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:

  1. julia> typeintersect(Int, Float64)
  2. Union{}
  4. julia> Union{Int, Float64}
  5. Union{Float64, Int64}
  7. julia> typejoin(Int, Float64)
  8. Real
  10. julia> typeintersect(Signed, Union{UInt8, Int8})
  11. Int8
  13. julia> Union{Signed, Union{UInt8, Int8}}
  14. Union{UInt8, Signed}
  16. julia> typejoin(Signed, Union{UInt8, Int8})
  17. Integer
  19. julia> typeintersect(Tuple{Integer, Float64}, Tuple{Int, Real})
  20. Tuple{Int64, Float64}
  22. julia> Union{Tuple{Integer, Float64}, Tuple{Int, Real}}
  23. Union{Tuple{Int64, Real}, Tuple{Integer, Float64}}
  25. julia> typejoin(Tuple{Integer, Float64}, Tuple{Int, Real})
  26. 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, 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:

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

The signature - as described in Function calls - of f3 is a UnionAll type wrapping a tuple type: Tuple{typeof(f3), Array{T}} where T. 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:

  1. julia> dump(Array)
  2. UnionAll
  3.   var: TypeVar
  4.     name: Symbol T
  5.     lb: Union{}
  6.     ub: Any
  7.   body: UnionAll
  8.     var: TypeVar
  9.       name: Symbol N
  10.       lb: Union{}
  11.       ub: Any
  12.     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:

  1. julia> TypeVar(:V, Signed, Real)
  2. 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

  1. let T = TypeVar(:T,Integer)
  2.     UnionAll(T, Array{T})
  3. 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:

  1. julia> TV, NV = TypeVar(:T), TypeVar(:N)
  2. (T, N)
  4. julia> Array
  5. Array
  7. julia> Array{TV, NV}
  8. Array{T, N}

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

  1. julia> dump(Array{Int,1}.name)
  2. TypeName
  3.   name: Symbol Array
  4.   module: Module Core
  5.   names: empty SimpleVector
  6.   wrapper: UnionAll
  7.     var: TypeVar
  8.       name: Symbol T
  9.       lb: Union{}
  10.       ub: Any
  11.     body: UnionAll
  12.       var: TypeVar
  13.         name: Symbol N
  14.         lb: Union{}
  15.         ub: Any
  16.       body: Array{T, N} <: DenseArray{T, N}
  17.   cache: SimpleVector
  18.     ...
  20.   linearcache: SimpleVector
  21.     ...
  23.   hash: Int64 -7900426068641098781
  24.   mt: MethodTable
  25.     name: Symbol Array
  26.     defs: Nothing nothing
  27.     cache: Nothing nothing
  28.     max_args: Int64 0
  29.     kwsorter: #undef
  30.     module: Module Core
  31.     : Int64 0
  32.     : 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.

  1. julia> pointer_from_objref(Array)
  2. Ptr{Cvoid} @0x00007fcc7de64850
  4. julia> pointer_from_objref(Array.body.body.name.wrapper)
  5. Ptr{Cvoid} @0x00007fcc7de64850
  7. julia> pointer_from_objref(Array{TV,NV})
  8. Ptr{Cvoid} @0x00007fcc80c4d930
  10. julia> pointer_from_objref(Array{TV,NV}.name.wrapper)
  11. Ptr{Cvoid} @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:

  1. julia> struct MyType{T,N} end
  3. julia> MyType{Int,2}
  4. MyType{Int64, 2}
  6. julia> MyType{Float32, 5}
  7. MyType{Float32, 5}

When you instantiate a parametric type, each concrete type gets saved in a type cache (MyType.body.body.name.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:

  1. julia> Tuple
  2. Tuple
  4. julia> Tuple.parameters
  5. 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:

  1. julia> typeintersect(Tuple, Tuple{Int,Float64})
  2. Tuple{Int64, Float64}
  4. julia> typeintersect(Tuple{Vararg{Any}}, Tuple{Int,Float64})
  5. Tuple{Int64, Float64}

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

  1. julia> typeintersect(Tuple{Vararg{T} where T}, Tuple{Int,Float64})
  2. Tuple{Int64, Float64}
  4. julia> typeintersect(Tuple{Vararg{T}} where T, Tuple{Int,Float64})
  5. 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:

  1. julia> Tuple{}
  2. Tuple{}
  4. julia> Tuple{}.parameters
  5. svec()
  7. julia> typeintersect(Tuple{}, Tuple{Int})
  8. Union{}

What is the “primary” tuple-type?

  1. julia> pointer_from_objref(Tuple)
  2. Ptr{Cvoid} @0x00007f5998a04370
  4. julia> pointer_from_objref(Tuple{})
  5. Ptr{Cvoid} @0x00007f5998a570d0
  7. julia> pointer_from_objref(Tuple.name.wrapper)
  8. Ptr{Cvoid} @0x00007f5998a04370
  10. julia> pointer_from_objref(Tuple{}.name.wrapper)
  11. Ptr{Cvoid} @0x00007f5998a04370

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

Diagonal types

Consider the type Tuple{T,T} where T. A method with this signature would look like:

  1. f(x::T, y::T) where {T} = ...

According to the usual interpretation of a UnionAll type, this T ranges over all types, including Any, so this type should be equivalent to Tuple{Any,Any}. However, this interpretation causes some practical problems.

First, a value of T needs to be available inside the method definition. For a call like f(1, 1.0), it’s not clear what T should be. It could be Union{Int,Float64}, or perhaps Real. Intuitively, we expect the declaration x::T to mean T === typeof(x). To make sure that invariant holds, we need typeof(x) === typeof(y) === T in this method. That implies the method should only be called for arguments of the exact same type.

It turns out that being able to dispatch on whether two values have the same type is very useful (this is used by the promotion system for example), so we have multiple reasons to want a different interpretation of Tuple{T,T} where T. To make this work we add the following rule to subtyping: if a variable occurs more than once in covariant position, it is restricted to ranging over only concrete types. (“Covariant position” means that only Tuple and Union types occur between an occurrence of a variable and the UnionAll type that introduces it.) Such variables are called “diagonal variables” or “concrete variables”.

So for example, Tuple{T,T} where T can be seen as Union{Tuple{Int8,Int8}, Tuple{Int16,Int16}, ...}, where T ranges over all concrete types. This gives rise to some interesting subtyping results. For example Tuple{Real,Real} is not a subtype of Tuple{T,T} where T, because it includes some types like Tuple{Int8,Int16} where the two elements have different types. Tuple{Real,Real} and Tuple{T,T} where T have the non-trivial intersection Tuple{T,T} where T<:Real. However, Tuple{Real} is a subtype of Tuple{T} where T, because in that case T occurs only once and so is not diagonal.

Next consider a signature like the following:

  1. f(a::Array{T}, x::T, y::T) where {T} = ...

In this case, T occurs in invariant position inside Array{T}. That means whatever type of array is passed unambiguously determines the value of T – we say T has an equality constraint on it. Therefore in this case the diagonal rule is not really necessary, since the array determines T and we can then allow x and y to be of any subtypes of T. So variables that occur in invariant position are never considered diagonal. This choice of behavior is slightly controversial – some feel this definition should be written as

  1. f(a::Array{T}, x::S, y::S) where {T, S<:T} = ...

to clarify whether x and y need to have the same type. In this version of the signature they would, or we could introduce a third variable for the type of y if x and y can have different types.

The next complication is the interaction of unions and diagonal variables, e.g.

  1. f(x::Union{Nothing,T}, y::T) where {T} = ...

Consider what this declaration means. y has type T. x then can have either the same type T, or else be of type Nothing. So all of the following calls should match:

  1. f(1, 1)
  2. f("", "")
  3. f(2.0, 2.0)
  4. f(nothing, 1)
  5. f(nothing, "")
  6. f(nothing, 2.0)

These examples are telling us something: when x is nothing::Nothing, there are no extra constraints on y. It is as if the method signature had y::Any. Indeed, we have the following type equivalence:

  1. (Tuple{Union{Nothing,T},T} where T) == Union{Tuple{Nothing,Any}, Tuple{T,T} where T}

The general rule is: a concrete variable in covariant position acts like it’s not concrete if the subtyping algorithm only uses it once. When x has type Nothing, we don’t need to use the T in Union{Nothing,T}; we only use it in the second slot. This arises naturally from the observation that in Tuple{T} where T restricting T to concrete types makes no difference; the type is equal to Tuple{Any} either way.

However, appearing in invariant position disqualifies a variable from being concrete whether that appearance of the variable is used or not. Otherwise types can behave differently depending on which other types they are compared to, making subtyping not transitive. For example, consider

  1. Tuple{Int,Int8,Vector{Integer}} <: Tuple{T,T,Vector{Union{Integer,T}}} where T

If the T inside the Union is ignored, then T is concrete and the answer is “false” since the first two types aren’t the same. But consider instead

  1. Tuple{Int,Int8,Vector{Any}} <: Tuple{T,T,Vector{Union{Integer,T}}} where T

Now we cannot ignore the T in the Union (we must have T == Any), so T is not concrete and the answer is “true”. That would make the concreteness of T depend on the other type, which is not acceptable since a type must have a clear meaning on its own. Therefore the appearance of T inside Vector is considered in both cases.

Subtyping diagonal variables

The subtyping algorithm for diagonal variables has two components: (1) identifying variable occurrences, and (2) ensuring that diagonal variables range over concrete types only.

The first task is accomplished by keeping counters occurs_inv and occurs_cov (in src/subtype.c) for each variable in the environment, tracking the number of invariant and covariant occurrences, respectively. A variable is diagonal when occurs_inv == 0 && occurs_cov > 1.

The second task is accomplished by imposing a condition on a variable’s lower bound. As the subtyping algorithm runs, it narrows the bounds of each variable (raising lower bounds and lowering upper bounds) to keep track of the range of variable values for which the subtype relation would hold. When we are done evaluating the body of a UnionAll type whose variable is diagonal, we look at the final values of the bounds. Since the variable must be concrete, a contradiction occurs if its lower bound could not be a subtype of a concrete type. For example, an abstract type like AbstractArray cannot be a subtype of a concrete type, but a concrete type like Int can be, and the empty type Bottom can be as well. If a lower bound fails this test the algorithm stops with the answer false.

For example, in the problem Tuple{Int,String} <: Tuple{T,T} where T, we derive that this would be true if T were a supertype of Union{Int,String}. However, Union{Int,String} is an abstract type, so the relation does not hold.

This concreteness test is done by the function is_leaf_bound. Note that this test is slightly different from jl_is_leaf_type, since it also returns true for Bottom. Currently this function is heuristic, and does not catch all possible concrete types. The difficulty is that whether a lower bound is concrete might depend on the values of other type variable bounds. For example, Vector{T} is equivalent to the concrete type Vector{Int} only if both the upper and lower bounds of T equal Int. We have not yet worked out a complete algorithm for this.

Introduction to the internal machinery

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:

  1. julia> function mysubtype(a,b)
  2.            ccall(:jl_breakpoint, Cvoid, (Any,), nothing)
  3.            a <: b
  4.        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:

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

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

  1. 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). Specificity is strict; if a is more specific than b, then a does not equal b and b is not more specific than a.

If a is a strict 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:

  1. 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.