On 2007-09-29, Roberto Zunino
Graham Hutton wrote:
Readers of this list may enjoy the following note by Martin Escardo, which shows how to write a number of "seemingly impossible Haskell programs" that perform exhaustive searches over spaces of infinite size, by exploiting some ideas from topology:
http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/#...
Very nice! The note shows that you can actually compute (forall p) where p is a *total* predicate over streams of Bool (a.k.a. infinite lazy lists). Indeed, (writing [] for streams)
forall :: ([Bool] -> Bool) -> Bool
seems impossible to compute.
Here's what I understood: (which might be wrong, of course!)
This apparent magic relies on the fact that any total predicate over (total) bool streams, i.e.
p :: [Bool] -> Bool
can only inspect a _finite_ prefix of the input list.
Well, any /computable/ total predicate. This distinction isn't that relevant when we're talking about predicates we might want to implement and run, but there is a mathematical distinction. -- Aaron Denney -><-