On 12/01/2013, at 6:30 AM, Raoul Duke wrote:
> On Fri, Jan 11, 2013 at 11:25 AM, john skaller
> <skal...@users.sourceforge.net> wrote:
>> The abstract properties of numbers are defined here:
>> http://felix-lang.org/$/usr/local/lib/felix/felix-latest/lib/std//algebraic.flx
>> as a mathematician you will understand this code, despite the fact it
>> is using type classes. It's just defining groups and rings. The "Float"
>> variant is there because floating point math isn't associative.
>> Its the best I could think of to mean "approximately" since there's
>> no universal way to say that for floats.
>
> ieee floating point isn't ever going to be real math, right? is it
> dangerous to try to pretend it kinda is? :-)
The Float type classes define the operations like + - * /
which floats do support, whatever they actually mean.
They do obey some laws, for example addition is symmetrical:
a + b = b + a
even for floats. So you see that rule in FloatAddgrp,
and you see Addgrp (additive group) is a FloatAddgrp
plus associativity.
In fact this is a cheat too if we bind C-land integers (since they overflow).
However floats really are "approximately associative most of the time",
it's just hard to state. And of course IEEE floats actually do form a
finite set of discrete values that are bound by rigid mathematical laws,
they're just too complicated to state in the weak logic language
supported by Felix, and stating them probably wouldn't be useful
(because they're really complicated -- I've actually seen a model)
On the other hand the semantic rules for integers are a bit easier
to understand, even if they're not easy to state. Integers form a
"locally approximate group" which means all the laws hold
provided the integer representation is big enough and we look
close enough to 0. How big and how close isn't relevant to the
maths because its expressed in terms of limits.
Also, some laws hold with certain representations like twos-complement,
for example:
if a + b - b is representable, then it equals a
that is, the cancellation law holds, even if an intermediate
calculation overflows. You can ignore the overflow.
Ultimately we can say that, for example, gmp_z, really obeys all the
integer laws because for any practical purposes, although it's actually
finite and bounded by address space, it won't give the wrong
answer because to do so would not only require more memory
than you could possibly have, the failing calculation would probably
take more than the expected life of the universe to complete
at todays processor speeds :)
--
john skaller
skal...@users.sourceforge.net
http://felix-lang.org
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