Tutorial: Objects and Classes in Python and Sage

Author: Florent Hivert <florent.hivert@univ-rouen.fr>

This tutorial is an introduction to object-oriented programming in Python and Sage. It requires basic knowledge about imperative/procedural programming (the most common programming style) – that is, conditional instructions, loops, functions (see the “Programming” section of the Sage tutorial) – but no further knowledge about objects and classes is assumed. It is designed as an alternating sequence of formal introduction and exercises. Solutions to the exercises are given at the end.

Foreword: variables, names and objects

As an object-oriented language, Python’s ‘’variables’’ behavior may be surprising for people used to imperative languages like C or Maple. The reason is that they are not variables but names.

The following explanation is borrowed from David Goodger.

Other languages have “variables”

In many other languages, assigning to a variable puts a value into a box.

int a = 1;

Box “a” now contains an integer 1.

Assigning another value to the same variable replaces the contents of the box:

a = 2;

Now box “a” contains an integer 2.

Assigning one variable to another makes a copy of the value and puts it in the new box:

int b = a;

“b” is a second box, with a copy of integer 2. Box “a” has a separate copy.

Python has “names”

In Python, a “name” or “identifier” is like a parcel tag (or nametag) attached to an object.

a = 1

Here, an integer 1 object has a tag labelled “a”.

If we reassign to “a”, we just move the tag to another object:

a = 2

Now the name “a” is attached to an integer 2 object.

The original integer 1 object no longer has a tag “a”. It may live on, but we can’t get to it through the name “a”. (When an object has no more references or tags, it is removed from memory.)

If we assign one name to another, we’re just attaching another nametag to an existing object:

b = a

The name “b” is just a second tag bound to the same object as “a”.

Although we commonly refer to “variables” even in Python (because it’s common terminology), we really mean “names” or “identifiers”. In Python, “variables” are nametags for values, not labelled boxes.

Warning

As a consequence, when there are two tags “a” and “b” on the same object, modifying the object tagged “b” also modifies the object tagged “a”:

sage: a = [1,2,3]
sage: b = a
sage: b[1] = 0
sage: a
[1, 0, 3]

Note that reassigning the tag “b” (rather than modifying the object with that tag) doesn’t affect the object tagged “a”:

sage: b = 7
sage: b
7
sage: a
[1, 0, 3]

Object-oriented programming paradigm

The object-oriented programming paradigm relies on the two following fundamental rules:

1. Anything of the real (or mathematical) world which needs to be manipulated by the computer is modeled by an object.
2. Each object is an instance of some class.

At this point, those two rules are a little meaningless, so let’s give some more or less precise definitions of the terms:

---

object

a portion of memory which contains the information needed to model the real world thing.

class

defines the data structure used to store the objects which are instances of the class together with their behavior.

---

Let’s start with some examples: We consider the vector space over \(\QQ\) whose basis is indexed by permutations, and a particular element in it:

sage: F = CombinatorialFreeModule(QQ, Permutations())
sage: el = 3*F([1,3,2])+ F([1,2,3])
sage: el
B[[1, 2, 3]] + 3*B[[1, 3, 2]]

(For each permutation, say [1, 3, 2], the corresponding element in F is denoted by B[[1, 3, 2]] – in a CombinatorialFreeModule, if an element is indexed by x, then by default its print representation is B[x].)

In Python, everything is an object so there isn’t any difference between types and classes. One can get the class of the object el by:

sage: type(el)
<class 'sage.combinat.free_module.CombinatorialFreeModule_with_category.element_class'>

As such, this is not very informative. We’ll come back to it later. The data associated to objects are stored in so-called attributes. They are accessed through the syntax obj.attribute_name. For an element of a combinatorial free module, the main attribute is called _monomial_coefficients. It is a dictionary associating coefficients to indices:

sage: el._monomial_coefficients
{[1, 2, 3]: 1, [1, 3, 2]: 3}

Modifying the attribute modifies the objects:

sage: el._monomial_coefficients[Permutation([3,2,1])] = 1/2
sage: el
B[[1, 2, 3]] + 3*B[[1, 3, 2]] + 1/2*B[[3, 2, 1]]

Warning

as a user, you are not supposed to do such a modification by yourself (see note on private attributes below).

As an element of a vector space, el has a particular behavior:

sage: 2*el
2*B[[1, 2, 3]] + 6*B[[1, 3, 2]] + B[[3, 2, 1]]
sage: sorted(el.support())
[[1, 2, 3], [1, 3, 2], [3, 2, 1]]
sage: el.coefficient([1, 2, 3])
1

The behavior is defined through methods (support, coefficient). Note that this is true even for equality, printing or mathematical operations. For example, the call a == b actually is translated to the method call a.__eq__(b). The names of those special methods which are usually called through operators are fixed by the Python language and are of the form __name__. Examples include __eq__ and __le__ for operators == and <=, __repr__ (see Sage specifics about classes) for printing, __add__ and __mult__ for operators + and *. See http://docs.python.org/library/ for a complete list.

sage: el.__eq__(F([1,3,2]))
False
sage: el.__repr__()
'B[[1, 2, 3]] + 3*B[[1, 3, 2]] + 1/2*B[[3, 2, 1]]'
sage: el.__mul__(2)
2*B[[1, 2, 3]] + 6*B[[1, 3, 2]] + B[[3, 2, 1]]

Note

We now create a custom Element class to explain the details of how attributes work in Python (you can ignore the parent in the code below, that is not relevant here):

sage: from sage.structure.element import Element
sage: class MyElt(Element):
....:     def __init__(self, parent, val):
....:         super(MyElt, self).__init__(parent)
....:         self.value = val
sage: el = MyElt(val=42, parent=ZZ)
sage: el
Generic element of a structure

The class is stored in a particular attribute called __class__, and the normal attributes are stored in a dictionary called __dict__:

sage: el.__dict__
{'value': 42}
sage: el.__class__
<class '__main__.MyElt'>

Some particular actions modify the attributes of el:

sage: el.rename("bla")
sage: el
bla
sage: el.__dict__
{'__custom_name': 'bla', 'value': 42}

Lots of Sage objects are not Python objects but compiled Cython objects. Python sees them as builtin objects and you do not have access to some of their internal data structure. For example, the base class Element stores the parent of el as a Cython attribute _parent but it does not appear in the __dict__ and we cannot access it from Python.

Some examples of Cython classes (technically, extension types) in Sage include integers and permutation group elements. These do not have a __dict__ at all:

sage: e = Integer(9)
sage: type(e)
<type 'sage.rings.integer.Integer'>
sage: e.__dict__
Traceback (most recent call last):
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute '__dict__'

sage: id4 = SymmetricGroup(4).one()
sage: type(id4)
<type 'sage.groups.perm_gps.permgroup_element.SymmetricGroupElement'>
sage: id4.__dict__
Traceback (most recent call last):
...
AttributeError: 'sage.groups.perm_gps.permgroup_element.SymmetricGroupElement' object has no attribute '__dict__'

Note

Each object corresponds to a portion of memory called its identity in Python. You can get the identity using id:

sage: el = Integer(9)
sage: id(el)  # random
139813642977744
sage: el1 = el; id(el1) == id(el)
True
sage: el1 is el
True

In Python (and therefore in Sage), two objects with the same identity will be equal, but the converse is not true in general. Thus the identity function is different from mathematical identity:

sage: el2 = Integer(9)
sage: el2 == el1
True
sage: el2 is el1
False
sage: id(el2) == id(el)
False

Summary

To define some object, you first have to write a class. The class will define the methods and the attributes of the object.

method

particular kind of function associated with an object used to get information about the object or to manipulate it.

attribute

variable where information about the object is stored.

An example: glass of beverage in a restaurant

Let’s write a small class about glasses in a restaurant:

sage: class Glass(object):
....:     def __init__(self, size):
....:         assert size > 0
....:         self._size = float(size)  # an attribute
....:         self._content = float(0.0)  # another attribute
....:     def __repr__(self):
....:         if self._content == 0.0:
....:             return "An empty glass of size %s"%(self._size)
....:         else:
....:             return "A glass of size %s cl containing %s cl of water"%(
....:                     self._size, self._content)
....:     def fill(self):
....:         self._content = self._size
....:     def empty(self):
....:         self._content = float(0.0)

Let’s create a small glass:

sage: myGlass = Glass(10); myGlass
An empty glass of size 10.0
sage: myGlass.fill(); myGlass
A glass of size 10.0 cl containing 10.0 cl of water
sage: myGlass.empty(); myGlass
An empty glass of size 10.0

Some comments:

1. The definition of the class Glass defines two attributes, _size and _content. It defines four methods, __init__, __repr__, fill, and empty. (Any instance of this class will also have other attributes and methods, inherited from the class object. See Inheritance below.)
2. The method __init__ is used to initialize the object: it is used by the so-called constructor of the class that is executed when calling Glass(10).
3. The method __repr__ returns a string which is used to print the object, for example in this case when evaluating myGlass.

Note

Private Attributes

• Most of the time, in order to ensure consistency of the data structures, the user is not supposed to directly change certain attributes of an object. Those attributes are called private. Since there is no mechanism to ensure privacy in Python, the convention is the following: private attributes have names beginning with an underscore.
• As a consequence, attribute access is only made through methods. Methods for reading or writing a private attribute are called accessors.
• Methods which are only for internal use are also prefixed with an underscore.

Exercises

1. Add a method is_empty which returns true if a glass is empty.
2. Define a method drink with a parameter amount which allows one to partially drink the water in the glass. Raise an error if one asks to drink more water than there is in the glass or a negative amount of water.
3. Allows the glass to be filled with wine, beer or another beverage. The method fill should accept a parameter beverage. The beverage is stored in an attribute _beverage. Update the method __repr__ accordingly.
4. Add an attribute _clean and methods is_clean and wash. At the creation a glass is clean, as soon as it’s filled it becomes dirty, and it must be washed to become clean again.
5. Test everything.
6. Make sure that everything is tested.
7. Test everything again.

Inheritance

The problem: objects of different classes may share a common behavior.

For example, if one wants to deal with different dishes (forks, spoons, …), then there is common behavior (becoming dirty and being washed). So the different classes associated to the different kinds of dishes should have the same clean, is_clean and wash methods. But copying and pasting code is very bad for maintenance: mistakes are copied, and to change anything one has to remember the location of all the copies. So there is a need for a mechanism which allows the programmer to factorize the common behavior. It is called inheritance or sub-classing: one writes a base class which factorizes the common behavior and then reuses the methods from this class.

We first write a small class ‘’AbstractDish’’ which implements the “clean-dirty-wash” behavior:

sage: class AbstractDish(object):
....:     def __init__(self):
....:         self._clean = True
....:     def is_clean(self):
....:         return self._clean
....:     def state(self):
....:         return "clean" if self.is_clean() else "dirty"
....:     def __repr__(self):
....:         return "An unspecified %s dish"%self.state()
....:     def _make_dirty(self):
....:         self._clean = False
....:     def wash(self):
....:         self._clean = True

Now one can reuse this behavior within a class Spoon:

sage: class Spoon(AbstractDish):  # Spoon inherits from AbstractDish
....:     def __repr__(self):
....:         return "A %s spoon"%self.state()
....:     def eat_with(self):
....:         self._make_dirty()

Let’s test it:

sage: s = Spoon(); s
A clean spoon
sage: s.is_clean()
True
sage: s.eat_with(); s
A dirty spoon
sage: s.is_clean()
False
sage: s.wash(); s
A clean spoon

Summary

1. Any class can reuse the behavior of another class. One says that the subclass inherits from the superclass or that it derives from it.

2. Any instance of the subclass is also an instance of its superclass:

   sage: type(s)
   <class '__main__.Spoon'>
   sage: isinstance(s, Spoon)
   True
   sage: isinstance(s, AbstractDish)
   True
   

3. If a subclass redefines a method, then it replaces the former one. One says that the subclass overloads the method. One can nevertheless explicitly call the hidden superclass method.

   sage: s.__repr__()
   'A clean spoon'
   sage: Spoon.__repr__(s)
   'A clean spoon'
   sage: AbstractDish.__repr__(s)
   'An unspecified clean dish'
   

Note

Advanced superclass method call

Sometimes one wants to call an overloaded method without knowing in which class it is defined. To do this, use the super operator:

sage: super(Spoon, s).__repr__()
'An unspecified clean dish'

A very common usage of this construct is to call the __init__ method of the superclass:

sage: class Spoon(AbstractDish):
....:     def __init__(self):
....:         print("Building a spoon")
....:         super(Spoon, self).__init__()
....:     def __repr__(self):
....:         return "A %s spoon"%self.state()
....:     def eat_with(self):
....:         self._make_dirty()
sage: s = Spoon()
Building a spoon
sage: s
A clean spoon

Exercises

1. Modify the class Glasses so that it inherits from Dish.
2. Write a class Plate whose instance can contain any meal together with a class Fork. Avoid as much as possible code duplication (hint: you can write a factorized class ContainerDish).
3. Test everything.

Sage specifics about classes

Compared to Python, Sage has particular ways to handle objects:

• Any classes for mathematical objects in Sage should inherit from SageObject rather than from object. Most of the time, they actually inherit from a subclass such as Parent or Element.
• Printing should be done through _repr_ instead of __repr__ to allow for renaming.
• More generally, Sage-specific special methods are usually named _meth_ rather than __meth__. For example, lots of classes implement _hash_ which is used and cached by __hash__. In the same vein, elements of a group usually implement _mul_, so that there is no need to take care about coercions as they are done in __mul__.

For more details, see the Sage Developer’s Guide.

Solutions to the exercises

1. Here is a solution to the first exercise:

   sage: class Glass(object):
   ....:     def __init__(self, size):
   ....:         assert size > 0
   ....:         self._size = float(size)
   ....:         self.wash()
   ....:     def __repr__(self):
   ....:         if self._content == 0.0:
   ....:             return "An empty glass of size %s"%(self._size)
   ....:         else:
   ....:             return "A glass of size %s cl containing %s cl of %s"%(
   ....:                     self._size, self._content, self._beverage)
   ....:     def content(self):
   ....:         return self._content
   ....:     def beverage(self):
   ....:         return self._beverage
   ....:     def fill(self, beverage = "water"):
   ....:         if not self.is_clean():
   ....:             raise ValueError("Don't want to fill a dirty glass")
   ....:         self._clean = False
   ....:         self._content = self._size
   ....:         self._beverage = beverage
   ....:     def empty(self):
   ....:         self._content = float(0.0)
   ....:     def is_empty(self):
   ....:         return self._content == 0.0
   ....:     def drink(self, amount):
   ....:         if amount <= 0.0:
   ....:             raise ValueError("amount must be positive")
   ....:         elif amount > self._content:
   ....:             raise ValueError("not enough beverage in the glass")
   ....:         else:
   ....:             self._content -= float(amount)
   ....:     def is_clean(self):
   ....:         return self._clean
   ....:     def wash(self):
   ....:         self._content = float(0.0)
   ....:         self._beverage = None
   ....:         self._clean = True
   

2. Let’s check that everything is working as expected:

   sage: G = Glass(10.0)
   sage: G
   An empty glass of size 10.0
   sage: G.is_empty()
   True
   sage: G.drink(2)
   Traceback (most recent call last):
   ...
   ValueError: not enough beverage in the glass
   sage: G.fill("beer")
   sage: G
   A glass of size 10.0 cl containing 10.0 cl of beer
   sage: G.is_empty()
   False
   sage: G.is_clean()
   False
   sage: G.drink(5.0)
   sage: G
   A glass of size 10.0 cl containing 5.0 cl of beer
   sage: G.is_empty()
   False
   sage: G.is_clean()
   False
   sage: G.drink(5)
   sage: G
   An empty glass of size 10.0
   sage: G.is_clean()
   False
   sage: G.fill("orange juice")
   Traceback (most recent call last):
   ...
   ValueError: Don't want to fill a dirty glass
   sage: G.wash()
   sage: G
   An empty glass of size 10.0
   sage: G.fill("orange juice")
   sage: G
   A glass of size 10.0 cl containing 10.0 cl of orange juice
   

3. Here is the solution to the second exercice:

   sage: class AbstractDish(object):
   ....:     def __init__(self):
   ....:         self._clean = True
   ....:     def is_clean(self):
   ....:         return self._clean
   ....:     def state(self):
   ....:         return "clean" if self.is_clean() else "dirty"
   ....:     def __repr__(self):
   ....:         return "An unspecified %s dish"%self.state()
   ....:     def _make_dirty(self):
   ....:         self._clean = False
   ....:     def wash(self):
   ....:         self._clean = True
   
   
   sage: class ContainerDish(AbstractDish):
   ....:     def __init__(self, size):
   ....:         assert size > 0
   ....:         self._size = float(size)
   ....:         self._content = float(0)
   ....:         super(ContainerDish, self).__init__()
   ....:     def content(self):
   ....:         return self._content
   ....:     def empty(self):
   ....:         self._content = float(0.0)
   ....:     def is_empty(self):
   ....:         return self._content == 0.0
   ....:     def wash(self):
   ....:         self._content = float(0.0)
   ....:         super(ContainerDish, self).wash()
   
   
   sage: class Glass(ContainerDish):
   ....:     def __repr__(self):
   ....:         if self._content == 0.0:
   ....:             return "An empty glass of size %s"%(self._size)
   ....:         else:
   ....:             return "A glass of size %s cl containing %s cl of %s"%(
   ....:                     self._size, self._content, self._beverage)
   ....:     def beverage(self):
   ....:         return self._beverage
   ....:     def fill(self, beverage = "water"):
   ....:         if not self.is_clean():
   ....:             raise ValueError("Don't want to fill a dirty glass")
   ....:         self._make_dirty()
   ....:         self._content = self._size
   ....:         self._beverage = beverage
   ....:     def drink(self, amount):
   ....:         if amount <= 0.0:
   ....:             raise ValueError("amount must be positive")
   ....:         elif amount > self._content:
   ....:             raise ValueError("not enough beverage in the glass")
   ....:         else:
   ....:             self._content -= float(amount)
   ....:     def wash(self):
   ....:         self._beverage = None
   ....:         super(Glass, self).wash()
   

4. Let’s check that everything is working as expected:

   sage: G = Glass(10.0)
   sage: G
   An empty glass of size 10.0
   sage: G.is_empty()
   True
   sage: G.drink(2)
   Traceback (most recent call last):
   ...
   ValueError: not enough beverage in the glass
   sage: G.fill("beer")
   sage: G
   A glass of size 10.0 cl containing 10.0 cl of beer
   sage: G.is_empty()
   False
   sage: G.is_clean()
   False
   sage: G.drink(5.0)
   sage: G
   A glass of size 10.0 cl containing 5.0 cl of beer
   sage: G.is_empty()
   False
   sage: G.is_clean()
   False
   sage: G.drink(5)
   sage: G
   An empty glass of size 10.0
   sage: G.is_clean()
   False
   sage: G.fill("orange juice")
   Traceback (most recent call last):
   ...
   ValueError: Don't want to fill a dirty glass
   sage: G.wash()
   sage: G
   An empty glass of size 10.0
   sage: G.fill("orange juice")
   sage: G
   A glass of size 10.0 cl containing 10.0 cl of orange juice
   

Todo

give the example of the class Plate.

That all folks !