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 many cases

The :: operator is read as “is an instance of” and 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)::FloatingPoint
type error: typeassert: expected FloatingPoint, got Int64

julia> (1+2)::Int
3

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

When attached to a variable, 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 the convert function:

julia> function foo()
         x::Int8 = 1000
         x
       end

julia> foo()
-24

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.

The “declaration” behavior only occurs in specific contexts:

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

In value contexts, such as f(x::Int8), the :: is a type assertion again and not a declaration. Note that these declarations cannot be used in global scope currently, in the REPL, since julia does not yet have constant-type globals.

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 Números Inteiros e de Ponto Flutuante, we introduced a variety of concrete types of numeric values: Int8, Uint8, Int16, Uint16, Int32, Uint32, Int64, Uint64, Float32, and Float64. These are all bits types, which we will discuss in the next section. Although they have different representation sizes, Int8, Int16, Int32 and Int64 all have in common that they are signed integer types. Likewise Uint8, Uint16, Uint32 and Uint64 are all unsigned integer types, while 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, as long as the appropriate low-level implementations of integer operations are used. 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 keyword. The general syntaxes for declaring an abstract type are:

abstract «name»
abstract «name» <: «supertype»

The abstract 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 called None. It is the exact opposite of Any: no object is an instance of None and all types are supertypes of None.

As a specific example, let’s consider a subset of the abstract types that make up Julia’s numerical hierarchy:

abstract Number
abstract Real     <: Number
abstract FloatingPoint <: Real
abstract Integer  <: Real
abstract Signed   <: Integer
abstract Unsigned <: Integer

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 FloatingPoint, 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 Julia’s rationals. Hence, FloatingPoint 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 <: FloatingPoint
false

Since abstract types have no instantiations and serve as no more than nodes in the type graph, there is not much more to say about them until we introduce parametric abstract types later on in Parametric Types.

Bits Types¶

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

bitstype 32 Float32 <: FloatingPoint
bitstype 64 Float64 <: FloatingPoint

bitstype 8  Bool <: Integer
bitstype 32 Char <: Integer

bitstype 8  Int8   <: Signed
bitstype 8  Uint8  <: Unsigned
bitstype 16 Int16  <: Signed
bitstype 16 Uint16 <: Unsigned
bitstype 32 Int32  <: Signed
bitstype 32 Uint32 <: Unsigned
bitstype 64 Int64  <: Signed
bitstype 64 Uint64 <: Unsigned

The general syntaxes for declaration of a bitstypes are:

bitstype «bits» «name»
bitstype «bits» «name» <: «supertype»

The number of bits indicates how much storage the type requires and the name gives the new type a name. A bits 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. Another 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, structures (“structs” in C), 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 Python and Ruby, 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, as in Python and Ruby, 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 by association with function objects rather than simply having named bags of methods “inside” each object ends up being a highly beneficial aspect of the language design.

Since composite types are the most common form of user-defined concrete type, they are simply introduced with the type keyword followed by a block of field names, optionally annotated with types using the :: operator:

type 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 composite 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

Since the bar field is unconstrained in type, any value will do; the value for baz must be an Int and qux must be a Float64. The signature of the default constructor is taken directly from the field type declarations (Any,Int,Float64), so arguments must match this implied type signature:

julia> Foo((), 23.5, 1)
no method Foo((),Float64,Int64)

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

You can also change the values as one would expect:

julia> foo.qux = 2
2.0

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

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

type NoFields
end

julia> is(NoFields(), NoFields())
true

The is function confirms that the “two” constructed instances of NoFields are actually one and the same.

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.

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,String)
Union(Int,String)

julia> 1 :: IntOrString
1

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

julia> 1.0 :: IntOrString
type error: typeassert: expected Union(Int,String), got Float64

The compilers for many languages have an internal union construct for reasoning about types; Julia simply exposes it to the programmer. The union of no types is the “bottom” type, None:

julia> Union()
None

Recall from the discussion above that None is the abstract type which is the subtype of all other types, and which no object is an instance of. Since a zero-argument Union call has no argument types for objects to be instances of, it should produce a type which no objects are instances of — i.e. None.

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. The type of a tuple of values is the tuple of types of values:

julia> typeof((1,"foo",2.5))
(Int64,ASCIIString,Float64)

Accordingly, a tuple of types can be used anywhere a type is expected:

julia> (1,"foo",2.5) :: (Int64,String,Any)
(1,"foo",2.5)

julia> (1,"foo",2.5) :: (Int64,String,Float32)
type error: typeassert: expected (Int64,String,Float32), got (Int64,ASCIIString,Float64)

If one of the components of the tuple is not a type, however, you will get an error:

julia> (1,"foo",2.5) :: (Int64,String,3)
type error: typeassert: expected Type{T}, got (BitsKind,AbstractKind,Int64)

Note that the empty tuple () is its own type:

julia> typeof(())
()

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.

The only kinds of types that are declared are abstract types, bits types, and composite types. All such types can be parameterized, with the same syntax in each case. We will discuss them in in the following order: first, parametric composite types, then parametric abstract types, and finally parametric bits types.

type Point{T}
  x::T
  y::T
end

julia> Point{Float64}
Point{Float64}

julia> Point{String}
Point{String}

julia> Point
Point{T}

julia> Point{Float64} <: Point
true

julia> Point{String} <: Point
true

julia> Float64 <: Point
false

julia> String <: Point
false

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

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

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

julia> typeof(ans)
Point{Float64}

julia> Point{Float64}(1.0)
no method Point(Float64,)

julia> Point{Float64}(1.0,2.0,3.0)
no method Point(Float64,Float64,Float64)

julia> Point(1.0,2.0)
Point(1.0,2.0)

julia> typeof(ans)
Point{Float64}

julia> Point(1,2)
Point(1,2)

julia> typeof(ans)
Point{Int64}

julia> Point(1,2.5)
no method Point(Int64,Float64)

abstract Pointy{T}

julia> Pointy{Int64} <: Pointy
true

julia> Pointy{1} <: Pointy
true

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

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

type Point{T} <: Pointy{T}
  x::T
  y::T
end

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

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

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

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

type DiagPoint{T} <: Pointy{T}
  x::T
end

abstract Pointy{T<:Real}

julia> Pointy{Float64}
Pointy{Float64}

julia> Pointy{Real}
Pointy{Real}

julia> Pointy{String}
type error: Pointy: in T, expected Real, got AbstractKind

julia> Pointy{1}
type error: Pointy: in T, expected Real, got Int64

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

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

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

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

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

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

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

julia> isa(Float64,Type)
true

julia> isa(Real,Type)
true

julia> isa(1,Type)
false

julia> isa("foo",Type)
false

# 32-bit system:
bitstype 32 Ptr{T}

# 64-bit system:
bitstype 64 Ptr{T}

julia> Ptr{Float64} <: Ptr
true

julia> Ptr{Int64} <: Ptr
true

Type Aliases¶

Sometimes it is convenient to introduce a new name for an already expressible type. For such occasions, Julia provides the typealias mechanism. For example, Uint is type 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 is(Int,Int64)
    typealias Uint Uint64
else
    typealias Uint Uint32
end

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

For parametric types, typealias can be convenient for providing a new parametric types name where one of the parameter choices is fixed. Julia’s arrays have type Array{T,n} where T is the element type and n is the number of array dimensions. For convenience, writing Array{Float64} allows one to specify the element type without specifying the dimension:

julia> Array{Float64,1} <: Array{Float64} <: Array
true

However, there is no way to equally simply restrict just the dimension but not the element type. Yet, one often needs to ensure an object is a vector or a matrix (imposing restrictions on the number of dimensions). For that reason, the following type aliases are provided:

typealias Vector{T} Array{T,1}
typealias Matrix{T} Array{T,2}

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 be always specified in full, this is not especially helpful, but in Julia, this allows one to write just Matrix for the abstract type including all two-dimensional dense arrays of any element type.

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,FloatingPoint)
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. Here we apply typeof to an instance of each of the kinds of types discussed above:

julia> typeof(Real)
AbstractKind

julia> typeof(Float64)
BitsKind

julia> typeof(Rational)
CompositeKind

julia> typeof(Union(Real,Float64,Rational))
UnionKind

julia> typeof((Real,Float64,Rational,None))
(AbstractKind,BitsKind,CompositeKind,UnionKind)

As you can see, the types of types are called, by convention, “kinds”:

• Abstract types have type AbstractKind
• Bits types have type BitsKind
• Composite types have type CompositeKind
• Unions have type UnionKind
• Tuples of types have a type that is the tuple of their respective kinds.

What if we repeat the process? What is the type of a kind? Kinds, as it happens, are all composite values and thus all have a type of CompositeKind:

julia> typeof(AbstractKind)
CompositeKind

julia> typeof(BitsKind)
CompositeKind

julia> typeof(CompositeKind)
CompositeKind

julia> typeof(UnionKind)
CompositeKind

The reader may note that CompositeKind shares with the empty tuple (see above), the distinction of being its own type (i.e. a fixed point of the typeof function). This leads any number of tuple types recursively built with () and CompositeKind as their only atomic values, which are their own type:

julia> typeof(())
()

julia> typeof(CompositeKind)
CompositeKind

julia> typeof(((),))
((),)

julia> typeof((CompositeKind,))
(CompositeKind,)

julia> typeof(((),CompositeKind))
((),CompositeKind)

All fixed points of the typeof function are like this.

Another operation that applies to some kinds of types is super. Only abstract types (AbstractKind), bits types (BitsKind), and composite types (CompositeKind) have a supertype, so these are the only kinds of types that the super function applies to:

julia> super(Float64)
FloatingPoint

julia> super(Number)
Any

julia> super(String)
Any

julia> super(Any)
Any

If you apply super to other type objects (or non-type objects), a “no method” error is raised:

julia> super(Union(Float64,Int64))
no method super(UnionKind,)

julia> super(None)
no method super(UnionKind,)

julia> super((Float64,Int64))
no method super((BitsKind,BitsKind),)