Types

# 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 programs without ever explicitly using types. When additional expressiveness is needed, however, it is easy to gradually introduce explicit type annotations into previously "untyped" code. Doing so will typically increase both the performance and robustness of these systems, and perhaps somewhat counterintuitively, often significantly simplify them.

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: 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.)

## 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 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()
    Stacktrace:
     [1] convert(::Type{Int64}, ::Float64) at ./float.jl:679
     [2] Foo(::Tuple{}, ::Float64, ::Int64) at ./none:2

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

    julia> fieldnames(foo)
    3-element Array{Symbol,1}:
     :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.

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.

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.

## 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:

- An object with an immutable type is passed around (both in assignment statements and in function calls) by copying, whereas a mutable type is passed around by reference.

- It is not permitted to modify the fields of a composite immutable type.

It is instructive, particularly for readers whose background is C/C++, to consider why these two properties go hand in hand. If they were separated, i.e., if the fields of objects passed around by copying could be modified, then it would become more difficult to reason about certain instances of generic code. For example, suppose `x` is a function argument of an abstract type, and suppose that the function changes a field: `x.isprocessed = true`. Depending on whether `x` is passed by copying or by reference, this statement may or may not alter the actual argument in the calling routine. Julia sidesteps the possibility of creating functions with unknown effects in this scenario by forbidding modification of fields of objects passed around by copying.

## Declared Types

The three kinds of types discussed in the previous three 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 bits 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` function:

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

    julia> 1 :: IntOrString
    1

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

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

The compilers for many languages have an internal union construct for reasoning about types; Julia simply exposes it to the programmer.

## 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 bits 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{T<:Real}(p::Point{T})` 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, 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: Cannot `convert` an object of type Float64 to an object of type Point{Float64}
    This may have arisen from a call to the constructor Point{Float64}(...),
    since type constructors fall back to convert methods.
    Stacktrace:
     [1] Point{Float64}(::Float64) at ./sysimg.jl:77

    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.

### 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: Pointy: in T, expected T<:Real, got Type{AbstractString}

    julia> Pointy{1}
    ERROR: TypeError: 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`.

#### 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, 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 `Number`s that are able to contain `Int`s (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, 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:41
      supertype(!Matched::UnionAll) at operators.jl:46

## 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 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{T}(io::IO, ::MIME"text/plain", z::Polar{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{T}(io::IO, ::MIME"text/html", z::Polar{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 e4.0 i

## "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{T}`, 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{T}
           end

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

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

    julia> firstlast(::Type{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 `Val` type rather than creating an instance, 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.

## Nullable Types: Representing Missing Values

In many settings, you need to interact with a value of type `T` that may or may not exist. To handle these settings, Julia provides a parametric type called `Nullable{T}`, which can be thought of as a specialized container type that can contain either zero or one values. `Nullable{T}` provides a minimal interface designed to ensure that interactions with missing values are safe. At present, the interface consists of several possible interactions:

- Construct a `Nullable` object.

- Check if a `Nullable` object has a missing value.

- Access the value of a `Nullable` object with a guarantee that a `NullException` will be thrown if the object's value is missing.

- Access the value of a `Nullable` object with a guarantee that a default value of type `T` will be returned if the object's value is missing.

- Perform an operation on the value (if it exists) of a `Nullable` object, getting a `Nullable` result. The result will be missing if the original value was missing.

- Performing a test on the value (if it exists) of a `Nullable` object, getting a result that is missing if either the `Nullable` itself was missing, or the test failed.

- Perform general operations on single `Nullable` objects, propagating the missing data.

### Constructing `Nullable` objects

To construct an object representing a missing value of type `T`, use the `Nullable{T}()` function:

    julia> x1 = Nullable{Int64}()
    Nullable{Int64}()

    julia> x2 = Nullable{Float64}()
    Nullable{Float64}()

    julia> x3 = Nullable{Vector{Int64}}()
    Nullable{Array{Int64,1}}()

To construct an object representing a non-missing value of type `T`, use the `Nullable(x::T)` function:

    julia> x1 = Nullable(1)
    Nullable{Int64}(1)

    julia> x2 = Nullable(1.0)
    Nullable{Float64}(1.0)

    julia> x3 = Nullable([1, 2, 3])
    Nullable{Array{Int64,1}}([1, 2, 3])

Note the core distinction between these two ways of constructing a `Nullable` object: in one style, you provide a type, `T`, as a function parameter; in the other style, you provide a single value of type `T` as an argument.

### Checking if a `Nullable` object has a value

You can check if a `Nullable` object has any value using `isnull()`:

    julia> isnull(Nullable{Float64}())
    true

    julia> isnull(Nullable(0.0))
    false

### Safely accessing the value of a `Nullable` object

You can safely access the value of a `Nullable` object using `get()`:

    julia> get(Nullable{Float64}())
    ERROR: NullException()
    Stacktrace:
     [1] get(::Nullable{Float64}) at ./nullable.jl:92

    julia> get(Nullable(1.0))
    1.0

If the value is not present, as it would be for `Nullable{Float64}`, a `NullException` error will be thrown. The error-throwing nature of the `get()` function ensures that any attempt to access a missing value immediately fails.

In cases for which a reasonable default value exists that could be used when a `Nullable` object's value turns out to be missing, you can provide this default value as a second argument to `get()`:

    julia> get(Nullable{Float64}(), 0.0)
    0.0

    julia> get(Nullable(1.0), 0.0)
    1.0

Tip

Make sure the type of the default value passed to `get()` and that of the `Nullable` object match to avoid type instability, which could hurt performance. Use `convert()` manually if needed.

### Performing operations on `Nullable` objects

`Nullable` objects represent values that are possibly missing, and it is possible to write all code using these objects by first testing to see if the value is missing with `isnull()`, and then doing an appropriate action. However, there are some common use cases where the code could be more concise or clear by using a higher-order function.

The `map` function takes as arguments a function `f` and a `Nullable` value `x`. It produces a `Nullable`:

- If `x` is a missing value, then it produces a missing value;

- If `x` has a value, then it produces a `Nullable` containing `f(get(x))` as value.

This is useful for performing simple operations on values that might be missing if the desired behaviour is to simply propagate the missing values forward.

The `filter` function takes as arguments a predicate function `p` (that is, a function returning a boolean) and a `Nullable` value `x`. It produces a `Nullable` value:

- If `x` is a missing value, then it produces a missing value;

- If `p(get(x))` is true, then it produces the original value `x`;

- If `p(get(x))` is false, then it produces a missing value.

In this way, `filter` can be thought of as selecting only allowable values, and converting non-allowable values to missing values.

While `map` and `filter` are useful in specific cases, by far the most useful higher-order function is `broadcast`, which can handle a wide variety of cases, including making existing operations work and propagate `Nullable`s. An example will motivate the need for `broadcast`. Suppose we have a function that computes the greater of two real roots of a quadratic equation, using the quadratic formula:

    julia> root(a::Real, b::Real, c::Real) = (-b + √(b^2 - 4a*c)) / 2a
    root (generic function with 1 method)

We may verify that the result of `root(1, -9, 20)` is `5.0`, as we expect, since `5.0` is the greater of two real roots of the quadratic equation.

Suppose now that we want to find the greatest real root of a quadratic equations where the coefficients might be missing values. Having missing values in datasets is a common occurrence in real-world data, and so it is important to be able to deal with them. But we cannot find the roots of an equation if we do not know all the coefficients. The best solution to this will depend on the particular use case; perhaps we should throw an error. However, for this example, we will assume that the best solution is to propagate the missing values forward; that is, if any input is missing, we simply produce a missing output.

The `broadcast()` function makes this task easy; we can simply pass the `root` function we wrote to `broadcast`:

    julia> broadcast(root, Nullable(1), Nullable(-9), Nullable(20))
    Nullable{Float64}(5.0)

    julia> broadcast(root, Nullable(1), Nullable{Int}(), Nullable{Int}())
    Nullable{Float64}()

    julia> broadcast(root, Nullable{Int}(), Nullable(-9), Nullable(20))
    Nullable{Float64}()

If one or more of the inputs is missing, then the output of `broadcast()` will be missing.

There exists special syntactic sugar for the `broadcast()` function using a dot notation:

    julia> root.(Nullable(1), Nullable(-9), Nullable(20))
    Nullable{Float64}(5.0)

In particular, the regular arithmetic operators can be `broadcast()` conveniently using `.`-prefixed operators:

    julia> Nullable(2) ./ Nullable(3) .+ Nullable(1.0)
    Nullable{Float64}(1.66667)