[Haskell-cafe] looking for examples of non-full
Functional Dependencies
Martin Sulzmann
martin.sulzmann at gmail.com
Thu Apr 17 02:06:18 EDT 2008
Mark P Jones wrote:
> Martin Sulzmann wrote:
>> We're also looking for (practical) examples of "multi-range"
>> functional dependencies
>>
>> class C a b c | c -> a b
>>
>> Notice that there are multiple (two) parameters in the range of the FD.
>>
>> It's tempting to convert the above to
>>
>> class C a b c | c -> a, c -> b
>>
>> but this yields a weaker (in terms of type improvement) system.
>
> I agree with Iavor.
>
> In fact, the two sets of dependencies that you have given here
> are provably equivalent, so it would be decidedly odd to have
> a "type improvement" system that distinguishes between them.
>
Consider
class C a b c | a -> b, a -> c
instance C a b b => C [a] [b] [b]
Suppose we encounter the constraint
C [x] y z
What results can we expect from type improvement?
1) Single-range non-full FDs in GHC following the FD-CHR formulation:
The first FD C a b c | a -> b in combination with
the instance head C [a] [b] [b] will yield
C [x] y z improved by y = [b1] for some b1
A similar reasoning yields
C [x] y z improved by z = [b2] for some b2
So, overall we get
C [x] y z improved by y= [b1] and z = [b2]
Unfortunately, we couldn't establish that b1 and b2 are equal.
Hence, we can *not* apply the instance.
2) Alternative design:
We could now argue that the improvement implied by the FD
is only applicable if we are in the "right" context.
That is,
C [x] y z doesn't yield any improvement because
the context y is still underspecified (doesn't match
the instance).
In case of C [x] [y] z we get z = [y]
(and C [x] z [y] yields z = [y])
3) Multi-range FDs
Consider
class C a b c | a -> b c
instance C a b b => C [a] [b] [b]
This time it's straightforward.
C [x] y z yields the improvement y = [b] and z = [b]
which then allows us to apply the instance.
4) Summary.
With multi-range FDs we can derive "more" improvement.
C [x] y z yields y = [b] and z = [b]
Based on the FD-CHR formulation, for the single-range FD case we get
C [x] y z yields y = [b1] and z = [b2]
which is clearly weaker.
The alternative design is even weaker, from C [x] y z we can derive
any improvement.
So, I conclude that in the Haskell type improvement context
there's clearly a difference among single-range and multi-range FDs.
Of course, we could define multi-range FDs in terms of single-range FDs
which then trivially solves the "equivalence" problem (but some user
may be disappointed that their multi-range FDs yield weaker improvement).
Thanks for your pointers below but I believe that FDs in the Haskell context
can be quite different from FDs in the database context.
- Martin
> While you're looking for unusual examples of fundeps, you
> might also want to consider things like:
>
> class C a b c | a -> b, b -> c
>
> Note that this is subtly different from a -> b c because
>
> {a -> b, b -> c} |= {a -> b c}
>
> while the reverse does not hold. Instead of type classes,
> I'll give you some more down-to-earth examples of relations
> that satisfy these dependencies:
>
> {Paper, Conference, Year}
> {Professor, University, Country}
> {Person, FavoriteFood, FoodGroup}
> ...
>
> For further insight, you might want to take a look at the theory
> of relational databases to see how functional dependencies are
> used there. In that context, some sets of dependencies (such
> as {a -> b, b -> c}) can be interpreted as indicators of bad
> design. This, in turn, might give you some ideas about the kinds
> of dependencies you can expect to encounter in well-designed
> Haskell code. (Of course, you might still find examples in other
> Haskell code of dependencies that would make a database person
> wince :-)
>
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