ContT {runContT :: forall r1 . (forall r2 . a-> m r2) -> m r1} callCC can be defined, however, you can not run it, and reset couldn't type check
Indeed you cannot. As the articles http://okmij.org/ftp/continuations/undelimited.html explain, the answer of _undelimited_ continuation is not available to the program itself. You really cannot write runUndelimitedCont -- just as you cannot write runIO. Once you in the monad of undelimited continuations, you cannot get out of it -- just you cannot get out of IO. Since reset is the composition of runCont and return, reset is not expressible either. The article above explains that in detail (see the CPS2 attempt). The article also shows how to cheat. This exercise points out that undelimited continuations are really not useful. In fact, I don't know of any practical application of them. I'm deeply puzzled why people insist on using callCC given how useless it is by itself (without other effects such as mutation). If one uses callCC and runCont, one deals with _delimited_ continuation. Why not to use shift then, which has a bit more convenient interface. Do you have a specific code that you want to write using ContT? It is generally more productive to discuss a concrete example.