What's a suitable data type?

In the question, Seeking constructive criticism on monad implementation, abesto asked people to criticize his "Monad" which kept count of the number of开发者_开发问答 bind operations. It turned out that this was not actually a monad because it did not satisfy the first two monadic laws, but I found the example interesting. Is there any data type that would be suitable for such kinds of structures?

That's an interesting question, and has to do with the mathematical lineage of monads.

We could certainly create a typeclass called something like Monadish, which would look exactly like the Monad typeclass:

class Monadish m where
  returnish :: a -> m a
  bindish :: m a -> (a -> m b) -> m b

So the monad laws have nothing to do with the actual signature of the typeclass; they're extra information that an implementor has to enforce by themselves. So, in one sense, the answer is "of course"; just make another typeclass and say it doesn't have to satisfy any laws.

But is such a typeclass interesting? For a mathematician, the answer would be no: the lack of any laws means that there is no interesting structure by which to reason with. When we define a mathematical structure, we usually define some objects (check), some operations (check) and then some properties of the operations (...nope). We need all three of these to prove theorems about this class of objects, and, to take one example, abstract algebra is all about taking the same operations and adding more or fewer laws.

For a software engineer, the answer is a little more complex. Reasoning is not required: you can always just use a typeclass to overload syntax for your own nefarious purposes. We can use a typeclass to group things together that "feel" the same, even though we don't have any formal reasons for believing so. There are some benefits to doing this, but I personally feel this throws out a lot of the benefits of having laws, and leads to architecture astronauts who invent abstract structures without a whole lot of thought of their applicability. Maths is a safer bet: the monad laws correspond to left identity, right identity, and associativity, reasonably fundamental assumptions that even a non-mathematical person would be familiar with.