Julia 中文文档

version: 1.0.1

Built with Teadocs

Julia 类型

# Types

Type systems have traditionally fallen into two quite different camps: static type systems, where
every program expression must have a type computable before the execution of the program, and
dynamic type systems, where nothing is known about types until run time, when the actual values
manipulated by the program are available. Object orientation allows some flexibility in statically
typed languages by letting code be written without the precise types of values being known at
compile time. The ability to write code that can operate on different types is called polymorphism.
All code in classic dynamically typed languages is polymorphic: only by explicitly checking types,
or when objects fail to support operations at run-time, are the types of any values ever restricted.

Julia's type system is dynamic, but gains some of the advantages of static type systems by making
it possible to indicate that certain values are of specific types. This can be of great assistance
in generating efficient code, but even more significantly, it allows method dispatch on the types
of function arguments to be deeply integrated with the language. Method dispatch is explored in
detail in Methods, but is rooted in the type system presented here.

The default behavior in Julia when types are omitted is to allow values to be of any type. Thus,
one can write many useful Julia functions without ever explicitly using types. When additional
expressiveness is needed, however, it is easy to gradually introduce explicit type annotations
into previously "untyped" code. Adding annotations serves three primary purposes: to take advantage
of Julia's powerful multiple-dispatch mechanism, to improve human readability, and to catch
programmer errors.

Describing Julia in the lingo of type systems, it
is: dynamic, nominative and parametric. Generic types can be parameterized, and the hierarchical
relationships between types are explicitly declared,
rather than implied by compatible structure.
One particularly distinctive feature of Julia's type system is that concrete types may not subtype
each other: all concrete types are final and may only have abstract types as their supertypes.
While this might at first seem unduly restrictive, it has many beneficial consequences with surprisingly
few drawbacks. It turns out that being able to inherit behavior is much more important than being
able to inherit structure, and inheriting both causes significant difficulties in traditional
object-oriented languages. Other high-level aspects of Julia's type system that should be mentioned
up front are:

  • There is no division between object and non-object values: all values in Julia are true objects
    having a type that belongs to a single, fully connected type graph, all nodes of which are equally
    first-class as types.
  • There is no meaningful concept of a "compile-time type": the only type a value has is its actual
    type when the program is running. This is called a "run-time type" in object-oriented languages
    where the combination of static compilation with polymorphism makes this distinction significant.
  • Only values, not variables, have types -- variables are simply names bound to values.
  • Both abstract and concrete types can be parameterized by other types. They can also be parameterized
    by symbols, by values of any type for which isbits returns true (essentially, things
    like numbers and bools that are stored like C types or structs with no pointers to other objects),
    and also by tuples thereof. Type parameters may be omitted when they do not need to be referenced
    or restricted.

Julia's type system is designed to be powerful and expressive, yet clear, intuitive and unobtrusive.
Many Julia programmers may never feel the need to write code that explicitly uses types. Some
kinds of programming, however, become clearer, simpler, faster and more robust with declared types.

# Type Declarations

The :: operator can be used to attach type annotations to expressions and variables in programs.
There are two primary reasons to do this:

  1. As an assertion to help confirm that your program works the way you expect,
  2. To provide extra type information to the compiler, which can then improve performance in some
    cases

When appended to an expression computing a value, the :: operator is read as "is an instance
of". It can be used anywhere to assert that the value of the expression on the left is an instance
of the type on the right. When the type on the right is concrete, the value on the left must have
that type as its implementation -- recall that all concrete types are final, so no implementation
is a subtype of any other. When the type is abstract, it suffices for the value to be implemented
by a concrete type that is a subtype of the abstract type. If the type assertion is not true,
an exception is thrown, otherwise, the left-hand value is returned:

julia> (1+2)::AbstractFloat
ERROR: TypeError: in typeassert, expected AbstractFloat, got Int64

julia> (1+2)::Int
3

This allows a type assertion to be attached to any expression in-place.

When appended to a variable on the left-hand side of an assignment, or as part of a local declaration,
the :: operator means something a bit different: it declares the variable to always have the
specified type, like a type declaration in a statically-typed language such as C. Every value
assigned to the variable will be converted to the declared type using convert:

julia> function foo()
           x::Int8 = 100
           x
       end
foo (generic function with 1 method)

julia> foo()
100

julia> typeof(ans)
Int8

This feature is useful for avoiding performance "gotchas" that could occur if one of the assignments
to a variable changed its type unexpectedly.

This "declaration" behavior only occurs in specific contexts:

local x::Int8 # in a local declaration
x::Int8 = 10 # as the left-hand side of an assignment


and applies to the whole current scope, even before the declaration. Currently, type declarations
cannot be used in global scope, e.g. in the REPL, since Julia does not yet have constant-type
globals.

Declarations can also be attached to function definitions:


function sinc(x)::Float64
    if x == 0
        return 1
    end
    return sin(pi*x)/(pi*x)
end

Returning from this function behaves just like an assignment to a variable with a declared type:
the value is always converted to Float64.

# Abstract Types

Abstract types cannot be instantiated, and serve only as nodes in the type graph, thereby describing
sets of related concrete types: those concrete types which are their descendants. We begin with
abstract types even though they have no instantiation because they are the backbone of the type
system: they form the conceptual hierarchy which makes Julia's type system more than just a collection
of object implementations.

Recall that in Integers and Floating-Point Numbers, we introduced a variety of concrete
types of numeric values: Int8, UInt8, Int16, UInt16,
Int32, UInt32, Int64, UInt64, Int128,
UInt128, Float16, Float32, and Float64. Although
they have different representation sizes, Int8, Int16, Int32, Int64 and Int128
all have in common that they are signed integer types. Likewise UInt8, UInt16, UInt32,
UInt64 and UInt128 are all unsigned integer types, while Float16, Float32 and
Float64 are distinct in being floating-point types rather than integers. It is common for
a piece of code to make sense, for example, only if its arguments are some kind of integer,
but not really depend on what particular kind of integer. For example, the greatest common
denominator algorithm works for all kinds of integers, but will not work for floating-point
numbers. Abstract types allow the construction of a hierarchy of types, providing a context
into which concrete types can fit. This allows you, for example, to easily program to any type
that is an integer, without restricting an algorithm to a specific type of integer.

Abstract types are declared using the abstract type keyword. The general syntaxes for declaring an
abstract type are:

abstract type «name» end
abstract type «name» <: «supertype» end

The abstract type keyword introduces a new abstract type, whose name is given by «name». This
name can be optionally followed by <: and an already-existing type, indicating that the newly
declared abstract type is a subtype of this "parent" type.

When no supertype is given, the default supertype is Any -- a predefined abstract type that
all objects are instances of and all types are subtypes of. In type theory, Any is commonly
called "top" because it is at the apex of the type graph. Julia also has a predefined abstract
"bottom" type, at the nadir of the type graph, which is written as Union{}. It is the exact
opposite of Any: no object is an instance of Union{} and all types are supertypes of Union{}.

Let's consider some of the abstract types that make up Julia's numerical hierarchy:

abstract type Number end
abstract type Real     <: Number end
abstract type AbstractFloat <: Real end
abstract type Integer  <: Real end
abstract type Signed   <: Integer end
abstract type Unsigned <: Integer end

The Number type is a direct child type of Any, and Real is its child.
In turn, Real has two children (it has more, but only two are shown here; we'll get to
the others later): Integer and AbstractFloat, separating the world into
representations of integers and representations of real numbers. Representations of real
numbers include, of course, floating-point types, but also include other types, such as
rationals. Hence, AbstractFloat is a proper subtype of Real, including only
floating-point representations of real numbers. Integers are further subdivided into
Signed and Unsigned varieties.

The <: operator in general means "is a subtype of", and, used in declarations like this, declares
the right-hand type to be an immediate supertype of the newly declared type. It can also be used
in expressions as a subtype operator which returns true when its left operand is a subtype of
its right operand:

julia> Integer <: Number
true

julia> Integer <: AbstractFloat
false

An important use of abstract types is to provide default implementations for concrete types. To
give a simple example, consider:

function myplus(x,y)
    x+y
end

The first thing to note is that the above argument declarations are equivalent to x::Any and
y::Any. When this function is invoked, say as myplus(2,5), the dispatcher chooses the most
specific method named myplus that matches the given arguments. (See Methods for more
information on multiple dispatch.)

Assuming no method more specific than the above is found, Julia next internally defines and compiles
a method called myplus specifically for two Int arguments based on the generic function given
above, i.e., it implicitly defines and compiles:

function myplus(x::Int,y::Int)
    x+y
end

and finally, it invokes this specific method.

Thus, abstract types allow programmers to write generic functions that can later be used as the
default method by many combinations of concrete types. Thanks to multiple dispatch, the programmer
has full control over whether the default or more specific method is used.

An important point to note is that there is no loss in performance if the programmer relies on
a function whose arguments are abstract types, because it is recompiled for each tuple of argument
concrete types with which it is invoked. (There may be a performance issue, however, in the case
of function arguments that are containers of abstract types; see [Performance Tips](@ref man-performance-tips).)

# Primitive Types

A primitive type is a concrete type whose data consists of plain old bits. Classic examples of primitive types are integers and floating-point values. Unlike most languages, Julia lets you declare your own primitive types, rather than providing only a fixed set of built-in ones. In fact, the standard primitive types are all defined in the language itself:

primitive type Float16 <: AbstractFloat 16 end
primitive type Float32 <: AbstractFloat 32 end
primitive type Float64 <: AbstractFloat 64 end

primitive type Bool <: Integer 8 end
primitive type Char <: AbstractChar 32 end

primitive type Int8    <: Signed   8 end
primitive type UInt8   <: Unsigned 8 end
primitive type Int16   <: Signed   16 end
primitive type UInt16  <: Unsigned 16 end
primitive type Int32   <: Signed   32 end
primitive type UInt32  <: Unsigned 32 end
primitive type Int64   <: Signed   64 end
primitive type UInt64  <: Unsigned 64 end
primitive type Int128  <: Signed   128 end
primitive type UInt128 <: Unsigned 128 end

The general syntaxes for declaring a primitive type are:

primitive type «name» «bits» end
primitive type «name» <: «supertype» «bits» end

The number of bits indicates how much storage the type requires and the name gives the new type
a name. A primitive type can optionally be declared to be a subtype of some supertype. If a supertype
is omitted, then the type defaults to having Any as its immediate supertype. The declaration
of Bool above therefore means that a boolean value takes eight bits to store, and has
Integer as its immediate supertype. Currently, only sizes that are multiples of
8 bits are supported. Therefore, boolean values, although they really need just a single bit,
cannot be declared to be any smaller than eight bits.

The types Bool, Int8 and UInt8 all have identical representations:
they are eight-bit chunks of memory. Since Julia's type system is nominative, however, they
are not interchangeable despite having identical structure. A fundamental difference between
them is that they have different supertypes: Bool's direct supertype is Integer,
Int8's is Signed, and UInt8's is Unsigned. All other
differences between Bool, Int8, and UInt8 are matters of
behavior -- the way functions are defined to act when given objects of these types as
arguments. This is why a nominative type system is necessary: if structure determined type,
which in turn dictates behavior, then it would be impossible to make Bool behave
any differently than Int8 or UInt8.

# Composite Types

Composite types are called records, structs,
or objects in various languages. A composite type is a collection of named fields,
an instance of which can be treated as a single value. In many languages, composite types are
the only kind of user-definable type, and they are by far the most commonly used user-defined
type in Julia as well.

In mainstream object oriented languages, such as C++, Java, Python and Ruby, composite types also
have named functions associated with them, and the combination is called an "object". In purer
object-oriented languages, such as Ruby or Smalltalk, all values are objects whether they are
composites or not. In less pure object oriented languages, including C++ and Java, some values,
such as integers and floating-point values, are not objects, while instances of user-defined composite
types are true objects with associated methods. In Julia, all values are objects, but functions
are not bundled with the objects they operate on. This is necessary since Julia chooses which
method of a function to use by multiple dispatch, meaning that the types of all of a function's
arguments are considered when selecting a method, rather than just the first one (see Methods
for more information on methods and dispatch). Thus, it would be inappropriate for functions to
"belong" to only their first argument. Organizing methods into function objects rather than having
named bags of methods "inside" each object ends up being a highly beneficial aspect of the language
design.

Composite types are introduced with the struct keyword followed by a block of field names, optionally
annotated with types using the :: operator:

julia> struct Foo
           bar
           baz::Int
           qux::Float64
       end

Fields with no type annotation default to Any, and can accordingly hold any type of value.

New objects of type Foo are created by applying the Foo type object like a function
to values for its fields:

julia> foo = Foo("Hello, world.", 23, 1.5)
Foo("Hello, world.", 23, 1.5)

julia> typeof(foo)
Foo

When a type is applied like a function it is called a constructor. Two constructors are generated
automatically (these are called default constructors). One accepts any arguments and calls
convert to convert them to the types of the fields, and the other accepts arguments
that match the field types exactly. The reason both of these are generated is that this makes
it easier to add new definitions without inadvertently replacing a default constructor.

Since the bar field is unconstrained in type, any value will do. However, the value for baz
must be convertible to Int:

julia> Foo((), 23.5, 1)
ERROR: InexactError: Int64(Int64, 23.5)
Stacktrace:
[...]

You may find a list of field names using the fieldnames function.

julia> fieldnames(Foo)
(:bar, :baz, :qux)

You can access the field values of a composite object using the traditional foo.bar notation:

julia> foo.bar
"Hello, world."

julia> foo.baz
23

julia> foo.qux
1.5

Composite objects declared with struct are immutable; they cannot be modified
after construction. This may seem odd at first, but it has several advantages:

  • It can be more efficient. Some structs can be packed efficiently into arrays, and
    in some cases the compiler is able to avoid allocating immutable objects entirely.
  • It is not possible to violate the invariants provided by the type's constructors.
  • Code using immutable objects can be easier to reason about.

An immutable object might contain mutable objects, such as arrays, as fields. Those contained
objects will remain mutable; only the fields of the immutable object itself cannot be changed
to point to different objects.

Where required, mutable composite objects can be declared with the keyword mutable struct, to be
discussed in the next section.

Immutable composite types with no fields are singletons; there can be only one instance of such types:

julia> struct NoFields
       end

julia> NoFields() === NoFields()
true

The === function confirms that the "two" constructed instances of NoFields are actually one
and the same. Singleton types are described in further detail [below](@ref man-singleton-types).

There is much more to say about how instances of composite types are created, but that discussion
depends on both Parametric Types and on Methods, and is sufficiently important
to be addressed in its own section: [Constructors](@ref man-constructors).

# Mutable Composite Types

If a composite type is declared with mutable struct instead of struct, then instances of
it can be modified:

julia> mutable struct Bar
           baz
           qux::Float64
       end

julia> bar = Bar("Hello", 1.5);

julia> bar.qux = 2.0
2.0

julia> bar.baz = 1//2
1//2

In order to support mutation, such objects are generally allocated on the heap, and have
stable memory addresses.
A mutable object is like a little container that might hold different values over time,
and so can only be reliably identified with its address.
In contrast, an instance of an immutable type is associated with specific field values ---
the field values alone tell you everything about the object.
In deciding whether to make a type mutable, ask whether two instances
with the same field values would be considered identical, or if they might need to change independently
over time. If they would be considered identical, the type should probably be immutable.

To recap, two essential properties define immutability in Julia:

  • It is not permitted to modify the value of an immutable type.
    • For bits types this means that the bit pattern of a value once set will never change
      and that value is the identity of a bits type.
    • For composite types, this means that the identity of the values of its fields will
      never change. When the fields are bits types, that means their bits will never change,
      for fields whose values are mutable types like arrays, that means the fields will
      always refer to the same mutable value even though that mutable value's content may
      itself be modified.
  • An object with an immutable type may be copied freely by the compiler since its
    immutability makes it impossible to programmatically distinguish between the original
    object and a copy.
    • In particular, this means that small enough immutable values like integers and floats
      are typically passed to functions in registers (or stack allocated).
    • Mutable values, on the other hand are heap-allocated and passed to
      functions as pointers to heap-allocated values except in cases where the compiler
      is sure that there's no way to tell that this is not what is happening.

# Declared Types

The three kinds of types (abstract, primitive, composite) discussed in the previous
sections are actually all closely related. They share the same key properties:

  • They are explicitly declared.
  • They have names.
  • They have explicitly declared supertypes.
  • They may have parameters.

Because of these shared properties, these types are internally represented as instances of the
same concept, DataType, which is the type of any of these types:

julia> typeof(Real)
DataType

julia> typeof(Int)
DataType

A DataType may be abstract or concrete. If it is concrete, it has a specified size, storage
layout, and (optionally) field names. Thus a primitive type is a DataType with nonzero size, but
no field names. A composite type is a DataType that has field names or is empty (zero size).

Every concrete value in the system is an instance of some DataType.

# Type Unions

A type union is a special abstract type which includes as objects all instances of any of its
argument types, constructed using the special Union keyword:

julia> IntOrString = Union{Int,AbstractString}
Union{Int64, AbstractString}

julia> 1 :: IntOrString
1

julia> "Hello!" :: IntOrString
"Hello!"

julia> 1.0 :: IntOrString
ERROR: TypeError: in typeassert, expected Union{Int64, AbstractString}, got Float64

The compilers for many languages have an internal union construct for reasoning about types; Julia
simply exposes it to the programmer. The Julia compiler is able to generate efficient code in the
presence of Union types with a small number of types [^1], by generating specialized code
in separate branches for each possible type.

A particularly useful case of a Union type is Union{T, Nothing}, where T can be any type and
Nothing is the singleton type whose only instance is the object nothing. This pattern
is the Julia equivalent of Nullable, Option or Maybe
types in other languages. Declaring a function argument or a field as Union{T, Nothing} allows
setting it either to a value of type T, or to nothing to indicate that there is no value.
See [this FAQ entry](@ref faq-nothing) for more information.

# Parametric Types

An important and powerful feature of Julia's type system is that it is parametric: types can take
parameters, so that type declarations actually introduce a whole family of new types -- one for
each possible combination of parameter values. There are many languages that support some version
of generic programming, wherein data structures
and algorithms to manipulate them may be specified without specifying the exact types involved.
For example, some form of generic programming exists in ML, Haskell, Ada, Eiffel, C++, Java, C#,
F#, and Scala, just to name a few. Some of these languages support true parametric polymorphism
(e.g. ML, Haskell, Scala), while others support ad-hoc, template-based styles of generic programming
(e.g. C++, Java). With so many different varieties of generic programming and parametric types
in various languages, we won't even attempt to compare Julia's parametric types to other languages,
but will instead focus on explaining Julia's system in its own right. We will note, however, that
because Julia is a dynamically typed language and doesn't need to make all type decisions at compile
time, many traditional difficulties encountered in static parametric type systems can be relatively
easily handled.

All declared types (the DataType variety) can be parameterized, with the same syntax in each
case. We will discuss them in the following order: first, parametric composite types, then parametric
abstract types, and finally parametric primitive types.

# Parametric Composite Types

Type parameters are introduced immediately after the type name, surrounded by curly braces:

julia> struct Point{T}
           x::T
           y::T
       end

This declaration defines a new parametric type, Point{T}, holding two "coordinates" of type
T. What, one may ask, is T? Well, that's precisely the point of parametric types: it can be
any type at all (or a value of any bits type, actually, although here it's clearly used as a type).
Point{Float64} is a concrete type equivalent to the type defined by replacing T in the definition
of Point with Float64. Thus, this single declaration actually declares an unlimited
number of types: Point{Float64}, Point{AbstractString}, Point{Int64}, etc. Each of these
is now a usable concrete type:

julia> Point{Float64}
Point{Float64}

julia> Point{AbstractString}
Point{AbstractString}

The type Point{Float64} is a point whose coordinates are 64-bit floating-point values, while
the type Point{AbstractString} is a "point" whose "coordinates" are string objects (see Strings).

Point itself is also a valid type object, containing all instances Point{Float64}, Point{AbstractString},
etc. as subtypes:

julia> Point{Float64} <: Point
true

julia> Point{AbstractString} <: Point
true

Other types, of course, are not subtypes of it:

julia> Float64 <: Point
false

julia> AbstractString <: Point
false

Concrete Point types with different values of T are never subtypes of each other:

julia> Point{Float64} <: Point{Int64}
false

julia> Point{Float64} <: Point{Real}
false

!!! warning
This last point is very important: even though Float64 <: Real we DO NOT have Point{Float64} <: Point{Real}.

In other words, in the parlance of type theory, Julia's type parameters are invariant, rather
than being covariant (or even contravariant). This is for practical reasons: while any instance
of Point{Float64} may conceptually be like an instance of Point{Real} as well, the two types
have different representations in memory:

  • An instance of Point{Float64} can be represented compactly and efficiently as an immediate pair
    of 64-bit values;
  • An instance of Point{Real} must be able to hold any pair of instances of Real.
    Since objects that are instances of Real can be of arbitrary size and structure, in
    practice an instance of Point{Real} must be represented as a pair of pointers to
    individually allocated Real objects.

The efficiency gained by being able to store Point{Float64} objects with immediate values is
magnified enormously in the case of arrays: an Array{Float64} can be stored as a contiguous
memory block of 64-bit floating-point values, whereas an Array{Real} must be an array of pointers
to individually allocated Real objects -- which may well be
boxed
64-bit floating-point values, but also might be arbitrarily large, complex objects, which are
declared to be implementations of the Real abstract type.

Since Point{Float64} is not a subtype of Point{Real}, the following method can't be applied
to arguments of type Point{Float64}:

function norm(p::Point{Real})
    sqrt(p.x^2 + p.y^2)
end

A correct way to define a method that accepts all arguments of type Point{T} where T is
a subtype of Real is:

function norm(p::Point{<:Real})
    sqrt(p.x^2 + p.y^2)
end

(Equivalently, one could define function norm(p::Point{T} where T<:Real) or
function norm(p::Point{T}) where T<:Real; see UnionAll Types.)

More examples will be discussed later in Methods.

How does one construct a Point object? It is possible to define custom constructors for composite
types, which will be discussed in detail in [Constructors](@ref man-constructors), but in the absence of any special
constructor declarations, there are two default ways of creating new composite objects, one in
which the type parameters are explicitly given and the other in which they are implied by the
arguments to the object constructor.

Since the type Point{Float64} is a concrete type equivalent to Point declared with Float64
in place of T, it can be applied as a constructor accordingly:

julia> Point{Float64}(1.0, 2.0)
Point{Float64}(1.0, 2.0)

julia> typeof(ans)
Point{Float64}

For the default constructor, exactly one argument must be supplied for each field:

julia> Point{Float64}(1.0)
ERROR: MethodError: no method matching Point{Float64}(::Float64)
[...]

julia> Point{Float64}(1.0,2.0,3.0)
ERROR: MethodError: no method matching Point{Float64}(::Float64, ::Float64, ::Float64)
[...]

Only one default constructor is generated for parametric types, since overriding it is not possible.
This constructor accepts any arguments and converts them to the field types.

In many cases, it is redundant to provide the type of Point object one wants to construct, since
the types of arguments to the constructor call already implicitly provide type information. For
that reason, you can also apply Point itself as a constructor, provided that the implied value
of the parameter type T is unambiguous:

julia> Point(1.0,2.0)
Point{Float64}(1.0, 2.0)

julia> typeof(ans)
Point{Float64}

julia> Point(1,2)
Point{Int64}(1, 2)

julia> typeof(ans)
Point{Int64}

In the case of Point, the type of T is unambiguously implied if and only if the two arguments
to Point have the same type. When this isn't the case, the constructor will fail with a MethodError:

julia> Point(1,2.5)
ERROR: MethodError: no method matching Point(::Int64, ::Float64)
Closest candidates are:
  Point(::T, !Matched::T) where T at none:2

Constructor methods to appropriately handle such mixed cases can be defined, but that will not
be discussed until later on in [Constructors](@ref man-constructors).

# Parametric Abstract Types

Parametric abstract type declarations declare a collection of abstract types, in much the same
way:

julia> abstract type Pointy{T} end

With this declaration, Pointy{T} is a distinct abstract type for each type or integer value
of T. As with parametric composite types, each such instance is a subtype of Pointy:

julia> Pointy{Int64} <: Pointy
true

julia> Pointy{1} <: Pointy
true

Parametric abstract types are invariant, much as parametric composite types are:

julia> Pointy{Float64} <: Pointy{Real}
false

julia> Pointy{Real} <: Pointy{Float64}
false

The notation Pointy{<:Real} can be used to express the Julia analogue of a
covariant type, while Pointy{>:Int} the analogue of a contravariant type,
but technically these represent sets of types (see UnionAll Types).

julia> Pointy{Float64} <: Pointy{<:Real}
true

julia> Pointy{Real} <: Pointy{>:Int}
true

Much as plain old abstract types serve to create a useful hierarchy of types over concrete types,
parametric abstract types serve the same purpose with respect to parametric composite types. We
could, for example, have declared Point{T} to be a subtype of Pointy{T} as follows:

julia> struct Point{T} <: Pointy{T}
           x::T
           y::T
       end

Given such a declaration, for each choice of T, we have Point{T} as a subtype of Pointy{T}:

julia> Point{Float64} <: Pointy{Float64}
true

julia> Point{Real} <: Pointy{Real}
true

julia> Point{AbstractString} <: Pointy{AbstractString}
true

This relationship is also invariant:

julia> Point{Float64} <: Pointy{Real}
false

julia> Point{Float64} <: Pointy{<:Real}
true

What purpose do parametric abstract types like Pointy serve? Consider if we create a point-like
implementation that only requires a single coordinate because the point is on the diagonal line
x = y:

julia> struct DiagPoint{T} <: Pointy{T}
           x::T
       end

Now both Point{Float64} and DiagPoint{Float64} are implementations of the Pointy{Float64}
abstraction, and similarly for every other possible choice of type T. This allows programming
to a common interface shared by all Pointy objects, implemented for both Point and DiagPoint.
This cannot be fully demonstrated, however, until we have introduced methods and dispatch in the
next section, Methods.

There are situations where it may not make sense for type parameters to range freely over all
possible types. In such situations, one can constrain the range of T like so:

julia> abstract type Pointy{T<:Real} end

With such a declaration, it is acceptable to use any type that is a subtype of
Real in place of T, but not types that are not subtypes of Real:

julia> Pointy{Float64}
Pointy{Float64}

julia> Pointy{Real}
Pointy{Real}

julia> Pointy{AbstractString}
ERROR: TypeError: in Pointy, in T, expected T<:Real, got Type{AbstractString}

julia> Pointy{1}
ERROR: TypeError: in Pointy, in T, expected T<:Real, got Int64

Type parameters for parametric composite types can be restricted in the same manner:

struct Point{T<:Real} <: Pointy{T}
    x::T
    y::T
end

To give a real-world example of how all this parametric type machinery can be useful, here is
the actual definition of Julia's Rational immutable type (except that we omit the
constructor here for simplicity), representing an exact ratio of integers:

struct Rational{T<:Integer} <: Real
    num::T
    den::T
end

It only makes sense to take ratios of integer values, so the parameter type T is restricted
to being a subtype of Integer, and a ratio of integers represents a value on the
real number line, so any Rational is an instance of the Real abstraction.

# Tuple Types

Tuples are an abstraction of the arguments of a function -- without the function itself. The salient
aspects of a function's arguments are their order and their types. Therefore a tuple type is similar
to a parameterized immutable type where each parameter is the type of one field. For example,
a 2-element tuple type resembles the following immutable type:

struct Tuple2{A,B}
    a::A
    b::B
end

However, there are three key differences:

  • Tuple types may have any number of parameters.
  • Tuple types are covariant in their parameters: Tuple{Int} is a subtype of Tuple{Any}. Therefore
    Tuple{Any} is considered an abstract type, and tuple types are only concrete if their parameters
    are.
  • Tuples do not have field names; fields are only accessed by index.

Tuple values are written with parentheses and commas. When a tuple is constructed, an appropriate
tuple type is generated on demand:

julia> typeof((1,"foo",2.5))
Tuple{Int64,String,Float64}

Note the implications of covariance:

julia> Tuple{Int,AbstractString} <: Tuple{Real,Any}
true

julia> Tuple{Int,AbstractString} <: Tuple{Real,Real}
false

julia> Tuple{Int,AbstractString} <: Tuple{Real,}
false

Intuitively, this corresponds to the type of a function's arguments being a subtype of the function's
signature (when the signature matches).

# Vararg Tuple Types

The last parameter of a tuple type can be the special type Vararg, which denotes any number
of trailing elements:

julia> mytupletype = Tuple{AbstractString,Vararg{Int}}
Tuple{AbstractString,Vararg{Int64,N} where N}

julia> isa(("1",), mytupletype)
true

julia> isa(("1",1), mytupletype)
true

julia> isa(("1",1,2), mytupletype)
true

julia> isa(("1",1,2,3.0), mytupletype)
false

Notice that Vararg{T} corresponds to zero or more elements of type T. Vararg tuple types are
used to represent the arguments accepted by varargs methods (see Varargs Functions).

The type Vararg{T,N} corresponds to exactly N elements of type T. NTuple{N,T} is a convenient
alias for Tuple{Vararg{T,N}}, i.e. a tuple type containing exactly N elements of type T.

# Named Tuple Types

Named tuples are instances of the NamedTuple type, which has two parameters: a tuple of
symbols giving the field names, and a tuple type giving the field types.

julia> typeof((a=1,b="hello"))
NamedTuple{(:a, :b),Tuple{Int64,String}}

A NamedTuple type can be used as a constructor, accepting a single tuple argument.
The constructed NamedTuple type can be either a concrete type, with both parameters specified,
or a type that specifies only field names:

julia> NamedTuple{(:a, :b),Tuple{Float32, String}}((1,""))
(a = 1.0f0, b = "")

julia> NamedTuple{(:a, :b)}((1,""))
(a = 1, b = "")

If field types are specified, the arguments are converted. Otherwise the types of the arguments
are used directly.

# Singleton Types

There is a special kind of abstract parametric type that must be mentioned here: singleton types.
For each type, T, the "singleton type" Type{T} is an abstract type whose only instance is
the object T. Since the definition is a little difficult to parse, let's look at some examples:

julia> isa(Float64, Type{Float64})
true

julia> isa(Real, Type{Float64})
false

julia> isa(Real, Type{Real})
true

julia> isa(Float64, Type{Real})
false

In other words, isa(A,Type{B}) is true if and only if A and B are the same object
and that object is a type. Without the parameter, Type is simply an abstract type which has
all type objects as its instances, including, of course, singleton types:

julia> isa(Type{Float64}, Type)
true

julia> isa(Float64, Type)
true

julia> isa(Real, Type)
true

Any object that is not a type is not an instance of Type:

julia> isa(1, Type)
false

julia> isa("foo", Type)
false

Until we discuss Parametric Methods and [conversions](@ref conversion-and-promotion), it is difficult to explain
the utility of the singleton type construct, but in short, it allows one to specialize function
behavior on specific type values. This is useful for writing methods (especially parametric
ones) whose behavior depends on a type that is given as an explicit argument rather than implied
by the type of one of its arguments.

A few popular languages have singleton types, including Haskell, Scala and Ruby. In general usage,
the term "singleton type" refers to a type whose only instance is a single value. This meaning
applies to Julia's singleton types, but with that caveat that only type objects have singleton
types.

# Parametric Primitive Types

Primitive types can also be declared parametrically. For example, pointers are represented as
primitive types which would be declared in Julia like this:

# 32-bit system:
primitive type Ptr{T} 32 end

# 64-bit system:
primitive type Ptr{T} 64 end

The slightly odd feature of these declarations as compared to typical parametric composite types,
is that the type parameter T is not used in the definition of the type itself -- it is just
an abstract tag, essentially defining an entire family of types with identical structure, differentiated
only by their type parameter. Thus, Ptr{Float64} and Ptr{Int64} are distinct types, even though
they have identical representations. And of course, all specific pointer types are subtypes of
the umbrella Ptr type:

julia> Ptr{Float64} <: Ptr
true

julia> Ptr{Int64} <: Ptr
true

# UnionAll Types

We have said that a parametric type like Ptr acts as a supertype of all its instances
(Ptr{Int64} etc.). How does this work? Ptr itself cannot be a normal data type, since without
knowing the type of the referenced data the type clearly cannot be used for memory operations.
The answer is that Ptr (or other parametric types like Array) is a different kind of type called a
UnionAll type. Such a type expresses the iterated union of types for all values of some parameter.

UnionAll types are usually written using the keyword where. For example Ptr could be more
accurately written as Ptr{T} where T, meaning all values whose type is Ptr{T} for some value
of T. In this context, the parameter T is also often called a "type variable" since it is
like a variable that ranges over types.
Each where introduces a single type variable, so these expressions are nested for types with
multiple parameters, for example Array{T,N} where N where T.

The type application syntax A{B,C} requires A to be a UnionAll type, and first substitutes B
for the outermost type variable in A.
The result is expected to be another UnionAll type, into which C is then substituted.
So A{B,C} is equivalent to A{B}{C}.
This explains why it is possible to partially instantiate a type, as in Array{Float64}: the first
parameter value has been fixed, but the second still ranges over all possible values.
Using explicit where syntax, any subset of parameters can be fixed. For example, the type of all
1-dimensional arrays can be written as Array{T,1} where T.

Type variables can be restricted with subtype relations.
Array{T} where T<:Integer refers to all arrays whose element type is some kind of
Integer.
The syntax Array{<:Integer} is a convenient shorthand for Array{T} where T<:Integer.
Type variables can have both lower and upper bounds.
Array{T} where Int<:T<:Number refers to all arrays of Numbers that are able to
contain Ints (since T must be at least as big as Int).
The syntax where T>:Int also works to specify only the lower bound of a type variable,
and Array{>:Int} is equivalent to Array{T} where T>:Int.

Since where expressions nest, type variable bounds can refer to outer type variables.
For example Tuple{T,Array{S}} where S<:AbstractArray{T} where T<:Real refers to 2-tuples
whose first element is some Real, and whose second element is an Array of any
kind of array whose element type contains the type of the first tuple element.

The where keyword itself can be nested inside a more complex declaration. For example,
consider the two types created by the following declarations:

julia> const T1 = Array{Array{T,1} where T, 1}
Array{Array{T,1} where T,1}

julia> const T2 = Array{Array{T,1}, 1} where T
Array{Array{T,1},1} where T

Type T1 defines a 1-dimensional array of 1-dimensional arrays; each
of the inner arrays consists of objects of the same type, but this type may vary from one inner array to the next.
On the other hand, type T2 defines a 1-dimensional array of 1-dimensional arrays all of whose inner arrays must have the
same type. Note that T2 is an abstract type, e.g., Array{Array{Int,1},1} <: T2, whereas T1 is a concrete type. As a consequence, T1 can be constructed with a zero-argument constructor a=T1() but T2 cannot.

There is a convenient syntax for naming such types, similar to the short form of function
definition syntax:

Vector{T} = Array{T,1}

This is equivalent to const Vector = Array{T,1} where T.
Writing Vector{Float64} is equivalent to writing Array{Float64,1}, and the umbrella type
Vector has as instances all Array objects where the second parameter -- the number of array
dimensions -- is 1, regardless of what the element type is. In languages where parametric types
must always be specified in full, this is not especially helpful, but in Julia, this allows one
to write just Vector for the abstract type including all one-dimensional dense arrays of any
element type.

# Type Aliases

Sometimes it is convenient to introduce a new name for an already expressible type.
This can be done with a simple assignment statement.
For example, UInt is aliased to either UInt32 or UInt64 as is
appropriate for the size of pointers on the system:

# 32-bit system:
julia> UInt
UInt32

# 64-bit system:
julia> UInt
UInt64

This is accomplished via the following code in base/boot.jl:

if Int === Int64
    const UInt = UInt64
else
    const UInt = UInt32
end

Of course, this depends on what Int is aliased to -- but that is predefined to be the correct
type -- either Int32 or Int64.

(Note that unlike Int, Float does not exist as a type alias for a specific sized
AbstractFloat. Unlike with integer registers, where the size of Int
reflects the size of a native pointer on that machine, the floating point register sizes
are specified by the IEEE-754 standard.)

# Operations on Types

Since types in Julia are themselves objects, ordinary functions can operate on them. Some functions
that are particularly useful for working with or exploring types have already been introduced,
such as the <: operator, which indicates whether its left hand operand is a subtype of its right
hand operand.

The isa function tests if an object is of a given type and returns true or false:

julia> isa(1, Int)
true

julia> isa(1, AbstractFloat)
false

The typeof function, already used throughout the manual in examples, returns the type
of its argument. Since, as noted above, types are objects, they also have types, and we can ask
what their types are:

julia> typeof(Rational{Int})
DataType

julia> typeof(Union{Real,String})
Union

What if we repeat the process? What is the type of a type of a type? As it happens, types are
all composite values and thus all have a type of DataType:

julia> typeof(DataType)
DataType

julia> typeof(Union)
DataType

DataType is its own type.

Another operation that applies to some types is supertype, which reveals a type's
supertype. Only declared types (DataType) have unambiguous supertypes:

julia> supertype(Float64)
AbstractFloat

julia> supertype(Number)
Any

julia> supertype(AbstractString)
Any

julia> supertype(Any)
Any

If you apply supertype to other type objects (or non-type objects), a MethodError
is raised:

julia> supertype(Union{Float64,Int64})
ERROR: MethodError: no method matching supertype(::Type{Union{Float64, Int64}})
Closest candidates are:
  supertype(!Matched::DataType) at operators.jl:42
  supertype(!Matched::UnionAll) at operators.jl:47

# Custom pretty-printing

Often, one wants to customize how instances of a type are displayed. This is accomplished by
overloading the show function. For example, suppose we define a type to represent
complex numbers in polar form:

julia> struct Polar{T<:Real} <: Number
           r::T
           Θ::T
       end

julia> Polar(r::Real,Θ::Real) = Polar(promote(r,Θ)...)
Polar

Here, we've added a custom constructor function so that it can take arguments of different
Real types and promote them to a common type (see [Constructors](@ref man-constructors)
and [Conversion and Promotion](@ref conversion-and-promotion)).
(Of course, we would have to define lots of other methods, too, to make it act like a
Number, e.g. +, *, one, zero, promotion rules and so on.) By default,
instances of this type display rather simply, with information about the type name and
the field values, as e.g. Polar{Float64}(3.0,4.0).

If we want it to display instead as 3.0 * exp(4.0im), we would define the following method to
print the object to a given output object io (representing a file, terminal, buffer, etcetera;
see Networking and Streams):

julia> Base.show(io::IO, z::Polar) = print(io, z.r, " * exp(", z.Θ, "im)")

More fine-grained control over display of Polar objects is possible. In particular, sometimes
one wants both a verbose multi-line printing format, used for displaying a single object in the
REPL and other interactive environments, and also a more compact single-line format used for
print or for displaying the object as part of another object (e.g. in an array). Although
by default the show(io, z) function is called in both cases, you can define a different multi-line
format for displaying an object by overloading a three-argument form of show that takes the
text/plain MIME type as its second argument (see Multimedia I/O), for example:

julia> Base.show(io::IO, ::MIME"text/plain", z::Polar{T}) where{T} =
           print(io, "Polar{$T} complex number:\n   ", z)

(Note that print(..., z) here will call the 2-argument show(io, z) method.) This results in:

julia> Polar(3, 4.0)
Polar{Float64} complex number:
   3.0 * exp(4.0im)

julia> [Polar(3, 4.0), Polar(4.0,5.3)]
2-element Array{Polar{Float64},1}:
 3.0 * exp(4.0im)
 4.0 * exp(5.3im)

where the single-line show(io, z) form is still used for an array of Polar values. Technically,
the REPL calls display(z) to display the result of executing a line, which defaults to show(stdout, MIME("text/plain"), z),
which in turn defaults to show(stdout, z), but you should not define new display
methods unless you are defining a new multimedia display handler (see Multimedia I/O).

Moreover, you can also define show methods for other MIME types in order to enable richer display
(HTML, images, etcetera) of objects in environments that support this (e.g. IJulia). For example,
we can define formatted HTML display of Polar objects, with superscripts and italics, via:

julia> Base.show(io::IO, ::MIME"text/html", z::Polar{T}) where {T} =
           println(io, "<code>Polar{$T}</code> complex number: ",
                   z.r, " <i>e</i><sup>", z.Θ, " <i>i</i></sup>")

A Polar object will then display automatically using HTML in an environment that supports HTML
display, but you can call show manually to get HTML output if you want:

julia> show(stdout, "text/html", Polar(3.0,4.0))
<code>Polar{Float64}</code> complex number: 3.0 <i>e</i><sup>4.0 <i>i</i></sup>
<p>An HTML renderer would display this as: <code>Polar{Float64}</code> complex number: 3.0 <i>e</i><sup>4.0 <i>i</i></sup></p>

As a rule of thumb, the single-line show method should print a valid Julia expression for creating
the shown object. When this show method contains infix operators, such as the multiplication
operator (*) in our single-line show method for Polar above, it may not parse correctly when
printed as part of another object. To see this, consider the expression object (see Program
representation
) which takes the square of a specific instance of our Polar type:

julia> a = Polar(3, 4.0)
Polar{Float64} complex number:
   3.0 * exp(4.0im)

julia> print(:($a^2))
3.0 * exp(4.0im) ^ 2

Because the operator ^ has higher precedence than * (see Operator Precedence and Associativity), this
output does not faithfully represent the expression a ^ 2 which should be equal to (3.0 * exp(4.0im)) ^ 2. To solve this issue, we must make a custom method for Base.show_unquoted(io::IO, z::Polar, indent::Int, precedence::Int), which is called internally by the expression object when
printing:

julia> function Base.show_unquoted(io::IO, z::Polar, ::Int, precedence::Int)
           if Base.operator_precedence(:*) <= precedence
               print(io, "(")
               show(io, z)
               print(io, ")")
           else
               show(io, z)
           end
       end

julia> :($a^2)
:((3.0 * exp(4.0im)) ^ 2)

The method defined above adds parentheses around the call to show when the precedence of the
calling operator is higher than or equal to the precedence of multiplication. This check allows
expressions which parse correctly without the parentheses (such as :($a + 2) and :($a == 2)) to
omit them when printing:

julia> :($a + 2)
:(3.0 * exp(4.0im) + 2)

julia> :($a == 2)
:(3.0 * exp(4.0im) == 2)

In some cases, it is useful to adjust the behavior of show methods depending
on the context. This can be achieved via the IOContext type, which allows
passing contextual properties together with a wrapped IO stream.
For example, we can build a shorter representation in our show method
when the :compact property is set to true, falling back to the long
representation if the property is false or absent:

julia> function Base.show(io::IO, z::Polar)
           if get(io, :compact, false)
               print(io, z.r, "ℯ", z.Θ, "im")
           else
               print(io, z.r, " * exp(", z.Θ, "im)")
           end
       end

This new compact representation will be used when the passed IO stream is an IOContext
object with the :compact property set. In particular, this is the case when printing
arrays with multiple columns (where horizontal space is limited):

julia> show(IOContext(stdout, :compact=>true), Polar(3, 4.0))
3.0ℯ4.0im

julia> [Polar(3, 4.0) Polar(4.0,5.3)]
1×2 Array{Polar{Float64},2}:
 3.0ℯ4.0im  4.0ℯ5.3im

See the IOContext documentation for a list of common properties which can be used
to adjust printing.

# "Value types"

In Julia, you can't dispatch on a value such as true or false. However, you can dispatch
on parametric types, and Julia allows you to include "plain bits" values (Types, Symbols, Integers,
floating-point numbers, tuples, etc.) as type parameters. A common example is the dimensionality
parameter in Array{T,N}, where T is a type (e.g., Float64) but N is just an Int.

You can create your own custom types that take values as parameters, and use them to control dispatch
of custom types. By way of illustration of this idea, let's introduce a parametric type, Val{x},
and a constructor Val(x) = Val{x}(), which serves as a customary way to exploit this technique
for cases where you don't need a more elaborate hierarchy.

Val is defined as:

julia> struct Val{x}
       end

julia> Val(x) = Val{x}()
Val

There is no more to the implementation of Val than this. Some functions in Julia's standard
library accept Val instances as arguments, and you can also use it to write your own functions.
For example:

julia> firstlast(::Val{true}) = "First"
firstlast (generic function with 1 method)

julia> firstlast(::Val{false}) = "Last"
firstlast (generic function with 2 methods)

julia> firstlast(Val(true))
"First"

julia> firstlast(Val(false))
"Last"

For consistency across Julia, the call site should always pass a Valinstance rather than using
a type, i.e., use foo(Val(:bar)) rather than foo(Val{:bar}).

It's worth noting that it's extremely easy to mis-use parametric "value" types, including Val;
in unfavorable cases, you can easily end up making the performance of your code much worse.
In particular, you would never want to write actual code as illustrated above. For more information
about the proper (and improper) uses of Val, please read the more extensive discussion in the performance tips.

[1]
"Small" is defined by the MAX_UNION_SPLITTING constant, which is currently set to 4.