Make Numeric
Refine a new AdditiveArithmetic
Protocol
 Proposal: https://github.com/rxwei/swiftevolution/blob/master/proposals/0215additivearithmeticprotocol.md
 Author: Richard Wei
 Review Manager: TBD
 Status: Awaiting review
 Implementation: apple/swift#19989
Introduction
This proposal introduces a weakening of the existing Numeric
protocol named AdditiveArithmetic
, which defines additive arithmetic operators and a zero, making conforming types roughly correspond to the mathematic notion of an additive group. This makes it possible for vector types to share additive arithmetic operators with scalar types, which enables generic algorithms over AdditiveArithmetic
to apply to both scalars and vectors.
Discussion thread: Should Numeric not refine ExpressibleByIntegerLiteral
Motivation
The Numeric
protocol today refines ExpressibleByIntegerLiteral
and defines all arithmetic operators. The design makes it easy for scalar types to adopt arithmetic operators, but makes it hard for vector types to adopt arithmetic operators by conforming to this protocol.
What's wrong with Numeric
? Assuming that we need to conform to Numeric
to get basic arithmetic operators and generic algorithms, we have three problems.
1. Vectors conforming to Numeric
would be mathematically incorrect.
Numeric
roughly corresponds to a ring. Vector spaces are not rings. Multiplication is not defined between vectors. Requirements *
and *=
below would make vector types inconsistent with the mathematical definition.
static func * (lhs: Self, rhs: Self) > Self
static func *= (lhs: inout Self, rhs: Self)
2. Literal conversion is undefined for dynamically shaped vectors.
Vectors can be dynamically shaped, in which case the the shape needs to be provided when we initialize a vector from a scalar. Dynamically shaped vector types often have an initializer init(repeating:shape:)
.
Conforming to Numeric
requires a conformance to ExpressibleByIntegerLiteral
, which requires init(integerLiteral:)
. However, the conversion from a scalar to a dynamically shaped vector is not defined when there is no given shape.
struct Vector<Scalar: Numeric>: Numeric {
// Okay!
init(repeating: Scalar, shape: [Int]) { ... }
// What's the shape?
init(integerLiteral: Int)
}
3. Common operator overloading causes type checking ambiguity.
Vector types mathematically represent vector spaces. Vectors by definition do not have multiplication between each other, but they come with scalar multiplication.
static func * (lhs: Vector, rhs: Scalar) > Vector { ... }
By established convention in numerical computing communities such as machine learning, many libraries define a multiplication operator *
between vectors as elementwise multiplication. Given that scalar multiplication has to exist by definition, elementwise multiplication and scalar multiplication would overload the *
operator.
static func * (lhs: Vector, rhs: Vector) > Vector { ... }
static func * (lhs: Vector, rhs: Scalar) > Vector { ... }
This compiles, but does not work in practice. The following trivial use case would fail to compile, because literal 1
can be implicitly converted to both a Scalar
and a Vector
, and *
is overloaded for both Vector
and Scalar
.
let x = Vector<Int>(...)
x * 1 // Ambiguous! Can be both `x * Vector(integerLiteral: 1)` and `x * (1 as Int)`.
Proposed solution
We keep Numeric
's behavior and requirements intact, and introduce a new protocol that
 does not require
ExpressibleByIntegerLiteral
conformance, and  shares common properties and operators between vectors and scalars.
To achieve these, we can try to find a mathematical concept that is close enough to makes practical sense without depending on unnecessary algebraic abstractions. This concept is additive group, containing a zero and all additive operators that are defined on Numeric
today. Numeric
will refine this new protocol, and vector types/protocols will conform to/refine the new protocol as well.
Detailed design
We define a new protocol called AdditiveArithmetic
. This protocol requires all additive arithmetic operators that today's Numeric
requires, and a zero. Zero is a fundamental property of an additive group.
public protocol AdditiveArithmetic: Equatable {
/// A zero value.
static var zero: Self { get }
/// Adds two values and produces their sum.
///
/// The addition operator (`+`) calculates the sum of its two arguments. For
/// example:
///
/// 1 + 2 // 3
/// 10 + 15 // 5
/// 15 + 5 // 20
/// 21.5 + 3.25 // 24.75
///
/// You cannot use `+` with arguments of different types. To add values of
/// different types, convert one of the values to the other value's type.
///
/// let x: Int8 = 21
/// let y: Int = 1000000
/// Int(x) + y // 1000021
///
///  Parameters:
///  lhs: The first value to add.
///  rhs: The second value to add.
static func + (lhs: Self, rhs: Self) > Self
/// Adds two values and stores the result in the lefthandside variable.
///
///  Parameters:
///  lhs: The first value to add.
///  rhs: The second value to add.
static func += (lhs: inout Self, rhs: Self) > Self
/// Subtracts one value from another and produces their difference.
///
/// The subtraction operator (``) calculates the difference of its two
/// arguments. For example:
///
/// 8  3 // 5
/// 10  5 // 15
/// 100  5 // 105
/// 10.5  100.0 // 89.5
///
/// You cannot use `` with arguments of different types. To subtract values
/// of different types, convert one of the values to the other value's type.
///
/// let x: UInt8 = 21
/// let y: UInt = 1000000
/// y  UInt(x) // 999979
///
///  Parameters:
///  lhs: A numeric value.
///  rhs: The value to subtract from `lhs`.
static func  (lhs: Self, rhs: Self) > Self
/// Subtracts the second value from the first and stores the difference in the
/// lefthandside variable.
///
///  Parameters:
///  lhs: A numeric value.
///  rhs: The value to subtract from `lhs`.
static func = (lhs: inout Self, rhs: Self) > Self
}
Remove arithmetic operator requirements from Numeric
, and make Numeric
refine AdditiveArithmetic
.
public protocol Numeric: AdditiveArithmetic, ExpressibleByIntegerLiteral {
associatedtype Magnitude: Comparable, Numeric
init?<T>(exactly source: T) where T : BinaryInteger
var magnitude: Self.Magnitude { get }
static func * (lhs: Self, rhs: Self) > Self
static func *= (lhs: inout Self, rhs: Self) > Self
}
To make sure today's Numeric
conforming types do not have to define a zero
, we provide an extension to AdditiveArithmetic
constrained on Self: ExpressibleByIntegerLiteral
.
extension AdditiveArithmetic where Self: ExpressibleByIntegerLiteral {
public static var zero: Self {
return 0
}
}
In the existing standard library, prefix +
is provided by an extension to
Numeric
. Since additive arithmetics are now defined on AdditiveArithmetic
,
we change this extension to apply to AdditiveArithmetic
.
extension AdditiveArithmetic {
/// Returns the given number unchanged.
///
/// You can use the unary plus operator (`+`) to provide symmetry in your
/// code for positive numbers when also using the unary minus operator.
///
/// let x = 21
/// let y = +21
/// // x == 21
/// // y == 21
///
///  Returns: The given argument without any changes.
public static prefix func + (x: Self) > Self {
return x
}
}
Source compatibility
The proposed change is fully sourcecompatible.
Effect on ABI stability
The proposed change will affect the existing ABI of the standard library, because it changes the protocol hierarchy and protocol requirements. As such, this protocol must be considered before the Swift 5 branching date.
Effect on API resilience
The proposed change will affect the existing ABI, and there is no way to make it not affect the ABI because it changes the protocol hierarchy and protocol requirements.
Alternatives considered

Make
Numeric
no longer refineExpressibleByIntegerLiteral
and not introduce any new protocol. This can solve the type checking ambiguity problem in vector protocols, but will break existing code: Functions generic overNumeric
may use integer literals for initialization. Plus, Steve Canon also pointed out that it is not mathematically accurate  there's a canonical homomorphism from the integers to every ring with unity. Moreover, it makes sense for vector types to conform toNumeric
to get arithmetic operators, but it is uncommon to make vectors, esp. fixedrank vectors, be expressible by integer literal. 
On top of
AdditiveArithmetic
, add aMultiplicativeArithmetic
protocol that refinesAdditiveArithmetic
, and makeNumeric
refineMultiplicativeArithmetic
. This would be a natural extension toAdditiveArithmetic
, but the practical benefit of this is unclear. 
Instead of a
zero
static computed property requirement, aninit()
could be used instead, and this would align well with Swift's preference for initializers. However, this would force conforming types to have aninit()
, which in some cases could be confusing or misleading. For example, it would be unclear whetherMatrix()
is creating a zero matrix or an identity matrix. Spelling it aszero
eliminates that ambiguity.