[swift-evolution] protocol-oriented integers (take 2)

[Xiaodi Wu]

On Mon, Jan 16, 2017 at 12:02 PM, Stephen Canon <scanon at apple.com> wrote:

> On Jan 16, 2017, at 3:25 AM, Xiaodi Wu via swift-evolution <
> swift-evolution at swift.org> wrote:
>
>
> Unless I'm mistaken, after removing division, models of SignedArithmetic
> would have the mathematical properties of a ring. For every element a in
> ring R, there must exist an additive inverse -a in R such that a + (-a) =
> 0. Models of Arithmetic alone would not necessarily have that property.
>
>
> Closure under the arithmetic operations is a sticky point for all the
> finite integer models vs. the actual ring axioms.  No finite [non-modulo]
> integer type is closed, because of overflow. Similarly, additive inverses
> don’t exist for the most negative value of a signed type,
>

I think this goes back to the distinct mentioned earlier: imperfection in
how we model something, or a difference in what we're modeling? Finite
memory will dictate that any model that attempts to represent integers will
face constraints. Signed integer types represent a best-effort attempt at
exactly representing the greatest possible number of integers within a
given amount of memory such that the greatest proportion of those have an
additive inverse that can be also be represented in the same amount of
memory.


> or for any non-zero value of an unsigned type.
>

This is not fundamentally attributable to a limitation of how we model
something. Non-zero values of unsigned type do not have additive inverses
in the same way that non-one values of unsigned type do not have
multiplicative inverses.

The obvious way around this is to say that types conforming to Arithmetic
> model a subset of a ring that need not be closed under the operations.
>

If we don't remove division, type conforming to Arithmetic would also model
a subset of a field that need not be closed under the operations. I'm not
sure it'd be wise to put such a mathematical definition on it with a "need
not" like that. Better, IMO, to give these protocols semantics based on a
positive description of the axioms that do hold--with the caveat that the
result of addition and multiplication will hold to these axioms only
insofar as the result does not overflow.

– Steve
>
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