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, parametric and dependent. Generic types can be parameterized by other types and by integers, 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 paramaterized by other types and by integers. Type parameters may be completely omitted when they do not need to be explicitly 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:
- As an assertion to help confirm that your program works the way you expect,
- 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 Integers and Floating-Point Numbers, 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.
Parametric Composite Types¶
Type parameters are introduced immediately after the type name, surrounded by curly braces:
type 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 an integer, 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{String}
,
Point{Int64}
, etc. Each of these is now a usable concrete type:
julia> Point{Float64}
Point{Float64}
julia> Point{String}
Point{String}
The type Point{Float64}
is a point whose coordinates are 64-bit
floating-point values, while the type Point{String}
is a “point”
whose “coordinates” are string objects (see Strings).
However, Point
itself is also a valid type object:
julia> Point
Point{T}
Here the T
is the dummy type symbol used in the original declaration
of Point
. What does Point
by itself mean? It is an abstract type
that contains all the specific instances Point{Float64}
,
Point{String}
, etc.:
julia> Point{Float64} <: Point
true
julia> Point{String} <: Point
true
Other types, of course, are not subtypes of it:
julia> Float64 <: Point
false
julia> String <: 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
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 ofReal
. Since objects that are instances ofReal
can be of arbitrary size and structure, in practice an instance ofPoint{Real}
must be represented as a pair of pointers to individually allocatedReal
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.
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(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)
no method Point(Float64,)
julia> Point{Float64}(1.0,2.0,3.0)
no method Point(Float64,Float64,Float64)
The provided arguments need to match the field types exactly, in this
case (Float64,Float64)
, as with all composite type default
constructors.
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(1.0,2.0)
julia> typeof(ans)
Point{Float64}
julia> Point(1,2)
Point(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 no method error:
julia> Point(1,2.5)
no method Point(Int64,Float64)
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:
abstract Pointy{T}
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
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:
type 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{String} <: Pointy{String}
true
This relationship is also invariant:
julia> Point{Float64} <: Pointy{Real}
false
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:
type 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:
abstract Pointy{T<:Real}
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{String}
type error: Pointy: in T, expected Real, got AbstractKind
julia> Pointy{1}
type error: Pointy: in T, expected Real, got Int64
Type parameters for parametric composite types can be restricted in the same manner:
type Point{T<:Real} <: Pointy{T}
x::T
y::T
end
To give a couple of real-world examples of how all this parametric type
machinery can be useful, here is the actual definition of Julia’s
Rational
type, representing an exact ratio of integers:
type 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.
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, rather just kinds of types, which is all that would be possible in the absence of singleton types. 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, whereas in most languages with singleton types, every object has one.
Parametric Bits Types¶
Bits types can also be declared parametrically. For example, pointers are represented as boxed bits types which would be declared in Julia like this:
# 32-bit system:
bitstype 32 Ptr{T}
# 64-bit system:
bitstype 64 Ptr{T}
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 subtype
of the umbrella Ptr
type:
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),)