# Unique Representation¶

Abstract classes for cached and unique representation behavior.

AUTHORS:

• Nicolas M. Thiery (2008): Original version.
• Simon A. King (2013-02): Separate cached and unique representation.
• Simon A. King (2013-08): 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 look-up 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
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: https://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 integer-valued 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)
True
sage: b is BetterUsage(-3)
True


In our next example, we create a cached representation class C that returns an instance of a sub-class C1 or C2 depending on the given arguments. This is implemented in a static __classcall_private__ method of C, letting it choose the sub-class according to the given arguments. Since a __classcall_private__ method will be ignored on sub-classes, 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 sub-class C2b of C2, demonstrating that the __classcall__ method is used on the sub-class (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))
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 (or create_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
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 sub-class 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
(<__main__.MyFactory object at ...>, (...), (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 sub-section, 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 cached-representation classes by subclassing an existing cached-representation 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.

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 __eq__(self, other):
....:         if type(self) != type(other):
....:             return False
....:         return 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


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 to MyClass2.__classcall__(MyClass2, [1,2,3]). This is an extension of the standard Python behavior, needed by CachedRepresentation, and implemented by the ClasscallMetaclass. Then, MyClass2.__classcall__ does the desired transformations on the arguments. Finally, it uses super to call the default implementation of __classcall__ provided by CachedRepresentation. 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>) calls CachedRepresentation.__classcall__(<preprocessed_arguments>), then MyClass2.__classcall__(<preprocessed_arguments>) should also result in a call to CachedRepresentation.__classcall__(<preprocessed_arguments>).
• MyClass2.__classcall__ should return the result of CachedRepresentation.__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 class C return objects that are not instances of C. Of course, instances of a subclass of C are fine. Compare the examples in unique_representation.

We illustrate what is meant by an “idempotent” preprocessing. Imagine that one has instances that are created with an integer-valued 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)
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 of copy and deepcopy:

sage: copy(x) is x
True
sage: from copy import deepcopy
sage: deepcopy(x) is x
True


However, in contrast to UniqueRepresentation, using CachedRepresentation allows for a comparison that is not by identity:

sage: t = MyClass(3)
sage: z = MyClass(2)
sage: t.value = 2


Now t and z are non-identical, 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 in general, but strong references to the most recently created objects. Hence, when all other references to, say, MyClass(1) have been deleted, the instance is eventually deleted from memory (after enough other objects have been created to remove the strong reference to MyClass(1)). A later call to MyClass(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: class OtherClass(UniqueRepresentation):
....:     def __init__(self, i):
....:         pass
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
False
sage: L = [OtherClass(i) for i in range(200)]
deleting instance for argument 2
sage: del O
deleting instance for argument 1
sage: O = SomeClass(1)
creating new instance for argument 1
sage: del O
sage: del L
sage: L = [OtherClass(i) for i in range(200)]
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)
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}))
True


Note

The default implementation of __reduce__ in CachedRepresentation 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)
True
sage: x.__dict__
{'value': 1}


Migrating classes to CachedRepresentation and unpickling

We 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 from CachedRepresentation:

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.value
1


Caveat: unpickling instances of a formerly old-style 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 the ClasscallMetaclass 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

Classes derived from UniqueRepresentation inherit a unique representation behavior for their instances.

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)  # py2
Traceback (most recent call last):
...
TypeError: isinstance() arg 2 must be a class, type, or tuple of classes and types
sage: isinstance(GF(7), GF)  # py3
Traceback (most recent call last):
...
TypeError: isinstance() arg 2 must be a type or tuple of 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 in unique_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


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)


When comparing two instances of a unique representation with == or != comparison by identity is used:

sage: x == y
True
sage: x is y
True
sage: z = MyClass(2)
sage: x == z
False
sage: x is z
False
sage: x != y
False
sage: x != z
True


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 use CachedRepresentation 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 …