call last):
...
TypeError: unhashable type: 'list'
In addition, equivalent ways of providing the arguments are not automatically normalised when forming the cache key, and hence different but equivalent arguments may yield distinct instances:
sage: C(1) is C(1,0)
False
sage: C(1) is C(a=1)
False
sage: repr(C(1)) == repr(C(a=1))
True
It should also be noted that the arguments are compared by equality, not by
identity. This is often desired, but can imply subtle problems. For example,
since C(1)
already is in the cache, and since the unit elements in
different finite fields are all equal to the integer one, we find:
sage: GF(5)(1) == 1 == GF(3)(1)
True
sage: C(1) is C(GF(3)(1)) is C(GF(5)(1))
True
But C(2)
is not in the cache, and the number two is not equal in different
finite fields (i. e., GF(5)(2) == GF(3)(2)
returns as False
), even
though it is equal to the number two in the ring of integers (
GF(5)(2) == 2 == GF(3)(2)
returns as True
; equality is not transitive
when comparing elements of distinct algebraic structures!!). Hence, we
have:
sage: GF(5)(2) == GF(3)(2)
False
sage: C(GF(3)(2)) is C(GF(5)(2))
False
Normalising the arguments¶
CachedRepresentation
uses the metaclass
ClasscallMetaclass
. Its
__classcall__
method is a
WeakCachedFunction
. This function creates an
instance of the given class using the given arguments, unless it finds the
result in the cache. This has the following implications:
 The arguments must be valid dictionary keys (i.e., they must be hashable; see above).
 It is a weak cache, hence, if the user does not keep a reference to the resulting instance, then it may be removed from the cache during garbage collection.
 It is possible to preprocess the input arguments by implementing a
__classcall__
or a__classcall_private__
method, but in order to benefit from caching,CachedRepresentation.__classcall__()
should at some point be called.
Note
For technical reasons, it is needed that __classcall__
respectively
__classcall_private__
are “static methods”, i.e., they are callable
objects that do not bind to an instance or class. For example, a
cached_function
can be used here, because it
is callable, but does not bind to an instance or class, because it has no
__get__()
method. A usual Python function, however, has a
__get__()
method and would thus under normal circumstances bind to an
instance or class, and thus the instance or class would be passed to the
function as the first argument. To prevent a callable object from being
bound to the instance or class, one can prepend the @staticmethod
decorator to the definition; see staticmethod
.
For more on Python’s __get__()
method, see:
http://docs.python.org/2/howto/descriptor.html
Warning
If there is preprocessing, then the preprocessed arguments
passed to CachedRepresentation.__classcall__()
must be invariant
under the preprocessing. That is to say, preprocessing the input
arguments twice must have the same effect as preprocessing the input
arguments only once. That is to say, the preprocessing must be idempotent.
The reason for this warning lies in the way pickling is implemented. If the
preprocessed arguments are passed to
CachedRepresentation.__classcall__()
, then the resulting instance will
store the preprocessed arguments in some attribute, and will use them for
pickling. If the pickle is unpickled, then preprocessing is applied to the
preprocessed arguments—and this second round of preprocessing must not
change the arguments further, since otherwise a different instance would be
created.
We illustrate the warning by an example. Imagine that one has instances that are created with an integervalued argument, but only depend on the square of the argument. It would be a mistake to square the given argument during preprocessing:
sage: class WrongUsage(CachedRepresentation):
....: @staticmethod
....: def __classcall__(cls, n):
....: return super(WrongUsage,cls).__classcall__(cls, n^2)
....: def __init__(self, n):
....: self.n = n
....: def __repr__(self):
....: return "Something(%d)"%self.n
....:
sage: import __main__
sage: __main__.WrongUsage = WrongUsage # This is only needed in doctests
sage: w = WrongUsage(3); w
Something(9)
sage: w._reduction
(<class '__main__.WrongUsage'>, (9,), {})
Indeed, the reduction data are obtained from the preprocessed argument. By consequence, if the resulting instance is pickled and unpickled, the argument gets squared again:
sage: loads(dumps(w))
Something(81)
Instead, the preprocessing should only take the absolute value of the given
argument, while the squaring should happen inside of the __init__
method,
where it won’t mess with the cache:
sage: class BetterUsage(CachedRepresentation):
....: @staticmethod
....: def __classcall__(cls, n):
....: return super(BetterUsage, cls).__classcall__(cls, abs(n))
....: def __init__(self, n):
....: self.n = n^2
....: def __repr__(self):
....: return "SomethingElse(%d)"%self.n
....:
sage: __main__.BetterUsage = BetterUsage # This is only needed in doctests
sage: b = BetterUsage(3); b
SomethingElse(9)
sage: loads(dumps(b)) is b
True
sage: b is BetterUsage(3)
True
In our next example, we create a cached representation class C
that
returns an instance of a subclass C1
or C2
depending on the given
arguments. This is implemented in a static __classcall_private__
method of
C
, letting it choose the subclass according to the given arguments. Since
a __classcall_private__
method will be ignored on subclasses, the caching
of CachedRepresentation
is available to both C1
and C2
. But
for illustration, we overload the static __classcall__
method on C2
,
doing some argument preprocessing. We also create a subclass C2b
of
C2
, demonstrating that the __classcall__
method is used on the
subclass (in contrast to a __classcall_private__
method!).
sage: class C(CachedRepresentation):
....: @staticmethod
....: def __classcall_private__(cls, n, implementation=0):
....: if not implementation:
....: return C.__classcall__(cls, n)
....: if implementation==1:
....: return C1(n)
....: if implementation>1:
....: return C2(n,implementation)
....: def __init__(self, n):
....: self.n = n
....: def __repr__(self):
....: return "C(%d, 0)"%self.n
....:
sage: class C1(C):
....: def __repr__(self):
....: return "C1(%d)"%self.n
....:
sage: class C2(C):
....: @staticmethod
....: def __classcall__(cls, n, implementation=0):
....: if implementation:
....: return super(C2, cls).__classcall__(cls, (n,)*implementation)
....: return super(C2, cls).__classcall__(cls, n)
....: def __init__(self, t):
....: self.t = t
....: def __repr__(self):
....: return "C2(%s)"%repr(self.t)
....:
sage: class C2b(C2):
....: def __repr__(self):
....: return "C2b(%s)"%repr(self.t)
....:
sage: __main__.C2 = C2 # not needed in an interactive session
sage: __main__.C2b = C2b
In the above example, C
drops the argument implementation
if it
evaluates to False
, and since the cached __classcall__
is called in
this case, we have:
sage: C(1)
C(1, 0)
sage: C(1) is C(1,0)
True
sage: C(1) is C(1,0) is C(1,None) is C(1,[])
True
(Note that we were able to bypass the issue of arguments having to be
hashable by catching the empty list []
during preprocessing in the
__classcall_private__
method. Similarly, unhashable arguments can
be made hashable – e. g., lists normalized to tuples – in the
__classcall_private__
method before they are further delegated to
__classcall__
. See
TCrystal
for an
example.)
If we call C1
directly or if we provide implementation=1
to C
, we
obtain an instance of C1
. Since it uses the __classcall__
method
inherited from CachedRepresentation
, the resulting instances are
cached:
sage: C1(2)
C1(2)
sage: C(2, implementation=1)
C1(2)
sage: C(2, implementation=1) is C1(2)
True
The class C2
preprocesses the input arguments. Instances can, again, be
obtained directly or by calling C
:
sage: C(1, implementation=3)
C2((1, 1, 1))
sage: C(1, implementation=3) is C2(1,3)
True
The argument preprocessing of C2
is inherited by C2b
, since
__classcall__
and not __classcall_private__
is used. Pickling works,
since the preprocessing of arguments is idempotent:
sage: c2b = C2b(2,3); c2b
C2b((2, 2, 2))
sage: loads(dumps(c2b)) is c2b
True
Using UniqueFactory
¶
For creating a cached representation using a factory, one has to
 create a class separately from the factory. This class must inherit
from
object
. Its instances must allow attribute assignment.  write a method
create_key
(orcreate_key_and_extra_args
) that creates the cache key from the given arguments.  write a method
create_object
that creates an instance of the class from a given cache key.  create an instance of the factory with a name that allows to conclude where it is defined.
An example:
sage: class C(object):
....: def __init__(self, t):
....: self.t = t
....: def __repr__(self):
....: return "C%s"%repr(self.t)
....:
sage: from sage.structure.factory import UniqueFactory
sage: class MyFactory(UniqueFactory):
....: def create_key(self, n, m=None):
....: if isinstance(n, (tuple,list)) and m is None:
....: return tuple(n)
....: return (n,)*m
....: def create_object(self, version, key, **extra_args):
....: # We ignore version and extra_args
....: return C(key)
....:
Now, we define an instance of the factory, stating that it can be found under
the name "F"
in the __main__
module. By consequence, pickling works:
sage: F = MyFactory("__main__.F")
sage: __main__.F = F # not needed in an interactive session
sage: loads(dumps(F)) is F
True
We can now create cached instances of C
by calling the factory. The
cache only takes into account the key computed with the method create_key
that we provided. Hence, different given arguments may result in the same
instance. Note that, again, the cache is weak, hence, the instance might be
removed from the cache during garbage collection, unless an external reference
is preserved.
sage: a = F(1, 2); a
C(1, 1)
sage: a is F((1,1))
True
If the class of the returned instances is a subclass of object
,
and if the resulting instance allows attribute assignment, then pickling
of the resulting instances is automatically provided for, and respects the
cache.
sage: loads(dumps(a)) is a
True
This is because an attribute is stored that explains how the instance was created:
sage: a._factory_data
(<class '__main__.MyFactory'>, (...), (1, 1), {})
Note
If a class is used that does not inherit from object
then unique
pickling is not provided.
Caching is only available if the factory is called. If an instance of the class is directly created, then the cache is not used:
sage: C((1,1))
C(1, 1)
sage: C((1,1)) is a
False
Comparing the two ways of implementing a cached representation¶
In this subsection, we discuss advantages and disadvantages of the two ways of implementing a cached representation, depending on the type of application.
Simplicity and transparency¶
In many cases, turning a class into a cached representation requires nothing
more than adding CachedRepresentation
to the list of base classes of
this class. This is, of course, a very easy and convenient way. Writing a
factory would involve a lot more work.
If preprocessing of the arguments is needed, then we have seen how to do this
by a __classcall_private__
or __classcall__
method. But these are
double underscore methods and hence, for example, invisible in the
automatically created reference manual. Moreover, preprocessing and caching
are implemented in the same method, which might be confusing. In a unique
factory, these two tasks are cleanly implemented in two separate methods.
With a factory, it is possible to create the resulting instance by arguments
that are different from the key used for caching. This is significantly
restricted with CachedRepresentation due to the requirement that argument
preprocessing be idempotent.
Hence, if advanced preprocessing is needed, then
UniqueFactory
might be easier and more
transparent to use than CachedRepresentation
.
Class inheritance¶
Using CachedRepresentation
has the advantage that one has a class and
creates cached instances of this class by the usual Python syntax:
sage: G = SymmetricGroup(6)
sage: issubclass(SymmetricGroup, sage.structure.unique_representation.CachedRepresentation)
True
sage: isinstance(G, SymmetricGroup)
True
In contrast, a factory is just a callable object that returns something that has absolutely nothing to do with the factory, and may in fact return instances of quite different classes:
sage: isinstance(GF, sage.structure.factory.UniqueFactory)
True
sage: K5 = GF(5)
sage: type(K5)
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>
sage: K25 = GF(25, 'x')
sage: type(K25)
<class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>
sage: Kp = GF(next_prime_power(1000000)^2, 'x')
sage: type(Kp)
<class 'sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt_with_category'>
This can be confusing to the user. Namely, the user might determine the class
of an instance and try to create further instances by calling the class rather
than the factory—which is a mistake since it works around the cache (and
also since the class might be more restrictive than the factory – i. e., the
type of K5
in the above doctest cannot be called on a prime power which
is not a prime). This mistake can more easily be avoided by using
CachedRepresentation
.
We have seen above that one can easily create new cachedrepresentation classes by subclassing an existing cachedrepresentation class, even making use of an existing argument preprocess. This would be much more complicated with a factory. Namely, one would need to rewrite old factories making them aware of the new classes, and/or write new factories for the new classes.
Python versus extension classes¶
CachedRepresentation
uses a metaclass, namely
ClasscallMetaclass
. Hence, it can
currently not be a Cython extension class. Moreover, it is supposed to be used
by providing it as a base class. But in typical applications, one also has
another base class, say, Parent
. Hence, one
would like to create a class with at least two base classes, which is
currently impossible in Cython extension classes.
In other words, when using CachedRepresentation
, one must work with
Python classes. These can be defined in Cython code (.pyx
files) and can
thus benefit from Cython’s speed inside of their methods, but they must not be
cdef class
and can thus not use cdef
attributes or methods.
Such restrictions do not exist when using a factory. However, if attribute
assignment does not work, then the automatic pickling provided by
UniqueFactory
will not be available.
What is a unique representation?¶
Instances of a class have a unique instance behavior when instances of this
class evaluate equal if and only if they are identical. Sage provides the base
class WithEqualityById
, which provides
comparison by identity and a hash that is determined by the memory address of
the instance. Both the equality test and the hash are implemented in Cython
and are very fast, even when one has a Python class inheriting from
WithEqualityById
.
In many applications, one wants to combine unique instance and cached representation behaviour. This is called unique representation behaviour. We have seen above that symmetric groups have a cached representation behaviour. However, they do not show the unique representation behaviour, since they are equal to groups created in a totally different way, namely to subgroups:
sage: G = SymmetricGroup(6)
sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))])
sage: G is G3
False
sage: type(G) == type(G3)
False
sage: G == G3
True
The unique representation behaviour can conveniently be implemented with a
class that inherits from UniqueRepresentation
: By adding
UniqueRepresentation
to the base classes, the class will
simultaneously inherit from CachedRepresentation
and from
WithEqualityById
.
For example, a symmetric function algebra is uniquely determined by the base
ring. Thus, it is reasonable to use UniqueRepresentation
in this
case:
sage: isinstance(SymmetricFunctions(CC), SymmetricFunctions)
True
sage: issubclass(SymmetricFunctions, UniqueRepresentation)
True
UniqueRepresentation
differs from CachedRepresentation
only
by adding WithEqualityById
as a base
class. Hence, the above examples of argument preprocessing work for
UniqueRepresentation
as well.
Note that a cached representation created with
UniqueFactory
does not automatically
provide unique representation behaviour, in spite of its name! Hence, for
unique representation behaviour, one has to implement hash and equality test
accordingly, for example by inheriting from
WithEqualityById
.

class
sage.structure.unique_representation.
CachedRepresentation
¶ Bases:
object
Classes derived from CachedRepresentation inherit a weak cache for their instances.
Note
If this class is used as a base class, then instances are (weakly) cached, according to the arguments used to create the instance. Pickling is provided, of course by using the cache.
Note
Using this class, one can have arbitrary hash and comparison. Hence, unique representation behaviour is not provided.
See also
EXAMPLES:
Providing a class with a weak cache for the instances is easy: Just inherit from
CachedRepresentation
:sage: from sage.structure.unique_representation import CachedRepresentation sage: class MyClass(CachedRepresentation): ....: # all the rest as usual ....: pass
We start with a simple class whose constructor takes a single value as argument (TODO: find a more meaningful example):
sage: class MyClass(CachedRepresentation): ....: def __init__(self, value): ....: self.value = value ....: def __cmp__(self, other): ....: c = cmp(type(self),type(other)) ....: if c: return c ....: return cmp(self.value, other.value)
Two coexisting instances of
MyClass
created with the same argument data are guaranteed to share the same identity. Since trac ticket #12215, this is only the case if there is some strong reference to the returned instance, since otherwise it may be garbage collected:sage: x = MyClass(1) sage: y = MyClass(1) sage: x is y # There is a strong reference True sage: z = MyClass(2) sage: x is z False
In particular, modifying any one of them modifies the other (reference effect):
sage: x.value = 3 sage: x.value, y.value (3, 3) sage: y.value = 1 sage: x.value, y.value (1, 1)
The arguments can consist of any combination of positional or keyword arguments, as taken by a usual
__init__
function. However, all values passed in should be hashable:sage: MyClass(value = [1,2,3]) Traceback (most recent call last): ... TypeError: unhashable type: 'list'
Argument preprocessing
Sometimes, one wants to do some preprocessing on the arguments, to put them in some canonical form. The following example illustrates how to achieve this; it takes as argument any iterable, and canonicalizes it into a tuple (which is hashable!):
sage: class MyClass2(CachedRepresentation): ....: @staticmethod ....: def __classcall__(cls, iterable): ....: t = tuple(iterable) ....: return super(MyClass2, cls).__classcall__(cls, t) ....: ....: def __init__(self, value): ....: self.value = value ....: sage: x = MyClass2([1,2,3]) sage: y = MyClass2(tuple([1,2,3])) sage: z = MyClass2(i for i in [1,2,3]) sage: x.value (1, 2, 3) sage: x is y, y is z (True, True)
A similar situation arises when the constructor accepts default values for some of its parameters. Alas, the obvious implementation does not work:
sage: class MyClass3(CachedRepresentation): ....: def __init__(self, value = 3): ....: self.value = value ....: sage: MyClass3(3) is MyClass3() False
Instead, one should do:
sage: class MyClass3(UniqueRepresentation): ....: @staticmethod ....: def __classcall__(cls, value = 3): ....: return super(MyClass3, cls).__classcall__(cls, value) ....: ....: def __init__(self, value): ....: self.value = value ....: sage: MyClass3(3) is MyClass3() True
A bit of explanation is in order. First, the call
MyClass2([1,2,3])
triggers a call toMyClass2.__classcall__(MyClass2, [1,2,3])
. This is an extension of the standard Python behavior, needed byCachedRepresentation
, and implemented by theClasscallMetaclass
. Then,MyClass2.__classcall__
does the desired transformations on the arguments. Finally, it usessuper
to call the default implementation of__classcall__
provided byCachedRepresentation
. This one in turn handles the caching and, if needed, constructs and initializes a new object in the class using__new__
and__init__
as usual.Constraints:
__classcall__()
is a staticmethod (like, implicitly,__new__
) the preprocessing on the arguments should be idempotent. That is, if
MyClass2.__classcall__(<arguments>)
callsCachedRepresentation.__classcall__(<preprocessed_arguments>)
, thenMyClass2.__classcall__(<preprocessed_arguments>)
should also result in a call toCachedRepresentation.__classcall__(<preprocessed_arguments>)
. MyClass2.__classcall__
should return the result ofCachedRepresentation.__classcall__()
without modifying it.
Other than that
MyClass2.__classcall__
may play any tricks, like acting as a factory and returning objects from other classes.Warning
It is possible, but strongly discouraged, to let the
__classcall__
method of a classC
return objects that are not instances ofC
. Of course, instances of a subclass ofC
are fine. Compare the examples inunique_representation
.We illustrate what is meant by an “idempotent” preprocessing. Imagine that one has instances that are created with an integervalued argument, but only depend on the square of the argument. It would be a mistake to square the given argument during preprocessing:
sage: class WrongUsage(CachedRepresentation): ....: @staticmethod ....: def __classcall__(cls, n): ....: return super(WrongUsage,cls).__classcall__(cls, n^2) ....: def __init__(self, n): ....: self.n = n ....: def __repr__(self): ....: return "Something(%d)"%self.n ....: sage: import __main__ sage: __main__.WrongUsage = WrongUsage # This is only needed in doctests sage: w = WrongUsage(3); w Something(9) sage: w._reduction (<class '__main__.WrongUsage'>, (9,), {})
Indeed, the reduction data are obtained from the preprocessed arguments. By consequence, if the resulting instance is pickled and unpickled, the argument gets squared again:
sage: loads(dumps(w)) Something(81)
Instead, the preprocessing should only take the absolute value of the given argument, while the squaring should happen inside of the
__init__
method, where it won’t mess with the cache:sage: class BetterUsage(CachedRepresentation): ....: @staticmethod ....: def __classcall__(cls, n): ....: return super(BetterUsage, cls).__classcall__(cls, abs(n)) ....: def __init__(self, n): ....: self.n = n^2 ....: def __repr__(self): ....: return "SomethingElse(%d)"%self.n ....: sage: __main__.BetterUsage = BetterUsage # This is only needed in doctests sage: b = BetterUsage(3); b SomethingElse(9) sage: loads(dumps(b)) is b True sage: b is BetterUsage(3) True
Cached representation and mutability
CachedRepresentation
is primarily intended for implementing objects which are (at least semantically) immutable. This is in particular assumed by the default implementations ofcopy
anddeepcopy
:sage: copy(x) is x True sage: from copy import deepcopy sage: deepcopy(x) is x True
However, in contrast to
UniqueRepresentation
, usingCachedRepresentation
allows for a comparison that is not by identity:sage: t = MyClass(3) sage: z = MyClass(2) sage: t.value = 2
Now
t
andz
are nonidentical, but equal:sage: t.value == z.value True sage: t == z True sage: t is z False
More on cached representation and identity
CachedRepresentation
is implemented by means of a cache. This cache uses weak references. Hence, when all other references to, say,MyClass(1)
have been deleted, the instance is actually deleted from memory. A later call toMyClass(1)
reconstructs the instance from scratch.sage: class SomeClass(UniqueRepresentation): ....: def __init__(self, i): ....: print("creating new instance for argument %s" % i) ....: self.i = i ....: def __del__(self): ....: print("deleting instance for argument %s" % self.i) ....: sage: O = SomeClass(1) creating new instance for argument 1 sage: O is SomeClass(1) True sage: O is SomeClass(2) creating new instance for argument 2 deleting instance for argument 2 False sage: del O deleting instance for argument 1 sage: O = SomeClass(1) creating new instance for argument 1 sage: del O deleting instance for argument 1
Cached representation and pickling
The default Python pickling implementation (by reconstructing an object from its class and dictionary, see “The pickle protocol” in the Python Library Reference) does not preserve cached representation, as Python has no chance to know whether and where the same object already exists.
CachedRepresentation
tries to ensure appropriate pickling by implementing a__reduce__
method returning the arguments passed to the constructor:sage: import __main__ # Fake MyClass being defined in a python module sage: __main__.MyClass = MyClass sage: x = MyClass(1) sage: loads(dumps(x)) is x True
CachedRepresentation
uses the__reduce__
pickle protocol rather than__getnewargs__
because the latter does not handle keyword arguments:sage: x = MyClass(value = 1) sage: x.__reduce__() (<function unreduce at ...>, (<class '__main__.MyClass'>, (), {'value': 1})) sage: x is loads(dumps(x)) True
Note
The default implementation of
__reduce__
inCachedRepresentation
requires to store the constructor’s arguments in the instance dictionary upon construction:sage: x.__dict__ {'_reduction': (<class '__main__.MyClass'>, (), {'value': 1}), 'value': 1}
It is often easy in a derived subclass to reconstruct the constructor’s arguments from the instance data structure. When this is the case,
__reduce__
should be overridden; automagically the arguments won’t be stored anymore:sage: class MyClass3(UniqueRepresentation): ....: def __init__(self, value): ....: self.value = value ....: ....: def __reduce__(self): ....: return (MyClass3, (self.value,)) ....: sage: import __main__; __main__.MyClass3 = MyClass3 # Fake MyClass3 being defined in a python module sage: x = MyClass3(1) sage: loads(dumps(x)) is x True sage: x.__dict__ {'value': 1}
Migrating classes to
CachedRepresentation
and unpicklingWe check that, when migrating a class to
CachedRepresentation
, older pickles can still be reasonably unpickled. Let us create a (new style) class, and pickle one of its instances:sage: class MyClass4(object): ....: def __init__(self, value): ....: self.value = value ....: sage: import __main__; __main__.MyClass4 = MyClass4 # Fake MyClass4 being defined in a python module sage: pickle = dumps(MyClass4(1))
It can be unpickled:
sage: y = loads(pickle) sage: y.value 1
Now, we upgrade the class to derive from
UniqueRepresentation
, which inherits fromCachedRepresentation
:sage: class MyClass4(UniqueRepresentation, object): ....: def __init__(self, value): ....: self.value = value sage: import __main__; __main__.MyClass4 = MyClass4 # Fake MyClass4 being defined in a python module sage: __main__.MyClass4 = MyClass4
The pickle can still be unpickled:
sage: y = loads(pickle) sage: y.value 1
Note however that, for the reasons explained above, unique representation is not guaranteed in this case:
sage: y is MyClass4(1) False
Todo
Illustrate how this can be fixed on a case by case basis.
Now, we redo the same test for a class deriving from SageObject:
sage: class MyClass4(SageObject): ....: def __init__(self, value): ....: self.value = value sage: import __main__; __main__.MyClass4 = MyClass4 # Fake MyClass4 being defined in a python module sage: pickle = dumps(MyClass4(1)) sage: class MyClass4(UniqueRepresentation, SageObject): ....: def __init__(self, value): ....: self.value = value sage: __main__.MyClass4 = MyClass4 sage: y = loads(pickle) sage: y.value 1
Caveat: unpickling instances of a formerly oldstyle class is not supported yet by default:
sage: class MyClass4: ....: def __init__(self, value): ....: self.value = value sage: import __main__; __main__.MyClass4 = MyClass4 # Fake MyClass4 being defined in a python module sage: pickle = dumps(MyClass4(1)) sage: class MyClass4(UniqueRepresentation, SageObject): ....: def __init__(self, value): ....: self.value = value sage: __main__.MyClass4 = MyClass4 sage: y = loads(pickle) # todo: not implemented sage: y.value # todo: not implemented 1
Rationale for the current implementation
CachedRepresentation
and derived classes use theClasscallMetaclass
of the standard Python type. The following example explains why.We define a variant of
MyClass
where the calls to__init__
are traced:sage: class MyClass(CachedRepresentation): ....: def __init__(self, value): ....: print("initializing object") ....: self.value = value ....:
Let us create an object twice:
sage: x = MyClass(1) initializing object sage: z = MyClass(1)
As desired the
__init__
method was only called the first time, which is an important feature.As far as we can tell, this is not achievable while just using
__new__
and__init__
(as defined by type; see Section Basic Customization in the Python Reference Manual). Indeed,__init__
is called systematically on the result of__new__
whenever the result is an instance of the class.Another difficulty is that argument preprocessing (as in the example above) cannot be handled by
__new__
, since the unprocessed arguments will be passed down to__init__
.

class
sage.structure.unique_representation.
UniqueRepresentation
¶ Bases:
sage.structure.unique_representation.CachedRepresentation
,sage.misc.fast_methods.WithEqualityById
Classes derived from UniqueRepresentation inherit a unique representation behavior for their instances.
See also
EXAMPLES:
The short story: to construct a class whose instances have a unique representation behavior one just has to do:
sage: class MyClass(UniqueRepresentation): ....: # all the rest as usual ....: pass
Everything below is for the curious or for advanced usage.
What is unique representation?
Instances of a class have a unique representation behavior when instances evaluate equal if and only if they are identical (i.e., share the same memory representation), if and only if they were created using equal arguments. For example, calling twice:
sage: f = SymmetricFunctions(QQ) sage: g = SymmetricFunctions(QQ)
to create the symmetric function algebra over \(\QQ\) actually gives back the same object:
sage: f == g True sage: f is g True
This is a standard design pattern. It allows for sharing cached data (say representation theoretical information about a group) as well as for very fast hashing and equality testing. This behaviour is typically desirable for parents and categories. It can also be useful for intensive computations where one wants to cache all the operations on a small set of elements (say the multiplication table of a small group), and access this cache as quickly as possible.
UniqueRepresentation
is very easy to use: a class just needs to derive from it, or make sure some of its super classes does. Also, it groups together the class and the factory in a single gadget:sage: isinstance(SymmetricFunctions(CC), SymmetricFunctions) True sage: issubclass(SymmetricFunctions, UniqueRepresentation) True
This nice behaviour is not available when one just uses a factory:
sage: isinstance(GF(7), GF) Traceback (most recent call last): ... TypeError: isinstance() arg 2 must be a class, type, or tuple of classes and types sage: isinstance(GF, sage.structure.factory.UniqueFactory) True
In addition,
UniqueFactory
only provides the cached representation behaviour, but not the unique representation behaviour—the examples inunique_representation
explain this difference.On the other hand, the
UniqueRepresentation
class is more intrusive, as it imposes a behavior (and a metaclass) on all the subclasses. In particular, the unique representation behaviour is imposed on all subclasses (unless the__classcall__
method is overloaded and not called in the subclass, which is not recommended). Its implementation is also more technical, which leads to some subtleties.EXAMPLES:
We start with a simple class whose constructor takes a single value as argument. This pattern is similar to what is done in
sage.combinat.sf.sf.SymmetricFunctions
:sage: class MyClass(UniqueRepresentation): ....: def __init__(self, value): ....: self.value = value ....: def __cmp__(self, other): ....: c = cmp(type(self),type(other)) ....: if c: return c ....: print("custom cmp") ....: return cmp(self.value, other.value) ....:
Two coexisting instances of
MyClass
created with the same argument data are guaranteed to share the same identity. Since trac ticket #12215, this is only the case if there is some strong reference to the returned instance, since otherwise it may be garbage collected:sage: x = MyClass(1) sage: y = MyClass(1) sage: x is y # There is a strong reference True sage: z = MyClass(2) sage: x is z False
In particular, modifying any one of them modifies the other (reference effect):
sage: x.value = 3 sage: x.value, y.value (3, 3) sage: y.value = 1 sage: x.value, y.value (1, 1)
Rich comparison by identity is used when possible (hence, for
==
, for!=
, and for identical arguments in the case of<
,<=
,>=
and>
), which is as fast as it can get. Only if identity is not enough to decide the answer of a comparison, the custom comparison is called:sage: x == y True sage: z = MyClass(2) sage: x == z, x is z (False, False) sage: x <= x True sage: x != z True sage: x <= z custom cmp True sage: x > z custom cmp False
A hash function equivalent to
object.__hash__()
is used, which is compatible with comparison by identity. However this means that the hash function may change in between Sage sessions, or even within the same Sage session.sage: hash(x) == object.__hash__(x) True
Warning
It is possible to inherit from
UniqueRepresentation
and then overload comparison in a way that destroys the unique representation property. We strongly recommend against it! You should useCachedRepresentation
instead.Mixing super types and super classes

sage.structure.unique_representation.
unreduce
(cls, args, keywords)¶ Calls a class on the given arguments:
sage: sage.structure.unique_representation.unreduce(Integer, (1,), {}) 1
Todo
should reuse something preexisting …