Parametric and Union Types

General Usage
1

Hi Community!

While playing around with types, some questions came up:

What is the difference between Type{<:Union{Int,String}} and Union{Type{Int}, Type{String}}?

Type{<:Union{Int,String}} |> supertype  # Any
Type{<:Union{Int,String}} |> eltype  # Any
Int isa Type{<:Union{Int,String}}  # true
Union{Type{Int}, Type{String}} |> supertype  # ERROR: MethodError
Union{Type{Int}, Type{String}} |> eltype  # Any
Int isa Union{Type{Int}, Type{String}}  # true
Type{<:Union{Int,String}} == Union{Type{Int}, Type{String}}  # false

Are parametric types without their parameters really types? Are they higher kinded types?
Why is Array{Int} <: Array true but, Array is nowhere to find when iteratively checking supertypes of Array{Int}?

Array{Int} <: Array  # true
Array{Int} |> supertype  # DenseArray{Int64,N} where N
Array{Int} |> supertype |> supertype  # AbstractArray{Int64,N} where N
Array{Int} |> supertype |> supertype |> supertype  # Any

Edit:

Is there a type that represents an Array that can only hold Int and String typed elements?

2 Likes
2

I’m going to attempt to rescue this thread before it gets completely buried. I might not be able to answer all your questions.

2

Parametric types with their parameters unspecified are abstract types. In particular, they are referred to as UnionAll Types. They represent the union of all types that are possible by assigning a concrete type to each type parameter.

Array is shorthand for Array{T,N} where N where T. Note that you can partially instantiate a parametric type:

julia> Array{Int64} == Array{Int64,N} where N
true

A partially instantiated parametric type is still an abstract type:

julia> isconcretetype(Array)
false

julia> isconcretetype(Array{Float64})
false

julia> isconcretetype(Array{Float64,1})
true

3

Yes, if we restrict ourselves to one-dimensional arrays, then we have the following concrete type:

Array{Union{Int,String},1}

If the number of dimensions is left unspecified, then we have an abstract type:

julia> IntStringArray{N} = Array{Union{Int, String}, N} where N
Array{Union{Int64, String},N} where N

julia> isconcretetype(IntStringArray)
false

julia> isconcretetype(IntStringArray{2})
true

julia> x = IntStringArray(undef, 0)
0-element Array{Union{Int64, String},1}

julia> push!(x, 4)
1-element Array{Union{Int64, String},1}:
 4

julia> push!(x, "hello")
2-element Array{Union{Int64, String},1}:
 4       
  "hello"

julia> push!(x, 1.2)
ERROR: MethodError: Cannot `convert` an object of type Float64 to an object of type Union{Int64, String}

1

I’m not sure if this is a point of confusion or not, but a type such as Type{Int64} is a sort of meta-type that represents the type object Int64 rather than an instance of Int64. For example, we can make two separate foo methods as follows:

julia> foo(::Int64) = 1
foo (generic function with 1 method)

julia> foo(::Type{Int64}) = 2
foo (generic function with 2 methods)

julia> methods(foo)
# 2 methods for generic function "foo":
[1] foo(::Int64) in Main at REPL[1]:1
[2] foo(::Type{Int64}) in Main at REPL[2]:1

julia> foo(100)
1

julia> foo(Int64)
2

Regarding the difference between

Type{T} where T<:Union{Int, String}

and

Union{Type{Int}, Type{String}}

…They do seem to be functionally equivalent, but Julia doesn’t seem to be able to do the algebra/set operations needed to declare them ==. I’m not sure precisely what’s going on there.

Edit:

It could be because the first is a UnionAll type and the second is a Union type:

julia> typeof(Type{T} where T<:Union{Int, String})
UnionAll

julia> typeof(Union{Type{Int}, Type{String}})
Union

It looks like Julia doesn’t have a specialized == method for comparing Union and UnionAll types:

julia> (Type{T} where T<:Union{Int, String}) == Union{Type{Int}, Type{String}}
false

julia> @which (Type{T} where T<:Union{Int, String}) == Union{Type{Int}, Type{String}}
==(T::Type, S::Type) in Base at operators.jl:162

It might be possible to implement a specialized ==(::Union, ::UnionAll) method that would catch this sort of scenario.

5 Likes
3

Thank you, that really helps!

Regarding 2.: So UnionAll types are unions which include all types defined by parametric types, but are not their supertype, correct?

I also wonder why Union and UnionAll are separate types. Maybe because the latter are infinite sets?

4

They are different beasts, UnionAll is for unions over parameters (... where ...), eg

julia> Vector
Array{T,1} where T

julia> typeof(Vector)
UnionAll

while Union is for enumerated types.

5

I’m not sure if I understand your exact meaning here, but to clarify:

Union and UnionAll are types of types. (The other common type of type is DataType.) So, Array{T,1} where T is a type of type UnionAll, but arrays of type Array{T,1} where T are not a subtype of UnionAll:

julia> typeof(Array{T,1} where T)
UnionAll

julia> (Array{T,1} where T) <: UnionAll
false

And similarly for Union types:

julia> typeof(Union{Int64, String})
Union

julia> Union{Int64, String} <: Union
false

In other words, Array{T,N} where N where T is a node on the hierarchy of types of arrays, whereas UnionAll is a node on the hierarchy of types of types. For example,

julia> Array{Int64,1} <: (Array{T,N} where N where T)
true

julia> (Array{T,N} where N where T) <: AbstractArray
true

julia> isconcretetype(UnionAll)
true

julia> supertype(UnionAll)
Type{T}

julia> supertype(supertype(UnionAll))
Any
6

What still confuses me is the snipped under my original 2:

Array{Int} <: Array  # true
Array{Int} |> supertype  # DenseArray{Int64,N} where N
Array{Int} |> supertype |> supertype  # AbstractArray{Int64,N} where N
Array{Int} |> supertype |> supertype |> supertype  # Any

Why is it that Array{Int} is a subtype of Array (according to <:) but Array is not a supertype of Array{Int}? I was thinking one implied the other.

In fact, Array does not even have subtypes:

Array |> subtypes  # 0-element Array{Type,1}

This made me think that a parametric type without its parameters specified is not a type. At least not of the same kind as other types.

7

It is my understanding that supertype() and subtypes() only return the explicitly declared supertype and subtypes, respectively. Array{Int,1} and Array{String,1} are not explicitly declared types. They are only declared implicitly through the parametric definition Array{T,N} where {T,N}. (Henceforth, I will drop the where {T,N} for brevity.) Therefore, subtypes(Array) returns an empty array because there are no explicitly declared subtypes of Array{T,N}.

The particular branch of the type hierarchy (tree) that we’re looking at is explicitly declared as the following:

AbstractArray ⟶ DenseArray ⟶ Array

Or with the type parameters made explicit:

AbstractArray{T,N} where {T,N}
    ⮑ DenseArray{T,N} where {T,N}
           ⮑ Array{T,N} where {T,N}

Note that Array{T,N} is a terminal node in the tree. There are no explicitly declared subtypes of Array{T,N}.

So, you could say that Array{Int,1} lives on the same node in the hierarchy as Array{T,N}. But Array{Int,1} is a subtype of Array{T,N} because it represents a subset of the various types that are implied by Array{T,N}.

Said another way, Array{T,N} is a supertype of Array{Int,1}, but it’s not returned by supertype(Array{Int,1}) because it lives on the same node as Array{Int,1}.

1 Like
8

Array{Int} <: Array && Array >: Array{Int}, those are symmetric operations.

9

Yea, I guess it works like that.
It does feel a bit unsatisfying on a formal level, though. :smile:

Thank you for your answers!
Oh and did you create those type hierarchy “visualizations” yourself or is there some function to print the type graph?

10

I get that, but I think it’s strange (or unsatisfying) that if A <: B is true, subtypes(B) does not list A.

11

I wrote them out by hand. I thought I saw a package once for making Julia type graphs, but I can’t find it now.

Array{T,N} has an infinite number of subtypes, so I think it’s reasonable to not list them all out. :slight_smile:

2 Likes
12

Fair enough. :smile: