# 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. A concrete type corresponds to a single entity in the space of
all possible types; an abstract type refers to a collection (set) of
concrete types. `Any`

is a type that describes the entire universe
of possible types; `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`

, or equivalently,
`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)Realjulia>typeintersect(Signed,Union{UInt8,Int8})Int8julia>Union{Signed,Union{UInt8,Int8}}Union{Signed,UInt8}julia>typejoin(Signed,Union{UInt8,Int8})Integerjulia>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
`typeintersect(args,sig)`

is not `Union{}`

. (Here, `args`

is a
tuple-type describing the types of the arguments, and `sig`

is a
tuple-type specifying the types in the function’s 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.

## TypeVars¶

Many types take parameters; an easy example is `Array`

, which takes
two parameters often written as `Array{T,N}`

. Let’s compare the
following methods:

`f1(A::Array)=1f2(A::Array{Int})=2f3{T}(A::Array{T})=3f4(A::Array{Any})=4f5{T<:Any}(A::Array{T})=5`

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>ArrayArray{T,N}julia>xdump(Array)Array{T,N}::DataType<:DenseArray{T,N}`

This indicates that `Array`

is a shorthand for `Array{T,N}`

. If
you type this at the REPL prompt—on its own, not while defining
a function or type—you get an error `Tnotdefined`

. So what,
exactly, are `T`

and `N`

? You can learn more by extracting these
parameters:

`julia>T,N=Array.parameterssvec(T,N)julia>xdump(T)TypeVarname:SymbolTlb:Union{}ub:Any::DataType<:Anybound:Boolfalse`

A `TypeVar`

is one of Julia’s built-in types—it’s defined in
`jltypes.c`

, although you can find a commented-out version in
`boot.jl`

. The `name`

field is straightforward: it’s what’s
printed when showing the object. `lb`

and `ub`

stand for “lower
bound” and “upper bound,” respectively: these are the sets that
constrain what types the TypeVar may represent. In this case, `T`

‘s
lower bound is `Union{}`

(i.e., `Bottom`

or the empty set); in
other words, this `TypeVar`

is not constrained from below. The
upper bound is `Any`

, so neither is it constrained from above.

In a method definition like:

`g{S<:Integer}(x::S)=0`

one can extract the underlying `TypeVar`

:

`g{S<:Integer}(x::S)=0m=start(methods(g))p=m.sig.parameterstv=p[1]xdump(tv)`

```
TypeVar
name: Symbol S
lb: Union{}
ub: Integer::DataType <: Real
bound: Bool true
```

Here `ub`

is `Integer`

, as specified in the function definition.

The last field of a `TypeVar`

is `bound`

. This boolean value
specifies whether the `TypeVar`

is defined as one of the function
parameters. For example:

`julia>h1(A::Array,b::Real)=1h1(genericfunction with1method)julia>h2{T<:Real}(A::Array,b::T)=1h2(genericfunction with1method)julia>h3{T<:Real}(A::Array{T},b::T)=1h3(genericfunction with1method)julia>p1=start(methods(h1)).sig.parameterssvec(Array{T,N},Real)julia>p2=start(methods(h2)).sig.parameterssvec(Array{T,N},T<:Real)julia>p3=start(methods(h3)).sig.parameterssvec(Array{T<:Real,N},T<:Real)julia>xdump(p1[1].parameters[1])TypeVarname:SymbolTlb:Union{}ub:Any::DataType<:Anybound:Boolfalsejulia>xdump(p3[1].parameters[1])TypeVarname:SymbolTlb:Union{}ub:Real::DataType<:Numberbound:Booltrue`

Note that `p2`

shows two objects called `T`

, but only one of them
has the upper bound `Real`

; in contrast, `p3`

shows both of them
bounded. This is because in `h3`

, the same type `T`

is used in
both places, whereas for `h2`

the `T`

inside the array is simply
the default symbol used for the first parameter of `Array`

.

One can construct `TypeVar`

s manually:

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

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

symbol.

Armed with this information, we can do some sneaky things that reveal a lot about how Julia does dispatch:

`julia>TV=TypeVar(:T,false)# bound = falseTjulia>candid{T}(A::Array{T},x::T)=0candid(genericfunction with1method)julia>@evalsneaky{T}(A::Array{T},x::$TV)=1sneaky(genericfunction with1method)julia>methods(candid)# 1 method for generic function "candid":candid{T}(A::Array{T,N},x::T)atnone:1julia>methods(sneaky)# 1 method for generic function "sneaky":sneaky{T}(A::Array{T,N},x::T)atnone:1`

These therefore print identically, but they have very different behavior:

`julia>candid([1],3.2)ERROR:MethodError:`candid`hasnomethodmatchingcandid(::Array{Int64,1},::Float64)Closestcandidatesare:candid{T}(::Array{T,N},!Matched::T)julia>sneaky([1],3.2)1`

To see what’s happening, it’s helpful to use Julia’s internal `jl_()`

function (defined in `builtins.c`

) for display, because it prints
bound `TypeVar`

objects with a hash (`#T`

instead of `T`

):

`julia>jl_(x)=ccall(:jl_,Void,(Any,),x)jl_(genericfunction with1method)`

`julia>jl_(start(methods(candid)))Method(sig=Tuple{Array{#T<:Any, N<:Any}, #T<:Any}, va=false, isstaged=false, tvars=#T<:Any, func=#<function>, invokes=nothing, next=nothing)julia>jl_(start(methods(sneaky)))Method(sig=Tuple{Array{#T<:Any, N<:Any}, T<:Any}, va=false, isstaged=false, tvars=#T<:Any, func=#<function>, invokes=nothing, next=nothing)`

Even though both print as `T`

, in `sneaky`

the second `T`

is
not bound, and hence it isn’t constrained to be the same type as the
element type of the `Array`

.

Some `TypeVar`

interactions depend on the `bound`

state, even when there are not two or more uses of the same `TypeVar`

. For example:

`julia>S=TypeVar(:S,false);T=TypeVar(:T,true)T# These would be the same no matter whether we used S or Tjulia>Array{Array{S}}<:Array{Array}truejulia>Array{Array{S}}<:Array{Array{S}}truejulia>Array{Array}<:Array{Array{S}}true# For these cases, it mattersjulia>Array{Array{Int}}<:Array{Array}falsejulia>Array{Array{Int}}<:Array{Array{S}}falsejulia>Array{Array{Int}}<:Array{Array{T}}true`

It’s this latter construction that allows function declarations like

`foo{T,N}(A::Array{Array{T,N}})=T,N`

to match despite the invariance of Julia’s type parameters.

## TypeNames¶

The following two `Array`

types are functionally equivalent, yet
print differently via `jl_()`

:

`julia>TV,NV=TypeVar(:T),TypeVar(:N)(T,N)julia>jl_(Array)Arrayjulia>jl_(Array{TV,NV})Array{T<:Any,N<:Any}`

These can be distinguished by examining the `name`

field of
the type, which is an object of type `TypeName`

:

`julia>xdump(Array.name)TypeNamename:SymbolArraymodule:ModuleCorenames:SimpleVectorlength:Int640primary:Array{T,N}::DataType<:DenseArray{T,N}cache:SimpleVectorlength:Int64135linearcache:SimpleVectorlength:Int6418uid:Int6437`

In this case, the relevant field is `primary`

, which holds a
reference to the “primary” instance of the type:

`julia>pointer_from_objref(Array)Ptr{Void}@0x00007fcc7de64850julia>pointer_from_objref(Array.name.primary)Ptr{Void}@0x00007fcc7de64850julia>pointer_from_objref(Array{TV,NV})Ptr{Void}@0x00007fcc80c4d930julia>pointer_from_objref(Array{TV,NV}.name.primary)Ptr{Void}@0x00007fcc7de64850`

The `primary`

field of `Array`

points to itself, but for
`Array{TV,NV}`

it points back to the default definition of the type.

What about the other fields? `uid`

assigns a unique 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}endjulia>MyType{Int,2}MyType{Int64,2}julia>MyType{Float32,5}MyType{Float32,5}julia>MyType.name.cachesvec(MyType{Float32,5},MyType{Int64,2},Evaluationsucceeded,butanerroroccurredwhileshowingvalueoftype SimpleVector:ERROR:UndefRefError:accesstoundefinedreferenceingetindexat./essentials.jl:211inshow_delim_arrayatshow.jl:229inshowatshow.jl:257inanonymousatshow.jl:1301inwith_output_limitat./show.jl:1278inshowlimitedatshow.jl:1300indisplayatmultimedia.jl:120[inlinedcode]frommultimedia.jl:151indisplayatmultimedia.jl:163`

(The error is triggered because 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 with `TypeVar`

parameters 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 be
able to accommodate any tuple. Let’s check the parameters:

`julia>TupleTuplejulia>Tuple.parameterssvec(Vararg{Any})`

It’s worth noting that the parameter is a type, `Any`

, rather than a
`TypeVarT<:Any`

: compare

`julia>jl_(Tuple.parameters)svec(Vararg{Any})julia>jl_(Array.parameters)svec(T<:Any,N<:Any)`

Unlike other types, tuple-types are covariant in their parameters, so
this definition permits `Tuple`

to match any type of tuple. This is
therefore equivalent to having an unbound `TypeVar`

but distinct
from a bound `TypeVar`

`julia>typeintersect(Tuple,Tuple{Int,Float64})Tuple{Int64,Float64}julia>typeintersect(Tuple{Vararg{Any}},Tuple{Int,Float64})Tuple{Int64,Float64}julia>T=TypeVar(:T,false)Tjulia>typeintersect(Tuple{Vararg{T}},Tuple{Int,Float64})Tuple{Int64,Float64}julia>T=TypeVar(:T,true)Tjulia>typeintersect(Tuple{Vararg{T}},Tuple{Int,Float64})Union{}`

Finally, it’s worth noting that `Tuple{}`

is distinct

`julia>Tuple{}Tuple{}julia>Tuple{}.parameterssvec()julia>typeintersect(Tuple{},Tuple{Int})Union{}`

What is the “primary” tuple-type?

`julia>pointer_from_objref(Tuple)Ptr{Void}@0x00007f5998a04370julia>pointer_from_objref(Tuple{})Ptr{Void}@0x00007f5998a570d0julia>pointer_from_objref(Tuple.name.primary)Ptr{Void}@0x00007f5998a04370julia>pointer_from_objref(Tuple{}.name.primary)Ptr{Void}@0x00007f5998a04370`

so `Tuple==Tuple{Vararg{Any}}`

is indeed the primary type.

## Introduction to the internal machinery: `jltypes.c`

¶

Many operations for dealing with types are found in the file
`jltypes.c`

. A good way to start is to watch type intersection in
action. Build Julia with `makedebug`

and fire up Julia within a
debugger. gdb debugging tips has some tips which may be useful.

Because the type intersection and matching 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 myintersect(a,b)ccall(:jl_breakpoint,Void,(Any,),nothing)typeintersect(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:

`myintersect(Tuple{Integer,Float64},Tuple{Int,Real})`

Set a breakpoint in `intersect_tuple`

and continue until it enters this function. You should be able to see something like this:

`Breakpoint2,intersect_tuple(a=0x7ffdf7409150,b=0x7ffdf74091b0,penv=0x7fffffffcc90,eqc=0x7fffffffcc70,var=covariant)atjltypes.c:405405{(gdb)calljl_(a)Tuple{Integer,Float64}(gdb)calljl_(b)Tuple{Int64,Real}`

The `var`

argument is either `covariant`

or `invariant`

, the
latter being used if you’re matching the type parameters of
`Array{T1}`

against `Array{T2}`

. The other two inputs to this
function (`penv`

and `eqc`

) may be currently mysterious, but we’ll
discuss them in a moment. For now, step through the code until you
get into the loop over the different entries in the tuple types `a`

and `b`

. The key call is:

`ce=jl_type_intersect(ae,be,penv,eqc,var);`

which, if you examine `ae`

, `be`

, and `ce`

, you’ll see is just
type intersection performed on these entries.

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

`julia>T=TypeVar(:T,true)Tjulia>myintersect(Tuple{Array{T},T},Tuple{Array{Int,2},Int8})Breakpoint1,jl_breakpoint(v=0x7ffdf35e8010)atbuiltins.c:15591559{(gdb)bintersect_tupleBreakpoint3at0x7ffff6dcb07d:filejltypes.c,line405.(gdb)cContinuing.Breakpoint3,intersect_tuple(a=0x7ffdf74d7a90,b=0x7ffdf74d7af0,penv=0x7fffffffcc90,eqc=0x7fffffffcc70,var=covariant)atjltypes.c:405405{(gdb)calljl_(a)Tuple{Array{＃T<:Any,N<:Any},＃T<:Any}(gdb)calljl_(b)Tuple{Array{Int64,2},Int8}`

Let’s watch how this bound `TypeVar`

gets handled. To follow this,
you’ll need to examine the variables `penv`

and `eqc`

, which are
defined as:

`typedefstruct{jl_value_t**data;size_tn;jl_svec_t*tvars;}cenv_t;`

These start out empty (with `penv->n==eqc->n==0`

). Once we get
into the loop and make the first call to `jl_type_intersect`

,
`eqc`

(which stands for “equality constraints”) has the following
value:

`(gdb)peqc->n$4=2(gdb)calljl_(eqc->data[0])＃T<:Any(gdb)calljl_(eqc->data[1])Int64`

This is just a `var`

, `value`

list of pairs, indicating that `T`

now has the value `Int64`

. If you now allow `intersect_tuple`

to
finish and keep progressing, you’ll eventually get to
`type_intersection_matching`

. This function contains a call to
`solve_tvar_constraints`

. Roughly speaking, `eqc`

defines `T=Int64`

, but `env`

defines it as `Int8`

; this conflict is detected
in `solve_tvar_constraints`

and the resulting return is
`jl_bottom_type`

, aka `Union{}`

.

## Subtyping and method sorting¶

Armed with this knowledge, you may find yourself surprised by the following:

`julia>typeintersect(Tuple{Array{Int},Float64},Tuple{Array{T},T})Union{}julia>Tuple{Array{Int},Float64}<:Tuple{Array{T},T}true`

where `T`

is a bound `TypeVar`

. In other words, `A<:B`

does
not imply that `typeintersect(A,B)==A`

. A little bit of digging
reveals the reason why: `jl_subtype_le`

does not use the `cenv_t`

constraints that we just saw in `typeintersect`

.

`jltypes.c`

contains three closely related collections of functions
for testing how types `a`

and `b`

are ordered:

- The
`subtype`

functions implement`a<:b`

. Among other uses, they serve in matching function arguments against method signatures in the function cache. - 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,0)`

really means “is`a`

at least as specific as`b`

?” and not “is`a`

strictly more specific than`b`

?” - The
`type_match`

functions are similar to`type_morespecific`

, but additionally accept (and employ) an environment to constrain typevars. The related`type_match_morespecific`

functions call`type_match`

with an argument`morespecific=1`

All three of these take an argument, `invariant`

, which is set to 1 when
comparing type parameters and otherwise is 0.

The rules for these are somewhat different. `subtype`

is sensitive
to the number 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:

`args_morespecific(a,b)=ccall(:jl_args_morespecific,Cint,(Any,Any),a,b)`

which allows you to test whether arg-tuple `a`

is more specific than
arg-tuple `b`

.