変換と昇格

変換と昇格

Juliaは、数学演算子の引数を共通の型に昇格させるシステムを持っています。このことは、整数と浮動小数点数、算術演算と初等関数など、型、およびメソッドなど、さまざまなセクションで言及されています。このセクションでは、この型の昇格システムの仕組みと、それを新しい型に拡張して、組み込みの数学演算子やそれ以外の関数に適用する方法について説明します。従来、プログラミング言語は、算術引数の昇格をどのように扱うかで、2 つの陣営に分類されます:

In a sense, Julia falls into the "no automatic promotion" category: mathematical operators are just functions with special syntax, and the arguments of functions are never automatically converted. However, one may observe that applying mathematical operations to a wide variety of mixed argument types is just an extreme case of polymorphic multiple dispatch – something which Julia's dispatch and type systems are particularly well-suited to handle. "Automatic" promotion of mathematical operands simply emerges as a special application: Julia comes with pre-defined catch-all dispatch rules for mathematical operators, invoked when no specific implementation exists for some combination of operand types. These catch-all rules first promote all operands to a common type using user-definable promotion rules, and then invoke a specialized implementation of the operator in question for the resulting values, now of the same type. User-defined types can easily participate in this promotion system by defining methods for conversion to and from other types, and providing a handful of promotion rules defining what types they should promote to when mixed with other types.

Conversion

The standard way to obtain a value of a certain type T is to call the type's constructor, T(x). However, there are cases where it's convenient to convert a value from one type to another without the programmer asking for it explicitly. One example is assigning a value into an array: if A is a Vector{Float64}, the expression A[1] = 2 should work by automatically converting the 2 from Int to Float64, and storing the result in the array. This is done via the convert function.

The convert function generally takes two arguments: the first is a type object and the second is a value to convert to that type. The returned value is the value converted to an instance of given type. The simplest way to understand this function is to see it in action:

julia> x = 12
12

julia> typeof(x)
Int64

julia> convert(UInt8, x)
0x0c

julia> typeof(ans)
UInt8

julia> convert(AbstractFloat, x)
12.0

julia> typeof(ans)
Float64

julia> a = Any[1 2 3; 4 5 6]
2×3 Array{Any,2}:
 1  2  3
 4  5  6

julia> convert(Array{Float64}, a)
2×3 Array{Float64,2}:
 1.0  2.0  3.0
 4.0  5.0  6.0

Conversion isn't always possible, in which case a no method error is thrown indicating that convert doesn't know how to perform the requested conversion:

julia> convert(AbstractFloat, "foo")
ERROR: MethodError: Cannot `convert` an object of type String to an object of type AbstractFloat
[...]

Some languages consider parsing strings as numbers or formatting numbers as strings to be conversions (many dynamic languages will even perform conversion for you automatically), however Julia does not: even though some strings can be parsed as numbers, most strings are not valid representations of numbers, and only a very limited subset of them are. Therefore in Julia the dedicated parse function must be used to perform this operation, making it more explicit.

When is convert called?

The following language constructs call convert:

Conversion vs. Construction

Note that the behavior of convert(T, x) appears to be nearly identical to T(x). Indeed, it usually is. However, there is a key semantic difference: since convert can be called implicitly, its methods are restricted to cases that are considered "safe" or "unsurprising". convert will only convert between types that represent the same basic kind of thing (e.g. different representations of numbers, or different string encodings). It is also usually lossless; converting a value to a different type and back again should result in the exact same value.

There are four general kinds of cases where constructors differ from convert:

Constructors for types unrelated to their arguments

Some constructors don't implement the concept of "conversion". For example, Timer(2) creates a 2-second timer, which is not really a "conversion" from an integer to a timer.

Mutable collections

convert(T, x) is expected to return the original x if x is already of type T. In contrast, if T is a mutable collection type then T(x) should always make a new collection (copying elements from x).

Wrapper types

For some types which "wrap" other values, the constructor may wrap its argument inside a new object even if it is already of the requested type. For example Some(x) wraps x to indicate that a value is present (in a context where the result might be a Some or nothing). However, x itself might be the object Some(y), in which case the result is Some(Some(y)), with two levels of wrapping. convert(Some, x), on the other hand, would just return x since it is already a Some.

Constructors that don't return instances of their own type

In very rare cases it might make sense for the constructor T(x) to return an object not of type T. This could happen if a wrapper type is its own inverse (e.g. Flip(Flip(x)) === x), or to support an old calling syntax for backwards compatibility when a library is restructured. But convert(T, x) should always return a value of type T.

Defining New Conversions

When defining a new type, initially all ways of creating it should be defined as constructors. If it becomes clear that implicit conversion would be useful, and that some constructors meet the above "safety" criteria, then convert methods can be added. These methods are typically quite simple, as they only need to call the appropriate constructor. Such a definition might look like this:

convert(::Type{MyType}, x) = MyType(x)

The type of the first argument of this method is a singleton type, Type{MyType}, the only instance of which is MyType. Thus, this method is only invoked when the first argument is the type value MyType. Notice the syntax used for the first argument: the argument name is omitted prior to the :: symbol, and only the type is given. This is the syntax in Julia for a function argument whose type is specified but whose value does not need to be referenced by name. In this example, since the type is a singleton, we already know its value without referring to an argument name.

All instances of some abstract types are by default considered "sufficiently similar" that a universal convert definition is provided in Julia Base. For example, this definition states that it's valid to convert any Number type to any other by calling a 1-argument constructor:

convert(::Type{T}, x::Number) where {T<:Number} = T(x)

This means that new Number types only need to define constructors, since this definition will handle convert for them. An identity conversion is also provided to handle the case where the argument is already of the requested type:

convert(::Type{T}, x::T) where {T<:Number} = x

Similar definitions exist for AbstractString, AbstractArray, and AbstractDict.

Promotion

Promotion refers to converting values of mixed types to a single common type. Although it is not strictly necessary, it is generally implied that the common type to which the values are converted can faithfully represent all of the original values. In this sense, the term "promotion" is appropriate since the values are converted to a "greater" type – i.e. one which can represent all of the input values in a single common type. It is important, however, not to confuse this with object-oriented (structural) super-typing, or Julia's notion of abstract super-types: promotion has nothing to do with the type hierarchy, and everything to do with converting between alternate representations. For instance, although every Int32 value can also be represented as a Float64 value, Int32 is not a subtype of Float64.

Promotion to a common "greater" type is performed in Julia by the promote function, which takes any number of arguments, and returns a tuple of the same number of values, converted to a common type, or throws an exception if promotion is not possible. The most common use case for promotion is to convert numeric arguments to a common type:

julia> promote(1, 2.5)
(1.0, 2.5)

julia> promote(1, 2.5, 3)
(1.0, 2.5, 3.0)

julia> promote(2, 3//4)
(2//1, 3//4)

julia> promote(1, 2.5, 3, 3//4)
(1.0, 2.5, 3.0, 0.75)

julia> promote(1.5, im)
(1.5 + 0.0im, 0.0 + 1.0im)

julia> promote(1 + 2im, 3//4)
(1//1 + 2//1*im, 3//4 + 0//1*im)

Floating-point values are promoted to the largest of the floating-point argument types. Integer values are promoted to the larger of either the native machine word size or the largest integer argument type. Mixtures of integers and floating-point values are promoted to a floating-point type big enough to hold all the values. Integers mixed with rationals are promoted to rationals. Rationals mixed with floats are promoted to floats. Complex values mixed with real values are promoted to the appropriate kind of complex value.

That is really all there is to using promotions. The rest is just a matter of clever application, the most typical "clever" application being the definition of catch-all methods for numeric operations like the arithmetic operators +, -, * and /. Here are some of the catch-all method definitions given in promotion.jl:

+(x::Number, y::Number) = +(promote(x,y)...)
-(x::Number, y::Number) = -(promote(x,y)...)
*(x::Number, y::Number) = *(promote(x,y)...)
/(x::Number, y::Number) = /(promote(x,y)...)

These method definitions say that in the absence of more specific rules for adding, subtracting, multiplying and dividing pairs of numeric values, promote the values to a common type and then try again. That's all there is to it: nowhere else does one ever need to worry about promotion to a common numeric type for arithmetic operations – it just happens automatically. There are definitions of catch-all promotion methods for a number of other arithmetic and mathematical functions in promotion.jl, but beyond that, there are hardly any calls to promote required in Julia Base. The most common usages of promote occur in outer constructors methods, provided for convenience, to allow constructor calls with mixed types to delegate to an inner type with fields promoted to an appropriate common type. For example, recall that rational.jl provides the following outer constructor method:

Rational(n::Integer, d::Integer) = Rational(promote(n,d)...)

This allows calls like the following to work:

julia> Rational(Int8(15),Int32(-5))
-3//1

julia> typeof(ans)
Rational{Int32}

For most user-defined types, it is better practice to require programmers to supply the expected types to constructor functions explicitly, but sometimes, especially for numeric problems, it can be convenient to do promotion automatically.

Defining Promotion Rules

Although one could, in principle, define methods for the promote function directly, this would require many redundant definitions for all possible permutations of argument types. Instead, the behavior of promote is defined in terms of an auxiliary function called promote_rule, which one can provide methods for. The promote_rule function takes a pair of type objects and returns another type object, such that instances of the argument types will be promoted to the returned type. Thus, by defining the rule:

promote_rule(::Type{Float64}, ::Type{Float32}) = Float64

one declares that when 64-bit and 32-bit floating-point values are promoted together, they should be promoted to 64-bit floating-point. The promotion type does not need to be one of the argument types, however; the following promotion rules both occur in Julia Base:

promote_rule(::Type{BigInt}, ::Type{Float64}) = BigFloat
promote_rule(::Type{BigInt}, ::Type{Int8}) = BigInt

In the latter case, the result type is BigInt since BigInt is the only type large enough to hold integers for arbitrary-precision integer arithmetic. Also note that one does not need to define both promote_rule(::Type{A}, ::Type{B}) and promote_rule(::Type{B}, ::Type{A}) – the symmetry is implied by the way promote_rule is used in the promotion process.

The promote_rule function is used as a building block to define a second function called promote_type, which, given any number of type objects, returns the common type to which those values, as arguments to promote should be promoted. Thus, if one wants to know, in absence of actual values, what type a collection of values of certain types would promote to, one can use promote_type:

julia> promote_type(Int8, Int64)
Int64

Internally, promote_type is used inside of promote to determine what type argument values should be converted to for promotion. It can, however, be useful in its own right. The curious reader can read the code in promotion.jl, which defines the complete promotion mechanism in about 35 lines.

Case Study: Rational Promotions

Finally, we finish off our ongoing case study of Julia's rational number type, which makes relatively sophisticated use of the promotion mechanism with the following promotion rules:

promote_rule(::Type{Rational{T}}, ::Type{S}) where {T<:Integer,S<:Integer} = Rational{promote_type(T,S)}
promote_rule(::Type{Rational{T}}, ::Type{Rational{S}}) where {T<:Integer,S<:Integer} = Rational{promote_type(T,S)}
promote_rule(::Type{Rational{T}}, ::Type{S}) where {T<:Integer,S<:AbstractFloat} = promote_type(T,S)

The first rule says that promoting a rational number with any other integer type promotes to a rational type whose numerator/denominator type is the result of promotion of its numerator/denominator type with the other integer type. The second rule applies the same logic to two different types of rational numbers, resulting in a rational of the promotion of their respective numerator/denominator types. The third and final rule dictates that promoting a rational with a float results in the same type as promoting the numerator/denominator type with the float.

This small handful of promotion rules, together with the type's constructors and the default convert method for numbers, are sufficient to make rational numbers interoperate completely naturally with all of Julia's other numeric types – integers, floating-point numbers, and complex numbers. By providing appropriate conversion methods and promotion rules in the same manner, any user-defined numeric type can interoperate just as naturally with Julia's predefined numerics.