类型

类型

通常，我们把程序语言中的类型系统划分成两类：静态类型和动态类型。对于静态类型系统，在程序运行之前，我们就知道每一个表达式的类型。而对于动态类型系统，我们只有通过运行那个程序，得到表达式具体的值，才能确定其具体的类型。在静态类型语言中，通常我们可以在不知道具体类型的情况下写一些代码，这种将一段代码用在多个类型的能力被称为多态性。在经典的动态类型语言中，所有的代码都是多态的，这意味着这些代码对于其中值的类型没有约束，除非在代码中去具体的判断一个值的类型，或者对对象做一些它不支持的操作。

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 @ref">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.
值有类型，变量没有类型——变量仅仅是绑定了值的名字而已。
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.

类型断言

::运算符可以用来在程序中给表达式和变量附加类型注释。这有两个主要原因：

作为断言，帮助程序确认能是否正常运行，
给编译器提供额外的类型信息，这可能帮助程序提升性能，在某些情况下
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

可以在任何表达式的所在位置做类型断言。

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

这个特性用于避免性能“陷阱”，即给一个变量赋值时意外更改了类型。

此“声明”行为仅发生在特定上下文中：

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

并适用于整个当前范围，甚至在声明之前。目前，声明类型不能用于全局范围，例如在 REPL 中就不可以，因为Julia 还没有定型的全局变量。

声明也可以附加到函数定义：

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

从此函数返回的值就像对具有声明类型的变量的赋值：值始终转换为Float64。

抽象类型

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 @ref">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.

使用abstract type关键词来声明抽象类型。抽象类型的一般语法时：

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

该abstract type 关键字引入了一个新的抽象类型，«name»为其名称。此名称后面可以跟<:和一个已存在的类型，表示新声明的抽象类型是此“父”类型的子类型。

如果没有给出父类型，则默认父类型为Any——所有对象和类型都是这个抽象类型的子类型。在类型理论中，Any通常称为"top"，因为它位于类型图的顶峰。Julia还有一个预定义的抽象"bottom"类型，在类型图的最低点，写成Union{}。这与Any完全相反：任何对象都不是Union{}的实例，所有的类型都是Union{}的父类型。

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

Number类型为Any类型的直接子类型，并且Real为它的子类型。反过来，Real有两个子类型（它还有更多的子类型，但这里只展示了两个，稍后将会看到更多的类型）： Integer 和AbstractFloat，将”数字世界“分为整数和实数的两部分。实数当然还包括浮点类型和其他类型，例如有理数。因此，AbstractFloat是一个Real的子类型，仅包括实数的浮点表示。整数被进一步细分为Signed和Unsigned 两类。

<:运算符的含义是”前者是后者的子类型“，被用于声明类型，它声明右侧是左侧新声明类型的直接父类型。也可以用来判断左侧是不是右侧的子类型，当其左侧是右侧的子类型时返回true

julia> Integer <: Number
true
julia> Integer <: AbstractFloat
false

抽象类型的一个重要用途是为具体类型提供默认实现。举一个简单的例子：

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 @ref">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

最后，调用这个具体的函数。

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.)

位类型

位类型是具体类型，其数据是由位构成。位类型的经典示例是整数和浮点数。与大多数语言不同，Julia允许您声明自己的位类型，而不是仅提供一组固定的内置类型。实际上，标准位类型都是在语言本身中定义的：

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

声明位类型的一般语法是：

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

bits表示该类型需要多少存储空间，name为新类型指定名称。可已经一个位类型声明为某个父类型的子类型。如果省略父类型，则默认Any为其直接父类型。上述声明中意味着Bool类型需要8位来储存，并且直接父类型为Integer。目前，仅支持8位倍数的大小。因此，布尔值虽然确实只需要一位，但不能声明为小于八位的值。

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.

复合类型

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 @ref">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.

struct关键词用于构造复合类型，后跟一个字段的名称，可选择使用::运算符注释类型：

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

没有类型注释的字段默认位Any类型，所以可以保存任何类型的值。

类型为Foo的新对象是通过将Foo类型对象（如函数）应用于其字段的值来创建的：

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.

由于bar字段在类型上不受限制，因此任何值都可以。但是baz的值必须可转换为Int类型：

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

可以使用fieldnames 函数找到字段名称列表.

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

可以使用传统的foo.bar表示法访问复合对象的字段值：

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

===函数用来确认”两个“构造的实例NoFields实际上是同一个。单态类型将在下面进一步详细描述。

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

可变复合类型

如果使用mutable struct而不是声明复合类型struct，则可以修改它的实例：

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).

在这个系统里的每一个具体的值都是某个DataType的实例。

Type Unions

类型共用体是一种特殊的抽象类型，它包含作为对象的任何参数类型的所有实例，使用特殊Union关键字构造：

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 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

当然，其他类型不是它的子类型：

julia> Float64 <: Point
false
julia> AbstractString <: Point
false

Point不同T值所声明的具体类型之间，不能互相作为子类型：

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

一种正确的方法来定义一个接受类型的所有参数的方法，Point{T}其中T是一个子类型Real：

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

(等效地, 另一种定义方法 function norm(p::Point{T} where T<:Real) 或function norm(p::Point{T}) where T<:Real; 查看UnionAll Types.)

稍后将在方法中讨论更多示例。

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, 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}

对于默认的构造函数，必须为每个字段提供一个参数：

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}

在上例中，当且仅当两个参数类型相同时，T的类型才能明确暗示，同时'Point具有相同的类型。如果不是这种情况，即参数类型不同时，构造函数将失败并显示@ref">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

可以定义适当处理此类混合情况的函数构造方法，将在后面的构造函数中讨论。

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

鉴于此类声明，对每个T ，都有 Point{T} 是 Pointy{T} 的子类型：

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

它们仍然不互为子类：

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, @ref">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.

元组类型

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

然而，有三个主要差异：

元组类型可以具有任意数量的参数。
Tuple types are covariant in their parameters: Tuple{Int} is a subtype of Tuple{Any}. Therefore Tuple{Any}被认为是一种抽象类型，而元组类型只有它们的参数有时才具体 are.
元组没有字段名称; 字段只能通过索引访问。
元组值用括号和逗号书写。构造元组时，会根据需要生成适当的元组类型：

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

元组类型的最后一个参数可以是特殊类型Vararg，它表示任意数量的尾随参数：

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 @ref">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.

单态类型

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

只有对象是类型时，才是 Type的实例：

julia> isa(1, Type)
false
julia> isa("foo", Type)
false

Until we discuss @ref">Parametric Methods and conversions, 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

在base/boot.jl中，通过以下代码实现:

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, the floating point register sizes are specified by the IEEE-754 standard. Whereas the size of Int reflects the size of a native pointer on that machine.)

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,Float64,Rational})
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 and 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 @ref">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 @ref">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 @ref">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>

An HTML renderer would display this as: Polar{Float64} complex number: 3.0 e^{4.0 i}

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 @ref">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 @ref">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.

原文: https://juliacn.github.io/JuliaZH.jl/latest/manual/types/

类型

类型