More about types

If you've used Julia for a while, you understand the fundamental role that types play. Here we try to get under the hood, focusing particularly on Parametric Types.

Types and sets (and Any and Union{}/Bottom)

It's perhaps easiest to conceive of Julia's type system in terms of sets. While programs manipulate individual values, a type refers to a set of values. This is not the same thing as a collection; for example a Set of values is itself a single Set value. Rather, a type describes a set of possible values, expressing uncertainty about which value we have.

A concrete type T describes the set of values whose direct tag, as returned by the typeof function, is T. An abstract type describes some possibly-larger set of values.

Any describes the entire universe of possible values. Integer is a subset of Any that includes Int, Int8, and other concrete types. Internally, Julia also makes heavy use of another type known as Bottom, which can also be written as Union{}. This corresponds to the empty set.

Julia's types support the standard operations of set theory: you can ask whether T1 is a "subset" (subtype) of T2 with T1 <: T2. Likewise, you intersect two types using typeintersect, take their union with Union, and compute a type that contains their union with typejoin:

julia> typeintersect(Int, Float64)
Union{}

julia> Union{Int, Float64}
Union{Float64, Int64}

julia> typejoin(Int, Float64)
Real

julia> typeintersect(Signed, Union{UInt8, Int8})
Int8

julia> Union{Signed, Union{UInt8, Int8}}
Union{Signed, UInt8}

julia> typejoin(Signed, Union{UInt8, Int8})
Integer

julia> typeintersect(Tuple{Integer,Float64}, Tuple{Int,Real})
Tuple{Int64,Float64}

julia> Union{Tuple{Integer,Float64}, Tuple{Int,Real}}
Union{Tuple{Int64,Real}, Tuple{Integer,Float64}}

julia> typejoin(Tuple{Integer,Float64}, Tuple{Int,Real})
Tuple{Integer,Real}

While these operations may seem abstract, they lie at the heart of Julia. For example, method dispatch is implemented by stepping through the items in a method list until reaching one for which the type of the argument tuple is a subtype of the method signature. For this algorithm to work, it's important that methods be sorted by their specificity, and that the search begins with the most specific methods. Consequently, Julia also implements a partial order on types; this is achieved by functionality that is similar to <:, but with differences that will be discussed below.

UnionAll types

Julia's type system can also express an iterated union of types: a union of types over all values of some variable. This is needed to describe parametric types where the values of some parameters are not known.

For example, :obj:Array has two parameters as in Array{Int,2}. If we did not know the element type, we could write Array{T,2} where T, which is the union of Array{T,2} for all values of T: Union{Array{Int8,2}, Array{Int16,2}, ...}.

Such a type is represented by a UnionAll object, which contains a variable (T in this example, of type TypeVar), and a wrapped type (Array{T,2} in this example).

Consider the following methods::

f1(A::Array) = 1
f2(A::Array{Int}) = 2
f3(A::Array{T}) where T<:Any = 3
f4(A::Array{Any}) = 4

The signature of f3 is a UnionAll type wrapping a tuple type. All but f4 can be called with a = [1,2]; all but f2 can be called with b = Any[1,2].

Let's look at these types a little more closely:

julia> dump(Array)
UnionAll
  var: TypeVar
    name: Symbol T
    lb: Core.BottomType Union{}
    ub: Any
  body: UnionAll
    var: TypeVar
      name: Symbol N
      lb: Core.BottomType Union{}
      ub: Any
    body: Array{T,N} <: DenseArray{T,N}

This indicates that Array actually names a UnionAll type. There is one UnionAll type for each parameter, nested. The syntax Array{Int,2} is equivalent to Array{Int}{2}; internally each UnionAll is instantiated with a particular variable value, one at a time, outermost-first. This gives a natural meaning to the omission of trailing type parameters; Array{Int} gives a type equivalent to Array{Int,N} where N.

A TypeVar is not itself a type, but rather should be considered part of the structure of a UnionAll type. Type variables have lower and upper bounds on their values (in the fields lb and ub). The symbol name is purely cosmetic. Internally, TypeVars are compared by address, so they are defined as mutable types to ensure that "different" type variables can be distinguished. However, by convention they should not be mutated.

One can construct TypeVars manually:

julia> TypeVar(:V, Signed, Real)
Signed<:V<:Real

There are convenience versions that allow you to omit any of these arguments except the name symbol.

The syntax Array{T} where T<:Integer is lowered to

let T = TypeVar(:T,Integer)
    UnionAll(T, Array{T})
end

so it is seldom necessary to construct a TypeVar manually (indeed, this is to be avoided).

Free variables

The concept of a free type variable is extremely important in the type system. We say that a variable V is free in type T if T does not contain the UnionAll that introduces variable V. For example, the type Array{Array{V} where V<:Integer} has no free variables, but the Array{V} part inside of it does have a free variable, V.

A type with free variables is, in some sense, not really a type at all. Consider the type Array{Array{T}} where T, which refers to all homogeneous arrays of arrays. The inner type Array{T}, seen by itself, might seem to refer to any kind of array. However, every element of the outer array must have the same array type, so Array{T} cannot refer to just any old array. One could say that Array{T} effectively "occurs" multiple times, and T must have the same value each "time".

For this reason, the function jl_has_free_typevars in the C API is very important. Types for which it returns true will not give meaningful answers in subtyping and other type functions.

TypeNames

The following two Array types are functionally equivalent, yet print differently:

julia> TV, NV = TypeVar(:T), TypeVar(:N)
(T, N)

julia> Array
Array

julia> Array{TV,NV}
Array{T,N}

These can be distinguished by examining the name field of the type, which is an object of type TypeName:

julia> dump(Array{Int,1}.name)
TypeName
  name: Symbol Array
  module: Module Core
  names: empty SimpleVector
  wrapper: UnionAll
    var: TypeVar
      name: Symbol T
      lb: Core.BottomType Union{}
      ub: Any
    body: UnionAll
      var: TypeVar
        name: Symbol N
        lb: Core.BottomType Union{}
        ub: Any
      body: Array{T,N} <: DenseArray{T,N}
  cache: SimpleVector
    ...

  linearcache: SimpleVector
    ...

  hash: Int64 -7900426068641098781
  mt: MethodTable
    name: Symbol Array
    defs: Void nothing
    cache: Void nothing
    max_args: Int64 0
    kwsorter: #undef
    module: Module Core
    : Int64 0
    : Int64 0

In this case, the relevant field is wrapper, which holds a reference to the top-level type used to make new Array types.

julia> pointer_from_objref(Array)
Ptr{Void} @0x00007fcc7de64850

julia> pointer_from_objref(Array.body.body.name.wrapper)
Ptr{Void} @0x00007fcc7de64850

julia> pointer_from_objref(Array{TV,NV})
Ptr{Void} @0x00007fcc80c4d930

julia> pointer_from_objref(Array{TV,NV}.name.wrapper)
Ptr{Void} @0x00007fcc7de64850

The wrapper field of Array points to itself, but for Array{TV,NV} it points back to the original definition of the type.

What about the other fields? hash assigns an integer to each type. To examine the cache field, it's helpful to pick a type that is less heavily used than Array. Let's first create our own type:

julia> type MyType{T,N} end

julia> MyType{Int,2}
MyType{Int64,2}

julia> MyType{Float32, 5}
MyType{Float32,5}

julia> MyType.body.body.name.cache
svec(MyType{Float32,5}, MyType{Int64,2}, #undef, #undef, #undef, #undef, #undef, #undef)

(The cache is pre-allocated to have length 8, but only the first two entries are populated.) Consequently, when you instantiate a parametric type, each concrete type gets saved in a type cache. However, instances containing free type variables are not cached.

Tuple types

Tuple types constitute an interesting special case. For dispatch to work on declarations like x::Tuple, the type has to be able to accommodate any tuple. Let's check the parameters:

julia> Tuple
Tuple

julia> Tuple.parameters
svec(Vararg{Any,N} where N)

Unlike other types, tuple types are covariant in their parameters, so this definition permits Tuple to match any type of tuple:

julia> typeintersect(Tuple, Tuple{Int,Float64})
Tuple{Int64,Float64}

julia> typeintersect(Tuple{Vararg{Any}}, Tuple{Int,Float64})
Tuple{Int64,Float64}

However, if a variadic (Vararg) tuple type has free variables it can describe different kinds of tuples:

julia> typeintersect(Tuple{Vararg{T} where T}, Tuple{Int,Float64})
Tuple{Int64,Float64}

julia> typeintersect(Tuple{Vararg{T}} where T, Tuple{Int,Float64})
Union{}

Notice that when T is free with respect to the Tuple type (i.e. its binding UnionAll type is outside the Tuple type), only one T value must work over the whole type. Therefore a heterogeneous tuple does not match.

Finally, it's worth noting that Tuple{} is distinct:

julia> Tuple{}
Tuple{}

julia> Tuple{}.parameters
svec()

julia> typeintersect(Tuple{}, Tuple{Int})
Union{}

What is the "primary" tuple-type?

julia> pointer_from_objref(Tuple)
Ptr{Void} @0x00007f5998a04370

julia> pointer_from_objref(Tuple{})
Ptr{Void} @0x00007f5998a570d0

julia> pointer_from_objref(Tuple.name.wrapper)
Ptr{Void} @0x00007f5998a04370

julia> pointer_from_objref(Tuple{}.name.wrapper)
Ptr{Void} @0x00007f5998a04370

so Tuple == Tuple{Vararg{Any}} is indeed the primary type.

Introduction to the internal machinery: jltypes.c

Most operations for dealing with types are found in the files jltypes.c and subtype.c. A good way to start is to watch subtyping in action. Build Julia with make debug and fire up Julia within a debugger. gdb debugging tips has some tips which may be useful.

Because the subtyping code is used heavily in the REPL itself–and hence breakpoints in this code get triggered often–it will be easiest if you make the following definition:

julia> function mysubtype(a,b)
           ccall(:jl_breakpoint, Void, (Any,), nothing)
           issubtype(a, b)
       end

and then set a breakpoint in jl_breakpoint. Once this breakpoint gets triggered, you can set breakpoints in other functions.

As a warm-up, try the following:

mysubtype(Tuple{Int,Float64}, Tuple{Integer,Real})

We can make it more interesting by trying a more complex case:

mysubtype(Tuple{Array{Int,2}, Int8}, Tuple{Array{T}, T} where T)

Subtyping and method sorting

The type_morespecific functions are used for imposing a partial order on functions in method tables (from most-to-least specific). Note that jl_type_morespecific(a,b) really means "is a at least as specific as b?" and not "is a strictly more specific than b?"

If a is a subtype of b, then it is automatically considered more specific. From there, type_morespecific employs some less formal rules. For example, subtype is sensitive to the number of arguments, but type_morespecific may not be. In particular, Tuple{Int,AbstractFloat} is more specific than Tuple{Integer}, even though it is not a subtype. (Of Tuple{Int,AbstractFloat} and Tuple{Integer,Float64}, neither is more specific than the other.) Likewise, Tuple{Int,Vararg{Int}} is not a subtype of Tuple{Integer}, but it is considered more specific. However, morespecific does get a bonus for length: in particular, Tuple{Int,Int} is more specific than Tuple{Int,Vararg{Int}}.

If you're debugging how methods get sorted, it can be convenient to define the function:

type_morespecific(a, b) = ccall(:jl_type_morespecific, Cint, (Any,Any), a, b)

which allows you to test whether tuple type a is more specific than tuple type b.