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Suppose we have a monad, defined by return , (>>=) and the set of laws. There is a data type newtype C m a = C { unC ∷ forall r. (a → m r) → m r } also known as Codensity . C m a ≅ m a given that m is a Monad , i.e. we can write two functions to ∷ Monad m ⇒ m a → C m a and from ∷ Monad m ⇒ C m a → m a to ∷ Monad m ⇒ m a → C m a to t = C $ \f → t >>= f from ∷ Monad m ⇒ C m a → m a from = ($ return) . unC and show that to ∘ from ≡ id and from ∘ to ≡ id by equational reasoning, for example: from . to = -- by definition of `(.)' \x → from (to x) = -- by definition of `to' \x → from (C $ \f → x >>= f) = -- by definition of `from' \x → ($ return) (unC (C $ \f → x >>= f)) = -- unC . C ≡ id \x → ($ return) (\f → x >>= f) = -- β-reduce \x → x >>= return = -- right identity law \x → x = -- by definition of `id' id So far so good. My questions are Given a type and a bunch of laws, how do we construct corresponding isomorphic CPS representation? Is this representation unique (I'd guess no)? If it's not unique, is there always the most "simple" (in the number of → s for example :) ) one?

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