5.2.1 Types and subtypes

Conceptually, a type is a set of values (i.e., possible states or objects) and a set of operations defined on the values in that set.

Similarly, a type S is (a behavioral) subtype of type T if the set of values of type S is a “subset” of the values in set T an set of operations of type S is a “superset” of the operations of type T. That is, we can safely substitute elements of subtype S for elements of type T because S’s operations behave the “same” as T’s operations.

This is known as the Liskov Substitution Principle [12,20].

Consider a type representing all furniture and a type representing all chairs. In general, we consider the set of chairs to be a subset of the set of furniture. A chair should have all the general characteristics of furniture, but it may have additional characteristics specific to chairs.

If we can perform an operation on furniture in general, we should be able to perform the same operation on a chair under the same circumstances and get the same result. Of course, there may be additional operations we can perform on chairs that are not applicable to furniture in general.

Thus the type of all chairs is a subtype of the type of all furniture according to the Liskov Substitution Principle.

5.2.2 Constants, variables, and expressions

Now consider the types of the basic program elements.

A constant has whatever types it is defined to have in the context in which it is used. For example, the constant symbol 1 might represent an integer, a real number, a complex number, a single bit, etc., depending upon the context.

A variable has whatever types its value has in a particular context and at a particular time during execution. The type may be constrained by a declaration of the variable.

An expression has whatever types its evaluation yields based on the types of the variables, constants, and operations from which it is constructed.

5.2.3 Static and dynamic

In a statically typed language, the types of a variable or expression can be determined from the program source code and checked at “compile time” (i.e., during the syntactic and semantic processing in the front-end of a language processor). Such languages may require at least some of the types of variables or expressions to be declared explicitly, while others may be inferred implicitly from the context.

Java, Scala, and Haskell are examples of statically typed languages.

In a dynamically typed language, the specific types of a variable or expression cannot be determined at “compile time” but can be checked at runtime.

Lisp, Python, JavaScript, and Lua are examples of dynamically typed languages.

Of course, most languages use a mixture of static and dynamic typing. For example, Java objects defined within an inheritance hierarchy must be bound dynamically to the appropriate operations at runtime. Also Java objects declared of type Object (the root class of all user-defined classes) often require explicit runtime checks or coercions.

5.2.4 Nominal and structural

In a language with nominal typing, the type of value is based on the type name assigned when the value is created. Two values have the same type if they have the same type name. A type S is a subtype of type T only if S is explicitly declared to be a subtype of T.

For example, Java is primarily a nominally typed language. It assigns types to an object based on the name of the class from which the object is instantiated and the superclasses extended and interfaces implemented by that class.

However, Java does not guarantee that subtypes satisfy the Liskov Substitution Principle. For example, a subclass might not implement an operation in a manner that is compatible with the superclass. (The behavior of subclass objects are this different from the behavior of superclass objects.) Ensuring that Java subclasses preserve the Substitution Principle is considered good programming practice in most circumstances.

In a language with structural typing, the type of a value is based on the structure of the value. Two values have the same type if they have the “same” structure; that is, they have the same public data attributes and operations and these are themselves of compatible types.

In structurally typed languages, a type S is a subtype of type T only if S has all the public data values and operations of type T and the data values and operations are themselves of compatible types. Subtype S may have additional data values and operations not in T.

Haskell is an example of a primarily structurally typed language.

5.2.5 Polymorphic operations

Polymorphism refers to the property of having “many shapes”. In programming languages, we are primarily interested in how polymorphic function names (or operator symbols) are associated with implementations of the functions (or operations).

In general, two primary kinds of polymorphic operations exist in programming languages:

1. Ad hoc polymorphism, in which the same function name (or operator symbol) can denote different implementations depending upon how it is used in an expression. That is, the implementation invoked depends upon the types of function’s arguments and return value.

   There are two subkinds of ad hoc polymorphism.

   1. Overloading refers to ad hoc polymorphism in which the language’s compiler or interpreter determines the appropriate implementation to invoke using information from the context. In statically typed languages, overloaded names and symbols can usually be bound to the intended implementation at compile time based on the declared types of the entities. They exhibit early binding.

      Consider the language Java. It overloads a few operator symbols, such as using the + symbol for both addition of numbers and concatenation of strings. Java also overloads calls of functions defined with the same name but different signatures (patterns of parameter types and return value). Java does not support user-defined operator overloading; C++ does.

      Haskell’s type class mechanism implements overloading polymorphism in Haskell. There are similar mechanisms in other languages such as Scala and Rust.

   2. Subtyping (also known as subtype polymorphism or inclusion polymorphism) refers to ad hoc polymorphism in which the appropriate implementation is determined by searching a hierarchy of types. The function may be defined in a supertype and redefined (overridden) in subtypes. Beginning with the actual types of the data involved, the program searches up the type hierarchy to find the appropriate implementation to invoke. This usually occurs at runtime, so this exhibits late binding.

      The object-oriented programming community often refers to inheritance-based subtype polymorphism as simply polymorphism. This is the polymorphism associated with the class structure in Java.

      Haskell does not support subtyping. Its type classes do support class extension, which enables one class to inherit the properties of another. However, Haskell’s classes are not types.

2. Parametric polymorphism, in which the same implementation can be used for many different types. In most cases, the function (or class) implementation is stated in terms of one or more type parameters. In statically typed languages, this binding can usually be done at compile time (i.e., exhibiting early binding).

   The object-oriented programming (e.g., Java) community often calls this type of polymorphism generics or generic programming.

   The functional programming (e.g., Haskell) community often calls this simply polymorphism.

5.2.6 Polymorphic variables

A polymorphic variable is a variable that can “hold” values of different types during program execution.

For example, a variable in a dynamically typed language (e.g., Python) is polymorphic. It can potentially “hold” any value. The variable takes on the type of whatever value it “holds” at a particular point during execution.

Also, a variable in a nominally and statically typed, object-oriented language (e.g., Java) is polymorphic. It can “hold” a value its declared type or of any of the subtypes of that type. The variable is declared with a static type; its value has a dynamic type.

A variable that is a parameter of a (parametrically) polymorphic function is polymorphic. It may be bound to different types on different calls of the function.

5.3.1 Objects

Python is object-based [7, Ch. 3]; it treats all data as objects.

A Python object has the following essential characteristics of objects [7, Ch. 3]:

1. a state (value) drawn from a set of possible values

   The state may consist of several distinct data attributes. In this case, the set of possible values is the Cartesian product of the sets of possible values of each attribute.

2. a set of operations that access and/or mutate the state

3. a unique identity (e.g., address in memory)

A Python pbject has one of the two important but nonessential characteristics of objects [7, Ch. 3]. Python does:

4. not enforce encapsulation of the state within the object, instead relying upon programming conventions and name obfuscation to hide private information

5. exhibit an independent lifecycle (i.e., has a different lifetime than the code that created it)

As we see in Chapter 7, each object has a distinct dictionary, the directory, that maps the local names to the data attributes and operations.

Python typically uses dot notation to access an object’s data attributes and operations:

• obj.data accesses the attribute data of obj

• obj.op accesses operation op of obj

• obj.op() invokes operation op of obj, passing any arguments in a comma-separated list between the parentheses

Some objects are immutable and others are mutable. The states (i.e., values) of:

• immutable objects (e.g., numbers, booleans, strings, and tuples) cannot be changed after creation

• mutable objects (e.g., lists, dictionaries, and sets) can be changed in place after creation

Caveat: We cannot modify a Python tuple’s structure (i.e., length) after its creation. However, if the components of a tuple are themselves mutable objects, they can be changed in-place.

All Python objects have a type.

5.3.2 Types

In terms of the discussion in Section {#sec:type-system-concepts}, all Python objects can be considered as having one or more conceptual types at a particular point in time. The types may change over time because the program can change the possible set of data attributes and operations associated with the object.

A Python variable is bound to an object by an assignment statement or its equivalent. Python variables are thus dynamically typed, as are Python expressions.

Although a Python program usually constructs an object within a particular nominal type hierarchy (e.g., as an instance of a class), this may not fully describe the type of the object, even initially. And the ability to dynamically add, remove, and modify attributes (both data fields and operations) means the type can change as the program executes.

The type of a Python object is determined by what it can do—what data it can hold and what operations it can perform on that data—rather than how it was created. We sometimes call this dynamic, structural typing approach duck typing. (If it walks like a duck and quacks like a duck, then it is a duck, even if is declared as a snake.)

For example, we can say that any object is of an iterable type if it implements an __iter__ operation that returns a valid iterator object. An iterator object must implement a __next__ operation that retrieves the next element of the “collection” and must raise a StopIteration exception if no more elements are available.

In Python, we sometimes refer to a type like iterable as a protocol. That is, it is an, perhaps informal, interface that objects are expected to satisfy in certain circumstances.

5.4.1 Singleton types

Python has single-element types used for special purposes. These include None and NotImplemented.

5.4.2 Number types

Core Python supports four types of numbers:

• integers
• real numbers
• complex numbers
• Booleans

    >>> type(1) 
    <class 'int'>
    >>> type(-14) 
    <class 'int'>
    >>> x = 2
    >>> type(x) 
    <class 'int'>
    >>> type(99999999999999999999999999999999)
    <class 'int'>

    >>> type(1.01) 
    <class 'float'>
    >>> type(-14.3) 
    <class 'float'>
    >>> x = 2 
    >>> type(x) 
    <class 'int'>
    >>> y = 2.0
    >>> type(y)
    <class 'float'>
    >>> x == y  # Note result of equality comparison
    True

    >>> type(complex('1+2j')) # real part 1, imaginary part 2
    <class 'complex'>
    >>> complex('1') == 1.0   # Note result of comparison
    True 
    >>> complex('1') == 1     # Note result of comparison
    True 

    >>> type(False) 
    <class 'bool'>
    >>> type(True) 
    <class 'bool'>
    >>> True == 1
    True

5.4.3 Sequence types

A sequence denotes a serially ordered collection of zero or more objects. An object may occur more than once in a sequence.

Python supports a number of core sequence types. Some sequences have immutable structures and some have mutable.

    >>> type('Hello world')
    <class 'str'>
    >>> type("Hi Earth")
    <class 'str'>
    >>> type('''
    ... Can have embedded newlines
    ... ''')
    <class 'str'>

    >>> type(())    # empty tuple
    <class 'tuple'>
    >>> type((1,))  # one-element tuple, note comma
    <class 'tuple'>
    >>> x = (1,'Ole Miss')  # mixed element types
    >>> type(x) 
    <class 'tuple'>
    >>> x[0]        # access element with index 0
    1
    >>> x[1]        # access element with index 1
    'Ole Miss'

    >>> list(range(5))
    [0, 1, 2, 3, 4]
    >>> list(range(1, 5))
    [1, 2, 3, 4]
    >>> list(range(0, 9, 3))
    [0, 3, 6]
    >>> list(range(0, 10, 3))
    [0, 3, 6, 9]
    >>> list(range(5, 0, -1))
    [5, 4, 3, 2, 1]
    >>> list(range(0))
    []
    >>> list(range(1, 0)) 
    []
    >>> list(range(0, 0)) 
    []

    >>> type(b'Hello\n   World!') 
    <class 'bytes>

    >>> type([])
    <class 'list'>
    >>> type([3])
    <class 'list'>
    >>> x = [1,2,3] + ['four','five'] # concatenation
    >>> x
    [1, 2, 3, 'four', 'five']
    >>> type(x)
    <class 'list'>
    >>> y = x[1:3] # get slice of list
    >>> y
    [2, 3]
    >>> y[0] = 3   # assign to list index 0
    [3, 3]

    >>> type(bytearray(b'Hello\n   World!')) 
    <class 'bytes>

5.4.4 Mapping types

Type dict (dictionary) denotes mutable finite sets of key-value pairs, where the key is an index into the set for the value with which it is paired.

The key can be any hashable object. That is, the key can be any immutable object or an object that always gives the same hash value. However, the associated value objects may be mutable and the membership in the set may change.

We can express dictionaries syntactically in various ways such as comma-separated lists of key-value pairs with braces.

    >>> x = { 1 : "one" }
    >>> x
    {1: 'one'}
    >>> type(x)
    <class 'dict'>
    >>> x[1]
    'one'
    >>> x.update({ 2 : "two" }) # add to dictionary
    >>> x
    {1: 'one', 2: 'two'}
    >>> type(x)
    <class 'dict'>
    >>> del x[1]  # delete element with key
    >>> x
    {2: 'two'}

5.4.5 Set Types

A set is an unordered collection of distinct hashable objects.

There are two built-in set types—set and frozenset.

    >>> sx = { 'Dijkstra', 'Hoare', 'Knuth' }
    >>> sx
    {'Knuth', 'Hoare', 'Dijkstra'}
    >>> sy = set(['Knuth', 'Dijkstra', 'Hoare'])
    >>> sy
<   {'Knuth', 'Hoare', 'Dijkstra'}      
    >>> sx == sy
    True
    >>> sx.add('Turing')  # add element to mutable set
    >>> sx
    {'Turing', 'Knuth', 'Hoare', 'Dijkstra'}

    >>> fx = frozenset(['Dijkstra', 'Hoare', 'Knuth'])
    >>> fx
    frozenset({'Knuth', 'Hoare', 'Dijkstra'})       
    >>> fy = frozenset(sy)
    >>> fy
    frozenset({'Knuth', 'Hoare', 'Dijkstra'})
    >>> fx.add('Turing')
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    AttributeError: 'frozenset' object has no attribute 'add'

5.4.6 Other object types

We discuss callable objects (e.g., functions), class objects, module objects, and user-defined types (classes) in later chapters.

TODO: Perhaps be more specific about later chapters.