# Natural numbers and the infinite search monad

My previous post explored the usage of infinite search techniques to exhaustively search the natural numbers to find a witness satisfying a total predicate, or decide when no such witness exists.  Turns out, these “searchable” spaces form a monad, available in the infinite search package on hackage.

To use the monad for search across the naturals, we have to first define the set of naturals.  Defining the set is all that is required to start performing infinite searches.  Here is one of my failed attempts at defining the naturals:

natural :: Set Nat
natural = foldr1 union $map (singleton)$ iterate S Z

What’s wrong with this definition?  It defines the set {Z, S Z, SS Z, …}, which represents the entire space of naturals.  Let’s try to do a search:

> search natural (\x -> x+2 == 1)
Nothing

> search natural (\x -> x*2 == 1)
(infinite loop)

Why does the second predicate above, which works just fine using search’ and lyingSearch from the previous post, fail on this predicate?  The issue is that the search’ function searches across the one-point compactification of the naturals (i.e. including infinity).  Our definition of natural above doesn’t include infinity because every element of the set ends in Z due to the iterate function.  Without infinity, the set is not compact, and hence not searchable.  Thanks goes out to Luke Palmer for providing the correct (and prettier!) definition:

natural :: Set Nat
natural = union (singleton Z) (S <\$> natural)

This recursive set definition defines infinity as part of the set, so now…

> search natural (\x -> x*2 == 1)
Nothing