Matus Tejiscak wrote:
On Pi, 2009-11-06 at 17:25 -0500, Brent Yorgey wrote:
On Fri, Nov 06, 2009 at 03:29:47PM +0000, Stephen Tetley wrote:
Hello all,
Are any of the of the more exotic recursion schemes definable without a least-fixed point /Mu/ type? Note that Haskell datatypes have a built-in implicit "mu" (that is,
I'd say Mu gets you a greatest-fixed point. I don't have a proof, I just note that Mu(ΛX. 1 + A x X) contains infinite lists, too.
Mu is the notation for least-fixed, Nu is the notation for greatest-fixed. In Haskell, the two fixed points coincide due to laziness and _|_.
I wonder whether this is a problem; if I want to define an initial algebra of a functor, I should take a least fixed point (if I understand it correctly) -- but all I have is Mu. Inductively defined functions will loop on infinite lists, which are included in the resulting type, and -- i imagine -- the type system might prevent such errors, by not enabling casting from greatest-Mu(F) to least-Mu(F). But that's probably too restrictive to be useful.
If you want the type system to catch infinite looping like this, then you'll need to switch to a total functional programming language which distinguishes data and codata (and hence what can be inducted on vs what needs to be coinducted on). -- Live well, ~wren