Open research problems
Efficiency of lazy functional languages
This is a problem that came up during IRC discussions. We consider a purely functional language L. By purely functional we mean a language that has value semantics. That is, there is no function such that after evaluation of the function the value that was referred to by something else changed. (Also known as "No Side Effects"). A value is "changed" when it is not the case during an evaluation that when the old value and the new value would both be fully evaluated, there wouldn't be the same result. This should make sure that laziness is allowed in the purely functional language.
The general problem is whether these purely functional languages can implement all algorithms that can be implemented in a language like C as efficient in an amortized sense ignoring space-usage.
As for a specific problem: Implement Data.STRef and Control.Monad.ST.runST without using the built-in ST/IO monad. This needs to happen with operations that all run in O(1) amortized time.