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
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