On Mon, Jan 16, 2017 at 12:02 PM, Stephen Canon <sca...@apple.com> wrote:
> On Jan 16, 2017, at 3:25 AM, Xiaodi Wu via swift-evolution < > swift-evolution@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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