On to Hydro

[Ben Laurie]

Tyler Close wrote:
> 
> Ben Laurie wrote:
> > Tyler Close wrote:
> > > >  But having partially ordered elements respond to "<=",
> > > > but not respond
> > > > to "<", freaks me out.
> > >
> > > It actually seems very natural to me. In order to search a linear
> > > container, you need to have either the "<", or the ">",
> > operator. A
> > > partial ordering does not let you search a linear
> > container, and so
> > > does not provide either the "<", or the ">" operator.
> > Pretty slick,
> > > no?
> >
> > No - this is, in effect, (a -> b) -> (a' -> b'), which is,
> > of course,
> > false.
> 
> Does formal proof logic apply to software interface design?
> 
> I do know that a fact of software interface design is that the
> interface defines the space of possible operations. My opinion is that
> if the operation is not in the space, then the interface should not
> include it.

This is Catch-22 - you redefine "<" and then claim that it isn't in the
space. Fine, but so what? I prefer "<" to behave like I expect it to
(i.e. to mean "<=" but not "==")

> 
> > Besides, "a < b" can be defined as "a <= b && a != b" if you "only"
> > provide "<=".
> 
> "a <= b && a != b" does not result in a total ordering. The contract
> for "a < b" would be that it provides a total ordering. If you tried
> to search a linear container using your partial order "<", then your
> search would be broken in the way defined by the "<=" contract.

Naturally if a set can only be partially ordered, then it can only be
partially ordered. I didn't claim to provide a total ordering, I merely
claimed that you can derive "<" from "<=" and "==".

> Note that your partial order "<" would allow you to search a
> multi-dimensional container like the Xanadu containers. A Xanadu
> container could be implemented in E using only the "<=" operator, as
> Dean has pointed out. So your syntactically plausible implementation
> is valid, so long as you only apply it within the space that it is
> defined to be valid for. By not providing your implementation by
> default, the language makes it clear that the implementation is only
> valid within the space defined by the "<=" operator.

It seems to me that this only helps if the programmer happens to write
his code in a way that uses the operator you want him to use to show the
lack of partial ordering. If the code happens to be written the other
way, nothing has been added. So, to make it obvious that you are talking
about partial ordering you should eschew _all_ the operators and use
some new ones (e.g. a.partiallyLessThan(b)). Then no-one gets a
surprise.

Cheers,

Ben.

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