In case this one went under the radar, at POPL'12, Martín Escardó gave a tutorial on seemingly impossible functional programs:
Programming language semantics is typically applied to
prove compiler correctness and allow (manual or automatic) program
verification. Certain kinds of semantics can also be applied to
discover programs that one wouldn't have otherwise thought of. This is
the case, in particular, for semantics that incorporate topological
ingredients (limits, continuity, openness, compactness). For example,
it turns out that some function types (X -> Y) with X infinite (but
compact) do have decidable equality, contradicting perhaps popular
belief, but certainly not (higher-type) computability theory. More
generally, one can often check infinitely many cases in finite time.
I will show you such programs, run them fast in surprising instances,
and introduce the theory behind their derivation and working. In
particular, I will study a single (very high type) program that (i)
optimally plays sequential games of unbounded length, (ii) implements
the Tychonoff Theorem from topology (and builds finite-time search
functions for infinite sets), (iii) realizes the double-negation shift
from proof theory (and allows us to extract programs from classical
proofs that use the axiom of countable choice). There will be several
examples in the languages Haskell and Agda.
A shorter version (coded in Haskell) appears in Andrej Bauer's blog.
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