Thanks for the detailed response...
On Thu, Dec 30, 2010 at 9:54 AM, Conor McBride
On 28 Dec 2010, at 23:29, Luke Palmer wrote:
Eta conversion corresponds to extensionality; i.e. there is nothing
more to a function than what it does to its argument. I think Conor McBride is the local expert on that subject.
...I suppose I might say something.
Dependent type theories have programs in types, and so require some notion of when it is safe to consider open terms equal in order to say when types match:
An attempt to recapitulate: The following is a dependent type example where equality of open terms comes up. z : (x : A) → B z = ... w : (y : A) → C w = z Here the compiler needs to show that the type B, with x free, is equivalent to C, with y free. This isn't always decidable, but eta rules, as an addition to beta and delta rules, make more of these equivalence checks possible (I'm assuming this is what extensionality means here). What would be example terms for B and C that would require invoking the eta rule for functions, for example? it's interesting to see how far one
can chuck eta into equality without losing decidability of conversion, messing up the "Geuvers property", or breaking type-preservation.
It's a minefield, so tread carefully. There are all sorts of bad interactions, e.g. with subtyping (if -> subtyping is too weak, (\x -> f x) can have more types than f), with strictness (if p = (fst p, snd p), then (case p of (x, y) -> True) = True, which breaks the Geuvers property on open terms), with reduction (there is no good way to orientate the unit type eta-rule, u = (), in a system of untyped reduction rules).
But the news is not all bad. It is possible to add some eta-rules without breaking the Geuvers property (for functions it's ok; for pairs and unit it's ok if you make their patterns irrefutable). You can then decide the beta-eta theory by postprocessing beta-normal forms with type-directed eta-expansion (or some equivalent type-directed trick). Epigram 2 has eta for functions, pairs, and logical propositions (seen as types with proofs as their indistinguishable inhabitants). I've spent a lot of time banging my head off these issues: my head has a lot of dents, but so have the issues.
So, indeed, eta-rules make conversion more extensional, which is unimportant for closed computation, but useful for reasoning and for comparing open terms. It's a fascinating, maddening game trying to add extensionality to conversion while keeping it decidable and ensuring that open computation is not too strict to deliver values.
Hoping this is useful, suspecting that it's TMI
Very useful. Not TMI at all. I find this fascinating. David -- David Sankel Sankel Software www.sankelsoftware.com