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PEP 483 -- The Theory of Type Hints

PEP: 483
Title: The Theory of Type Hints
Author: Guido van Rossum <guido at>, Ivan Levkivskyi <levkivskyi at>
Discussions-To: Python-Ideas < python-ideas at >
Status: Draft
Type: Informational
Created: 19-Dec-2014


This PEP lays out the theory referenced by PEP 484 .


This document lays out the theory of the new type hinting proposal for Python 3.5. It's not quite a full proposal or specification because there are many details that need to be worked out, but it lays out the theory without which it is hard to discuss more detailed specifications. We start by recalling basic concepts of type theory; then we explain gradual typing; then we state some general rules and define the new special types (such as Union ) that can be used in annotations; and finally we define the approach to generic types and pragmatic aspects of type hinting.

Notational conventions

  • t1 , t2 , etc. and u1 , u2 , etc. are types. Sometimes we write ti or tj to refer to "any of t1 , t2 , etc."
  • T , U etc. are type variables (defined with TypeVar() , see below).
  • Objects, classes defined with a class statement, and instances are denoted using standard PEP 8 conventions.
  • the symbol == applied to types in the context of this PEP means that two expressions represent the same type.
  • Note that PEP 484 makes a distinction between types and classes (a type is a concept for the type checker, while a class is a runtime concept). In this PEP we clarify this distinction but avoid unnecessary strictness to allow more flexibility in the implementation of type checkers.


There are many definitions of the concept of type in the literature. Here we assume that type is a set of values and a set of functions that one can apply to these values.

There are several ways to define a particular type:

  • By explicitly listing all values. E.g., True and False form the type bool .

  • By specifying functions which can be used with variables of a type. E.g. all objects that have a __len__ method form the type Sized . Both [1, 2, 3] and 'abc' belong to this type, since one can call len on them:

    len([1, 2, 3])  # OK
    len('abc')      # also OK
    len(42)         # not a member of Sized
  • By a simple class definition, for example if one defines a class:

    class UserID(int):

    then all instances of this class also form a type.

  • There are also more complex types. E.g., one can define the type FancyList as all lists containing only instances of int , str or their subclasses. The value [1, 'abc', UserID(42)] has this type.

It is important for the user to be able to define types in a form that can be understood by type checkers. The goal of this PEP is to propose such a systematic way of defining types for type annotations of variables and functions using PEP 3107 syntax. These annotations can be used to avoid many kind of bugs, for documentation purposes, or maybe even to increase speed of program execution. Here we only focus on avoiding bugs by using a static type checker.

Subtype relationships

A crucial notion for static type checker is the subtype relationship. It arises from the question: If first_var has type first_type , and second_var has type second_type , is it safe to assign first_var = second_var ?

A strong criterion for when it should be safe is:

  • every value from second_type is also in the set of values of first_type ; and
  • every function from first_type is also in the set of functions of second_type .

The relation defined thus is called a subtype relation.

By this definition:

  • Every type is a subtype of itself.
  • The set of values becomes smaller in the process of subtyping, while the set of functions becomes larger.

An intuitive example: Every Dog is an Animal , also Dog has more functions, for example it can bark, therefore Dog is a subtype of Animal . Conversely, Animal is not a subtype of Dog .

A more formal example: Integers are subtype of real numbers. Indeed, every integer is of course also a real number, and integers support more operations, such as, e.g., bitwise shifts << and >> :

lucky_number = 3.14    # type: float
lucky_number = 42      # Safe
lucky_number * 2       # This works
lucky_number << 5      # Fails

unlucky_number = 13    # type: int
unlucky_number << 5    # This works
unlucky_number = 2.72  # Unsafe

Let us also consider a tricky example: If List[int] denotes the type formed by all lists containing only integer numbers, then it is not a subtype of List[float] , formed by all lists that contain only real numbers. The first condition of subtyping holds, but appending a real number only works with List[float] so that the second condition fails:

def append_pi(lst: List[float]) -> None:
    lst += [3.14]

my_list = [1, 3, 5]  # type: List[int]

append_pi(my_list)   # Naively, this should be safe...

my_list[-1] << 5     # ... but this fails

There are two widespread approaches to declare subtype information to type checker.

In nominal subtyping, the type tree is based on the class tree, i.e., UserID is considered a subtype of int . This approach should be used under control of the type checker, because in Python one can override attributes in an incompatible way:

class Base:
    answer = '42' # type: str

class Derived(Base):
    answer = 5 # should be marked as error by type checker

In structural subtyping the subtype relation is deduced from the declared methods, i.e., UserID and int would be considered the same type. While this may occasionally cause confusion, structural subtyping is considered more flexible. We strive to provide support for both approaches, so that structural information can be used in addition to nominal subtyping.

Summary of gradual typing

Gradual typing allows one to annotate only part of a program, thus leverage desirable aspects of both dynamic and static typing.

We define a new relationship, is-consistent-with, which is similar to is-subtype-of, except it is not transitive when the new type Any is involved. (Neither relationship is symmetric.) Assigning a_value to a_variable is OK if the type of a_value is consistent with the type of a_variable . (Compare this to "... if the type of a_value is a subtype of the type of a_variable ", which states one of the fundamentals of OO programming.) The is-consistent-with relationship is defined by three rules:

  • A type t1 is consistent with a type t2 if t1 is a subtype of t2 . (But not the other way around.)
  • Any is consistent with every type. (But Any is not a subtype of every type.)
  • Every type is consistent with Any . (But every type is not a subtype of Any .)

That's all! See Jeremy Siek's blog post What is Gradual Typing for a longer explanation and motivation. Any can be considered a type that has all values and all methods. Combined with the definition of subtyping above, this places Any partially at the top (it has all values) and bottom (it has all methods) of the type hierarchy. Contrast this to object -- it is not consistent with most types (e.g. you can't use an object() instance where an int is expected). IOW both Any and object mean "any type is allowed" when used to annotate an argument, but only Any can be passed no matter what type is expected (in essence, Any declares a fallback to dynamic typing and shuts up complaints from the static checker).

Here's an example showing how these rules work out in practice:

Say we have an Employee class, and a subclass Manager :

class Employee: ...
class Manager(Employee): ...

Let's say variable worker is declared with type Employee :

worker = Employee()  # type: Employee

Now it's okay to assign a Manager instance to worker (rule 1):

worker = Manager()

It's not okay to assign an Employee instance to a variable declared with type Manager :

boss = Manager()  # type: Manager
boss = Employee()  # Fails static check

However, suppose we have a variable whose type is Any :

something = some_func()  # type: Any

Now it's okay to assign something to worker (rule 2):

worker = something  # OK

Of course it's also okay to assign worker to something (rule 3), but we didn't need the concept of consistency for that:

something = worker  # OK

Types vs. Classes

In Python, classes are object factories defined by the class statement, and returned by the type(obj) built-in function. Class is a dynamic, runtime concept.

Type concept is described above, types appear in variable and function type annotations, can be constructed from building blocks described below, and are used by static type checkers.

Every class is a type as discussed above. But it is tricky and error prone to implement a class that exactly represents semantics of a given type, and it is not a goal of PEP 484 . The static types described in PEP 484, should not be confused with the runtime classes. Examples:

  • int is a class and a type.

  • UserID is a class and a type.

  • Union[str, int] is a type but not a proper class:

    class MyUnion(Union[str, int]): ...  # raises TypeError
    Union[str, int]()  # raises TypeError

Typing interface is implemented with classes, i.e., at runtime it is possible to evaluate, e.g., Generic[T].__bases__ . But to emphasize the distinction between classes and types the following general rules apply:

  • No types defined below (i.e. Any , Union , etc.) can be instantiated, an attempt to do so will raise TypeError . (But non-abstract subclasses of Generic can be.)
  • No types defined below can be subclassed, except for Generic and classes derived from it.
  • All of these will raise TypeError if they appear in isinstance or issubclass (except for unparametrized generics).

Fundamental building blocks

  • Any . Every type is consistent with Any ; and it is also consistent with every type (see above).
  • Union[t1, t2, ...] . Types that are subtype of at least one of t1 etc. are subtypes of this.
    • Unions whose components are all subtypes of t1 etc. are subtypes of this. Example: Union[int, str] is a subtype of Union[int, float, str] .
    • The order of the arguments doesn't matter. Example: Union[int, str] == Union[str, int] .
    • If ti is itself a Union the result is flattened. Example: Union[int, Union[float, str]] == Union[int, float, str] .
    • If ti and tj have a subtype relationship, the less specific type survives. Example: Union[Employee, Manager] == Union[Employee] .
    • Union[t1] returns just t1 . Union[] is illegal, so is Union[()]
    • Corollary: Union[..., object, ...] returns object .
  • Optional[t1] . Alias for Union[t1, None] , i.e. Union[t1, type(None)] .
  • Tuple[t1, t2, ..., tn] . A tuple whose items are instances of t1 , etc. Example: Tuple[int, float] means a tuple of two items, the first is an int , the second is a float ; e.g., (42, 3.14) .
    • Tuple[u1, u2, ..., um] is a subtype of Tuple[t1, t2, ..., tn] if they have the same length n==m and each ui is a subtype of ti .
    • To spell the type of the empty tuple, use Tuple[()] .
    • A variadic homogeneous tuple type can be written Tuple[t1, ...] . (That's three dots, a literal ellipsis; and yes, that's a valid token in Python's syntax.)
  • Callable[[t1, t2, ..., tn], tr] . A function with positional argument types t1 etc., and return type tr . The argument list may be empty n==0 . There is no way to indicate optional or keyword arguments, nor varargs, but you can say the argument list is entirely unchecked by writing Callable[..., tr] (again, a literal ellipsis).

We might add:

  • Intersection[t1, t2, ...] . Types that are subtype of each of t1 , etc are subtypes of this. (Compare to Union , which has at least one instead of each in its definition.)
    • The order of the arguments doesn't matter. Nested intersections are flattened, e.g. Intersection[int, Intersection[float, str]] == Intersection[int, float, str] .
    • An intersection of fewer types is a supertype of an intersection of more types, e.g. Intersection[int, str] is a supertype of Intersection[int, float, str] .
    • An intersection of one argument is just that argument, e.g. Intersection[int] is int .
    • When argument have a subtype relationship, the more specific type survives, e.g. Intersection[str, Employee, Manager] is Intersection[str, Manager] .
    • Intersection[] is illegal, so is Intersection[()] .
    • Corollary: Any disappears from the argument list, e.g. Intersection[int, str, Any] == Intersection[int, str] . Intersection[Any, object] is object .
    • The interaction between Intersection and Union is complex but should be no surprise if you understand the interaction between intersections and unions of regular sets (note that sets of types can be infinite in size, since there is no limit on the number of new subclasses).

Generic types

The fundamental building blocks defined above allow to construct new types in a generic manner. For example, Tuple can take a concrete type float and make a concrete type Vector = Tuple[float, ...] , or it can take another type UserID and make another concrete type Registry = Tuple[UserID, ...] . Such semantics is known as generic type constructor, it is similar to semantics of functions, but a function takes a value and returns a value, while generic type constructor takes a type and "returns" a type.

It is common when a particular class or a function behaves in such a type generic manner. Consider two examples:

  • Container classes, such as list or dict , typically contain only values of a particular type. Therefore, a user might want to type annotate them as such:

    users = [] # type: List[UserID]
    users.append(UserID(42)) # OK
    users.append('Some guy') # Should be rejected by the type checker
    examples = {} # type: Dict[str, Any]
    examples['first example'] = object() # OK
    examples[2] = None                   # rejected by the type checker
  • The following function can take two arguments of type int and return an int , of take two arguments of type float and return a float , etc.:

    def add(x, y):
        return x + y
    add(1, 2) == 3
    add('1', '2') == '12'
    add(2.7, 3.5) == 6.2

To allow type annotations in situations from the first example, built-in containers and container abstract base classes are extended with type parameters, so that they behave as generic type constructors. Classes, that behave as generic type constructors are called generic types . Example:

from typing import Iterable

class Task:

def work(todo_list: Iterable[Task]) -> None:

Here Iterable is a generic type that takes a concrete type Task and returns a concrete type Iterable[Task] .

Functions that behave in the type generic manner (as in second example) are called generic functions . Type annotations of generic functions are allowed by type variables . Their semantics with respect to generic types is somewhat similar to semantics of parameters in functions. But one does not assign concrete types to type variables, it is the task of a static type checker to find their possible values and warn the user if it cannot find. Example:

def take_first(seq: Sequence[T]) -> T: # a generic function
    return seq[0]

accumulator = 0 # type: int

accumulator += take_first([1, 2, 3])   # Safe, T deduced to be int
accumulator += take_first((2.7, 3.5))  # Unsafe

Type variables are used extensively in type annotations, also internal machinery of the type inference in type checkers is typically build on type variables. Therefore, let us consider them in detail.

Type variables

X = TypeVar('X') declares a unique type variable. The name must match the variable name. By default, a type variable ranges over all possible types. Example:

def do_nothing(one_arg: T, other_arg: T) -> None:

do_nothing(1, 2)               # OK, T is int
do_nothing('abc', UserID(42))  # also OK, T is object

Y = TypeVar('Y', t1, t2, ...) . Ditto, constrained to t1 , etc. Behaves similar to Union[t1, t2, ...] . A constrained type variable ranges only over constrains t1 , etc. exactly ; subclasses of the constrains are replaced by the most-derived base class among t1 , etc. Examples:

  • Function type annotation with a constrained type variable:

    S = TypeVar('S', str, bytes)
    def longest(first: S, second: S) -> S:
        return first if len(first) >= len(second) else second
    result = longest('a', 'abc')  # The inferred type for result is str
    result = longest('a', b'abc')  # Fails static type check

    In this example, both arguments to longest() must have the same type ( str or bytes ), and moreover, even if the arguments are instances of a common str subclass, the return type is still str , not that subclass (see next example).

  • For comparison, if the type variable was unconstrained, the common subclass would be chosen as the return type, e.g.:

    S = TypeVar('S')
    def longest(first: S, second: S) -> S:
        return first if len(first) >= len(second) else second
    class MyStr(str): ...
    result = longest(MyStr('a'), MyStr('abc'))

    The inferred type of result is MyStr (whereas in the AnyStr example it would be str ).

  • Also for comparison, if a Union is used, the return type also has to be a Union :

    U = Union[str, bytes]
    def longest(first: U, second: U) -> U:
        return first if len(first) >= len(second) else second
    result = longest('a', 'abc')

    The inferred type of result is still Union[str, bytes] , even though both arguments are str .

    Note that the type checker will reject this function:

    def concat(first: U, second: U) -> U:
        return x + y  # Error: can't concatenate str and bytes

    For such cases where parameters could change their types only simultaneously one should use constrained type variables.

Defining and using generic types

Users can declare their classes as generic types using the special building block Generic . The definition class MyGeneric(Generic[X, Y, ...]): ... defines a generic type MyGeneric over type variables X , etc. MyGeneric itself becomes parameterizable, e.g. MyGeneric[int, str, ...] is a specific type with substitutions X -> int , etc. Example:

class CustomQueue(Generic[T]):

    def put(self, task: T) -> None:
    def get(self) -> T:

def communicate(queue: CustomQueue[str]) -> Optional[str]:

Classes that derive from generic types become generic. A class can subclass multiple generic types. However, classes derived from specific types returned by generics are not generic. Examples:

class TodoList(Iterable[T], Container[T]):
    def check(self, item: T) -> None:

def check_all(todo: TodoList[T]) -> None:  # TodoList is generic

class URLList(Iterable[bytes]):
    def scrape_all(self) -> None:

def search(urls: URLList) -> Optional[bytes]  # URLList is not generic

Subclassing a generic type imposes the subtype relation on the corresponding specific types, so that TodoList[t1] is a subtype of Iterable[t1] in the above example.

Generic types can be specialized (indexed) in several steps. Every type variable could be substituted by a specific type or by another generic type. If Generic appears in the base class list, then it should contain all type variables, and the order of type parameters is determined by the order in which they appear in Generic . Examples:

Table = Dict[int, T]     # Table is generic
Messages = Table[bytes]  # Same as Dict[int, bytes]

class BaseGeneric(Generic[T, S]):

class DerivedGeneric(BaseGeneric[int, T]): # DerivedGeneric has one parameter

SpecificType = DerivedGeneric[int]         # OK

class MyDictView(Generic[S, T, U], Iterable[Tuple[U, T]]):

Example = MyDictView[list, int, str]       # S -> list, T -> int, U -> str

If a generic type appears in a type annotation with a type variable omitted, it is assumed to be Any . Such form could be used as a fallback to dynamic typing and is allowed for use with issubclass and isinstance . All type information in instances is erased at runtime. Examples:

def count(seq: Sequence) -> int:      # Same as Sequence[Any]

class FrameworkBase(Generic[S, T]):

class UserClass:

issubclass(UserClass, FrameworkBase)  # This is OK

class Node(Generic[T]):

IntNode = Node[int]
my_node = IntNode()  # at runtime my_node.__class__ is Node
                     # inferred static type of my_node is Node[int]

Covariance and Contravariance

If t2 is a subtype of t1 , then a generic type constructor GenType is called:

  • Covariant, if GenType[t2] is a subtype of GenType[t1] for all such t1 and t2 .
  • Contravariant, if GenType[t1] is a subtype of GenType[t2] for all such t1 and t2 .
  • Invariant, if neither of the above is true.

To better understand this definition, let us make an analogy with ordinary functions. Assume that we have:

def cov(x: float) -> float:
    return 2*x

def contra(x: float) -> float:
    return -x

def inv(x: float) -> float:
    return x*x

If x1 < x2 , then always cov(x1) < cov(x2) , and contra(x2) < contra(x1) , while nothing could be said about inv . Replacing < with is-subtype-of, and functions with generic type constructor we get examples of covariant, contravariant, and invariant behavior. Let us now consider practical examples:

  • Union behaves covariantly in all its arguments. Indeed, as discussed above, Union[t1, t2, ...] is a subtype of Union[u1, u2, ...] , if t1 is a subtype of u1 , etc.
  • FrozenSet[T] is also covariant. Let us consider int and float in place of T . First, int is a subtype of float . Second, set of values of FrozenSet[int] is clearly a subset of values of FrozenSet[float] , while set of functions from FrozenSet[float] is a subset of set of functions from FrozenSet[int] . Therefore, by definition FrozenSet[int] is a subtype of FrozenSet[float] .
  • List[T] is invariant. Indeed, although set of values of List[int] is a subset of values of List[float] , only int could be appended to a List[int] , as discussed in section "Background". Therefore, List[int] is not a subtype of List[float] . This is a typical situation with mutable types, they are typically invariant.

One of the best examples to illustrate (somewhat counterintuitive) contravariant behavior is the callable type. It is covariant in the return type, but contravariant in the arguments. For two callable types that differ only in the return type, the subtype relationship for the callable types follows that of the return types. Examples:

  • Callable[[], int] is a subtype of Callable[[], float] .
  • Callable[[], Manager] is a subtype of Callable[[], Employee] .

While for two callable types that differ only in the type of one argument, the subtype relationship for the callable types goes in the opposite direction as for the argument types. Examples:

  • Callable[[float], None] is a subtype of Callable[[int], None] .
  • Callable[[Employee], None] is a subtype of Callable[[Manager], None] .

Yes, you read that right. Indeed, if a function that can calculate the salary for a manager is expected:

def calculate_all(lst: List[Manager], salary: Callable[[Manager], Decimal]):

then Callable[[Employee], Decimal] that can calculate a salary for any employee is also acceptable.

The example with Callable shows how to make more precise type annotations for functions: choose the most general type for every argument, and the most specific type for the return value.

It is possible to declare the variance for user defined generic types by using special keywords covariant and contravariant in the definition of type variables used as parameters. Types are invariant by default. Examples:

T = TypeVar('T')
T_co = TypeVar('T_co', covariant=True)
T_contra = TypeVar('T_contra', contravariant=True)

class LinkedList(Generic[T]):  # invariant by default
    def append(self, element: T) -> None:

class Box(Generic[T_co]):      #  this type is declared covariant
    def __init__(self, content: T_co) -> None:
        self._content = content
    def get_content(self) -> T_co:
        return self._content

class Sink(Generic[T_contra]): # this type is declared contravariant
    def send_to_nowhere(self, data: T_contra) -> None:
        with open(os.devnull, 'w') as devnull:
            print(data, file=devnull)

Note, that although the variance is defined via type variables, it is not a property of type variables, but a property of generic types. In complex definitions of derived generics, variance only determined from type variables used. A complex example:

T_co = TypeVar('T_co', Employee, Manager, covariant=True)
T_contra = TypeVar('T_contra', Employee, Manager, contravariant=True)

class Base(Generic[T_contra]):

class Derived(Base[T_co]):

A type checker finds from the second declaration that Derived[Manager] is a subtype of Derived[Employee] , and Derived[t1] is a subtype of Base[t1] . If we denote the is-subtype-of relationship with < , then the full diagram of subtyping for this case will be:

Base[Manager]    >  Base[Employee]
    v                   v
Derived[Manager] <  Derived[Employee]

so that a type checker will also find that, e.g., Derived[Manager] is a subtype of Base[Employee] .

For more information on type variables, generic types, and variance, see PEP 484 , the mypy docs on generics , and Wikipedia .


Some things are irrelevant to the theory but make practical use more convenient. (This is not a full list; I probably missed a few and some are still controversial or not fully specified.)

  • Where a type is expected, None can be substituted for type(None) ; e.g. Union[t1, None] == Union[t1, type(None)] .

  • Type aliases, e.g.:

    Point = Tuple[float, float]
    def distance(point: Point) -> float: ...
  • Forward references via strings, e.g.:

    class MyComparable:
        def compare(self, other: 'MyComparable') -> int: ...
  • If a default of None is specified, the type is implicitly Optional , e.g.:

    def get(key: KT, default: VT = None) -> VT: ...
  • Type variables can be declared in unconstrained, constrained, or bounded form. The variance of a generic type can also be indicated using a type variable declared with special keyword arguments, thus avoiding any special syntax, e.g.:

    T = TypeVar('T', bound=complex)
    def add(x: T, y: T) -> T:
        return x + y
    T_co = TypeVar('T_co', covariant=True)
    class ImmutableList(Generic[T_co]): ...
  • Type declaration in comments, e.g.:

    lst = []  # type: Sequence[int]
  • Casts using cast(T, obj) , e.g.:

    zork = cast(Any, frobozz())
  • Other things, e.g. overloading and stub modules, see PEP 484 .

Predefined generic types and Protocols in

(See also the module .)

  • Everything from (but Set renamed to AbstractSet ).
  • Dict , List , Set , FrozenSet , a few more.
  • re.Pattern[AnyStr] , re.Match[AnyStr] .
  • io.IO[AnyStr] , io.TextIO ~ io.IO[str] , io.BinaryIO ~ io.IO[bytes] .