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Today, a talk by Henning Kerstan ("Trace Semantics for Probabilistic Transition Systems") confronted me with category theory for the first time. He has built a theoretical framework for describing probablistic transition systems and their behaviour in a general way, i.e. with uncountably infinite state sets and different notions of traces. To this end, he goes up through several layers of abstraction to finally end up with the notion of monads which he combines with measure theory to build the model he needs.

In the end, it took him 45 minutes to (roughly) build a framework to describe a concept he initially explained in 5 minutes. I appreciate the beauty of the approach (it does generalise nicely over different notions of traces) but it strikes me as an odd balance nevertheless.

I struggle to see what a monad really is and how so general a concept can be useful in applications (both in theory and practice). Is it really worth the effort, result-wise?

Therefore this question:

Are there problems that are natural (in the sense of CS) on which the abstract notion of monads can be applied and helps (or is even instrumental) to derive desired results (at all or in a nicer way than without)?

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    $\begingroup$ Encoding states in a purely functional programming language? Is this a natural enough CS problem? $\endgroup$ – Stéphane Gimenez Mar 8 '12 at 21:10
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    $\begingroup$ The more general problem of handling effects in functional languages is the example I've seen most: for the theory, the monads for effects are sexy and for the practice, Haskell's IO monad is very handy. $\endgroup$ – jmad Mar 8 '12 at 21:20
  • $\begingroup$ And what would be the advantages compared to classic, relatively lightweight semantics? Are FP monads even the same thing as in category theory? Questions upon questions. $\endgroup$ – Raphael Mar 8 '12 at 22:51
  • $\begingroup$ See this question on cstheory.SE for a more general question after uses of category theory. $\endgroup$ – Raphael Jul 23 '12 at 10:12
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Asking whether an occurrence of monad is natural is similar to asking whether a group (in the sense of group theory) is natural. Once you formalise something, in this case as an endofunctor, either it satisfies the axioms of being a monad or not. If it does satisfy the axioms, then one gets a lot of technical machinery for free.

Moggi's paper Notions of computation and monads pretty much seals the deal: monads are incredibly natural and useful for describing computational effects in a unified way. Wadler and others translated these notions to deal with computational effects in functional programming languages, by using the connection that a functor is a data type constructor. This adds the icing on the cake. FP monads allow a treatment of computational effects such as IO which would be extremely unnatural without them. Monads have inspired related useful notion such as arrows and idioms which are also very useful for structuring functional programs. See the Wadler link for references. FP monads are the same thing as category theory monads in the sense that for an FP monad to work the same equations need to hold --- the compiler relies on these. Generally, the presentation of the monad differs (different but equivalent operations and equations), but this is a superficial difference.

A vast amount of the work of Bart Jacobs, to take but one example, uses monads. A lot of work stems from coalgebra, which is a general theory of systems. One of Jacobs' (many) contributions to the area is the development of a generic notion of trace semantics for systems (described as coalgebras) based on monads. One could argue that the notion of trace semantics is natural: What is the semantics of a system? The list of actions that can be observed!

One way to understand monads is to first program in Haskell using monads. Then find one of the many good tutorials available (via google). Start from the programming angle, then move to the theoretical side, starting with some basic category theory.

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