Set factories¶
A set factory \(F\) is a device for constructing some Parent
\(P\) that models subsets of a big set \(S\). Typically, each such parent
is constructed as the subset of \(S\) of all elements satisfying a
certain collection of constraints \(cons\). In such a hierarchy of
subsets, one needs an easy and flexible control on how elements are
constructed. For example, one may want to construct the elements of
\(P\) in some subclass of the class of the elements of \(S\). On other
occasions, one also often needs \(P\) to be a facade parent, whose
elements are represented as elements of \(S\) (see
FacadeSets
).
The role of a set factory is twofold:
 Manage a database of constructors for the different parents \(P = F(cons)\) depending on the various kinds of constraints \(cons\). Note: currently there is no real support for that. We are gathering use cases before fixing the interface.
 Ensure that the elements \(e = P(...)\) created by the different parents follows a consistent policy concerning their class and parent.
Basic usage: constructing parents through a factory
The file sage.structure.set_factories_example
shows an example of a
SetFactory
together with typical implementation. Note that the
written code is intentionally kept minimal, many things and in particular
several iterators could be written in a more efficient way.
Consider the set \(S\) of couples \((x,y)\) with \(x\) and \(y\) in \(I:=\{0,1,2,3,4\}\).
We represent an element of \(S\) as a 2elements tuple, wrapped in a class
XYPair
deriving from ElementWrapper
.
You can create a XYPair
with any
Parent
:
sage: from sage.structure.set_factories import *
sage: from sage.structure.set_factories_example import *
sage: p = XYPair(Parent(), (0,1)); p
(0, 1)
Now, given \((a, b)\in S\) we want to consider the following subsets of \(S\)
The constraints considered here are admittedly trivial. In a realistic example, there would be much more of them. And for some sets of constraints no good enumeration algorithms would be known.
In Sage, those sets are constructed by passing the constraints to the factory. We first create the set with no constraints at all:
sage: XYPairs
Factory for XY pairs
sage: S = XYPairs(); S.list()
[(0, 0), (1, 0), ..., (4, 0), (0, 1), (1, 1), ..., (3, 4), (4, 4)]
sage: S.cardinality()
25
Let us construct \(S_2\), \(S^3\) and \(S_2^3\):
sage: Sx2 = XYPairs(x=2); Sx2.list()
[(2, 0), (2, 1), (2, 2), (2, 3), (2, 4)]
sage: Sy3 = XYPairs(y=3); Sy3.list()
[(0, 3), (1, 3), (2, 3), (3, 3), (4, 3)]
sage: S23 = XYPairs(x=2, y=3); S23.list()
[(2, 3)]
Set factories provide an alternative way to build subsets of an
already constructed set: each set constructed by a factory has a
method subset()
which accept new
constraints. Sets constructed by the factory or the
subset()
methods are identical:
sage: Sx2s = S.subset(x=2); Sx2 is Sx2s
True
sage: Sx2.subset(y=3) is S23
True
It is not possible to change an already given constraint:
sage: S23.subset(y=5)
Traceback (most recent call last):
...
ValueError: Duplicate value for constraints 'y': was 3 now 5
Constructing custom elements: policies
We now come to the point of factories: constructing custom elements. The
writer of XYPairs()
decided that, by default,
the parents Sx2
, Sy3
and S23
are facade for parent S
. This
means that each element constructed by those subsets behaves as if they where
directly constructed by S
itself:
sage: Sx2.an_element().parent()
AllPairs
sage: el = Sx2.an_element()
sage: el.parent() is S
True
sage: type(el) is S.element_class
True
This is not always desirable. The device which decides how to construct an
element is called a policy (see SetFactoryPolicy
). Each factory
should have a default policy. Here is the policy of
XYPairs()
:
sage: XYPairs._default_policy
Set factory policy for <class 'sage.structure.set_factories_example.XYPair'> with parent AllPairs[=Factory for XY pairs(())]
This means that with the current policy, the parent builds elements with class
XYPair
and parent AllPairs
which is itself constructed by calling the
factory XYPairs()
with constraints ()
. There
is a lot of flexibility to change that. We now illustrate how to make a few
different choices.
1  In a first use case, we want to add some methods to the constructed
elements. As illustration, we add here a new method sum
which returns
\(x+y\). We therefore create a new class for the elements which inherits from
XYPair
:
sage: class NewXYPair(XYPair):
....: def sum(self):
....: return sum(self.value)
Here is an instance of this class (with a dummy parent):
sage: el = NewXYPair(Parent(), (2,3))
sage: el.sum()
5
We now want to have subsets generating those new elements while still having a
single real parent (the one with no constraint) for each element. The
corresponding policy is called TopMostParentPolicy
. It takes three
parameters:
 the factory;
 the parameters for void constraint;
 the class used for elements.
Calling the factory with this policy returns a new set which builds its elements with the new policy:
sage: new_policy = TopMostParentPolicy(XYPairs, (), NewXYPair)
sage: NewS = XYPairs(policy=new_policy)
sage: el = NewS.an_element(); el
(0, 0)
sage: el.sum()
0
sage: el.parent() is NewS
True
sage: isinstance(el, NewXYPair)
True
Newly constructed subsets inherit the policy:
sage: NewS2 = NewS.subset(x=2)
sage: el2 = NewS2.an_element(); el2
(2, 0)
sage: el2.sum()
2
sage: el2.parent() is NewS
True
2  In a second use case, we want the elements to remember which
parent created them. The corresponding policy is called
SelfParentPolicy
. It takes only two parameters:
 the factory;
 the class used for elements.
Here is an example:
sage: selfpolicy = SelfParentPolicy(XYPairs, NewXYPair)
sage: SelfS = XYPairs(policy=selfpolicy)
sage: el = SelfS.an_element()
sage: el.parent() is SelfS
True
Now all subsets are the parent of the elements that they create:
sage: SelfS2 = SelfS.subset(x=2)
sage: el2 = SelfS2.an_element()
sage: el2.parent() is NewS
False
sage: el2.parent() is SelfS2
True
3  Finaly, a common use case is to construct simple python object which are
not Sage sage.structure.Element
. As an example, we show how to build
a parent TupleS
which construct pairs as tuple. The corresponding policy
is called BareFunctionPolicy
. It takes two parameters:
 the factory;
 the function called to construct the elements.
Here is how to do it:
sage: cons = lambda t, check: tuple(t) # ignore the check parameter
sage: tuplepolicy = BareFunctionPolicy(XYPairs, cons)
sage: P = XYPairs(x=2, policy=tuplepolicy)
sage: P.list()
[(2, 0), (2, 1), (2, 2), (2, 3), (2, 4)]
sage: el = P.an_element()
sage: type(el)
<... 'tuple'>
Here are the currently implemented policies:
FacadeParentPolicy
: reuse an existing parent together with its element_classTopMostParentPolicy
: use a parent created by the factory itself and provide a classElement
for it. In this case, we need to specify the set of constraints which build this parent taking the ownership of all elements and the class which will be used for theElement
.SelfParentPolicy
: provide systematicallyElement
and element_class and ensure that the parent isself
.BareFunctionPolicy
: instead of calling a class constructor element are passed to a function provided to the policy.
Todo
Generalize TopMostParentPolicy
to be able to have several
topmost parents.
Technicalities: how policies inform parents
Parents built from factories should inherit from
ParentWithSetFactory
. This class provide a few methods
related to factories and policies. The __init__
method of
ParentWithSetFactory
must be provided with the set of
constraints and the policy. A parent built from a factory must create
elements through a call to the method _element_constructor_
. The
current way in which policies inform parents how to builds their elements is
set by a few attributes. So the class must accept attribute
adding. The precise details of which attributes are set may be subject
to change in the future.
How to write a set factory
See also
set_factories_example
for an example of a factory.
Here are the specifications:
A parent built from a factory should
 inherit from
ParentWithSetFactory
. It should accept apolicy
argument and pass it verbatim to the__init__
method ofParentWithSetFactory
together with the set of constraints;  create its elements through calls to the method
_element_constructor_
;  define a method
ParentWithSetFactory.check_element
which checks if a built element indeed belongs to it. The method should accept an extra keyword parameter calledcheck
specifying which level of check should be performed. It will only be called whenbool(check)
evaluates toTrue
.
The constructor of the elements of a parent from a factory should:
 receive the parent as first mandatory argument;
 accept an extra optional keyword parameter called
check
which is meant to tell if the input must be checked or not. The precise meaning ofcheck
is intentionally left vague. The only intent is that ifbool(check)
evaluates toFalse
, no check is performed at all.
A factory should
 define a method
__call__
which is responsible for calling the appropriate parent constructor given the constraints;  define a method overloading
SetFactory.add_constraints()
which is responsible of computing the union of two sets of constraints;  optionally define a method or an attribute
_default_policy
passed to theParentWithSetFactory
if no policy is given to the factory.
Todo
There is currently no support for dealing with sets of constraints. The set factory and the parents must cooperate to consistently handle them. More support, together with a generic mechanism to select the appropriate parent class from the constraints, will be added as soon as we have gathered sufficiently enough usecases.
AUTHORS:
 Florent Hivert (20112012): initial revision

class
sage.structure.set_factories.
BareFunctionPolicy
(factory, constructor)¶ Bases:
sage.structure.set_factories.SetFactoryPolicy
Policy where element are constructed using a bare function.
INPUT:
factory
– an instance ofSetFactory
constructor
– a function
Given a factory
F
and a functionc
, returns a policy for parentP
creating element using the functionf
.EXAMPLES:
sage: from sage.structure.set_factories import BareFunctionPolicy sage: from sage.structure.set_factories_example import XYPairs sage: cons = lambda t, check: tuple(t) # ignore the check parameter sage: tuplepolicy = BareFunctionPolicy(XYPairs, cons) sage: P = XYPairs(x=2, policy=tuplepolicy) sage: el = P.an_element() sage: type(el) <... 'tuple'>

element_constructor_attributes
(constraints)¶ Return the element constructor attributes as per
SetFactoryPolicy.element_constructor_attributes()
.INPUT:
constraints
– a bunch of constraints

class
sage.structure.set_factories.
FacadeParentPolicy
(factory, parent)¶ Bases:
sage.structure.set_factories.SetFactoryPolicy
Policy for facade parent.
INPUT:
factory
– an instance ofSetFactory
parent
– an instance ofParent
Given a factory
F
and a classE
, returns a policy for parentP
creating elements as if they were created byparent
.EXAMPLES:
sage: from sage.structure.set_factories import SelfParentPolicy, FacadeParentPolicy sage: from sage.structure.set_factories_example import XYPairs, XYPair
We create a custom standard parent
P
:sage: selfpolicy = SelfParentPolicy(XYPairs, XYPair) sage: P = XYPairs(x=2, policy=selfpolicy) sage: pol = FacadeParentPolicy(XYPairs, P) sage: P2 = XYPairs(x=2, y=3, policy=pol) sage: el = P2.an_element() sage: el.parent() is P True sage: type(el) is P.element_class True
If
parent
is itself a facade parent, then transitivity is correctly applied:sage: P = XYPairs() sage: P2 = XYPairs(x=2) sage: P2.category() Category of facade finite enumerated sets sage: pol = FacadeParentPolicy(XYPairs, P) sage: P23 = XYPairs(x=2, y=3, policy=pol) sage: el = P2.an_element() sage: el.parent() is P True sage: type(el) is P.element_class True

element_constructor_attributes
(constraints)¶ Return the element constructor attributes as per
SetFactoryPolicy.element_constructor_attributes()
.INPUT:
constraints
– a bunch of constraints

class
sage.structure.set_factories.
ParentWithSetFactory
(constraints, policy, category=None)¶ Bases:
sage.structure.parent.Parent
Abstract class for parent belonging to a set factory.
INPUT:
constraints
– a set of constraintspolicy
– the policy for element constructioncategory
– the category of the parent (default toNone
)
Depending on the constraints and the policy, initialize the parent in a proper category to set up element construction.
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs, PairsX_ sage: P = PairsX_(3, XYPairs._default_policy) sage: P is XYPairs(3) True sage: P.category() Category of facade finite enumerated sets

check_element
(x, check)¶ Check that
x
verifies the constraints ofself
.INPUT:
x
– an instance ofself.element_class
.check
– the level of checking to be performed (usually a boolean).
This method may assume that
x
was properly constructed byself
or a possible superset ofself
for whichself
is a facade. It should return nothing ifx
verifies the constraints and raise aValueError
explaining which constraintsx
fails otherwise.The method should accept an extra parameter check specifying which level of check should be performed. It will only be called when
bool(check)
evaluates toTrue
.Todo
Should we always call check element and let it decide which check has to be performed ?
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs sage: S = XYPairs() sage: el = S((2,3)) sage: S.check_element(el, True) sage: XYPairs(x=2).check_element(el, True) sage: XYPairs(x=3).check_element(el, True) Traceback (most recent call last): ... ValueError: Wrong first coordinate sage: XYPairs(y=4).check_element(el, True) Traceback (most recent call last): ... ValueError: Wrong second coordinate

constraints
()¶ Return the constraints defining
self
.Note
Currently there is no specification on how constraints are passed as arguments.
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs sage: XYPairs().constraints() () sage: XYPairs(x=3).constraints() (3, None) sage: XYPairs(y=2).constraints() (None, 2)

facade_policy
()¶ Return the policy for parent facade for
self
.EXAMPLES:
sage: from sage.structure.set_factories import SelfParentPolicy sage: from sage.structure.set_factories_example import XYPairs, XYPair
We create a custom standard parent
P
:sage: selfpolicy = SelfParentPolicy(XYPairs, XYPair) sage: P = XYPairs(x=2, policy=selfpolicy) sage: P.facade_policy() Set factory policy for facade parent {(2, b)  b in range(5)}
Now passing
P.facade_policy()
creates parent which are facade forP
:sage: P3 = XYPairs(x=2, y=3, policy=P.facade_policy()) sage: P3.facade_for() == (P,) True sage: el = P3.an_element() sage: el.parent() is P True

factory
()¶ Return the factory which built
self
.EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs sage: XYPairs().factory() is XYPairs True sage: XYPairs(x=3).factory() is XYPairs True

policy
()¶ Return the policy used when
self
was created.EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs sage: XYPairs().policy() Set factory policy for <class 'sage.structure.set_factories_example.XYPair'> with parent AllPairs[=Factory for XY pairs(())] sage: XYPairs(x=3).policy() Set factory policy for <class 'sage.structure.set_factories_example.XYPair'> with parent AllPairs[=Factory for XY pairs(())]

subset
(*args, **options)¶ Return a subset of
self
by adding more constraints.Note
Currently there is no specification on how constraints are passed as arguments.
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs sage: S = XYPairs() sage: S3 = S.subset(x=3) sage: S3.list() [(3, 0), (3, 1), (3, 2), (3, 3), (3, 4)]

class
sage.structure.set_factories.
SelfParentPolicy
(factory, Element)¶ Bases:
sage.structure.set_factories.SetFactoryPolicy
Policy where each parent is a standard parent.
INPUT:
factory
– an instance ofSetFactory
Element
– a subclass ofElement
Given a factory
F
and a classE
, returns a policy for parentP
creating elements in classE
and parentP
itself.EXAMPLES:
sage: from sage.structure.set_factories import SelfParentPolicy sage: from sage.structure.set_factories_example import XYPairs, XYPair, Pairs_Y sage: pol = SelfParentPolicy(XYPairs, XYPair) sage: S = Pairs_Y(3, pol) sage: el = S.an_element() sage: el.parent() is S True sage: class Foo(XYPair): pass sage: pol = SelfParentPolicy(XYPairs, Foo) sage: S = Pairs_Y(3, pol) sage: el = S.an_element() sage: isinstance(el, Foo) True

element_constructor_attributes
(constraints)¶ Return the element constructor attributes as per
SetFactoryPolicy.element_constructor_attributes()
INPUT:
constraints
– a bunch of constraints

class
sage.structure.set_factories.
SetFactory
¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.sage_object.SageObject
This class is currently just a stub that we will be using to add more structures on factories.

add_constraints
(cons, *args, **opts)¶ Add constraints to the set of constraints \(cons\).
Should return a set of constraints.
Note
Currently there is no specification on how constraints are passed as arguments.
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs sage: XYPairs.add_constraints((3,),((None, 2), {})) (3, 2) sage: XYPairs.add_constraints((3,),((None, None), {'y': 2})) (3, 2)


class
sage.structure.set_factories.
SetFactoryPolicy
(factory)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.sage_object.SageObject
Abstract base class for policies.
A policy is a device which is passed to a parent inheriting from
ParentWithSetFactory
in order to setup the element construction framework.INPUT:
factory
– aSetFactory
Warning
This class is a base class for policies, one should not try to create instances.

element_constructor_attributes
(constraints)¶ Element constructor attributes.
INPUT:
constraints
– a bunch of constraints
Should return the attributes that are prerequisite for element construction. This is coordinated with
ParentWithSetFactory._element_constructor_()
. Currently two standard attributes are provided infacade_element_constructor_attributes()
andself_element_constructor_attributes()
. You should return the one needed depending on the given constraints.EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs, XYPair sage: pol = XYPairs._default_policy sage: pol.element_constructor_attributes(()) {'Element': <class 'sage.structure.set_factories_example.XYPair'>, '_parent_for': 'self'} sage: pol.element_constructor_attributes((1)) {'_facade_for': AllPairs, '_parent_for': AllPairs, 'element_class': <class 'sage.structure.set_factories_example.AllPairs_with_category.element_class'>}

facade_element_constructor_attributes
(parent)¶ Element Constructor Attributes for facade parent.
The list of attributes which must be set during the init of a facade parent with factory.
INPUT:
parent
– the actual parent for the elements
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs, XYPair sage: pol = XYPairs._default_policy sage: pol.facade_element_constructor_attributes(XYPairs()) {'_facade_for': AllPairs, '_parent_for': AllPairs, 'element_class': <class 'sage.structure.set_factories_example.AllPairs_with_category.element_class'>}

factory
()¶ Return the factory for
self
.EXAMPLES:
sage: from sage.structure.set_factories import SetFactoryPolicy, SelfParentPolicy sage: from sage.structure.set_factories_example import XYPairs, XYPair sage: XYPairs._default_policy.factory() Factory for XY pairs sage: XYPairs._default_policy.factory() is XYPairs True

self_element_constructor_attributes
(Element)¶ Element Constructor Attributes for non facade parent.
The list of attributes which must be set during the init of a non facade parent with factory.
INPUT:
Element
– the class used for the elements
EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs, XYPair sage: pol = XYPairs._default_policy sage: pol.self_element_constructor_attributes(XYPair) {'Element': <class 'sage.structure.set_factories_example.XYPair'>, '_parent_for': 'self'}

class
sage.structure.set_factories.
TopMostParentPolicy
(factory, top_constraints, Element)¶ Bases:
sage.structure.set_factories.SetFactoryPolicy
Policy where the parent of the elements is the topmost parent.
INPUT:
factory
– an instance ofSetFactory
top_constraints
– the empty set of constraints.Element
– a subclass ofElement
Given a factory
F
and a classE
, returns a policy for parentP
creating element in classE
and parentfactory(*top_constraints, policy)
.EXAMPLES:
sage: from sage.structure.set_factories_example import XYPairs, XYPair sage: P = XYPairs(); P.policy() Set factory policy for <class 'sage.structure.set_factories_example.XYPair'> with parent AllPairs[=Factory for XY pairs(())]

element_constructor_attributes
(constraints)¶ Return the element constructor attributes as per
SetFactoryPolicy.element_constructor_attributes()
.INPUT:
constraints
– a bunch of constraints