## Non-determinism: a sublanguage rather than a monad

Non-determinism: a sublanguage rather than a monad

A puzzlingly named, exceedingly technical device introduced to structure the denotational semantics has by now achieved cult status. It has been married to effects -- more than once. It is compulsively looked for in all manner of things, including burritos. At least two ICFP papers brought it up without a rhyme or reason (or understanding), as the authors later admitted. I am talking about monads.

In truth, effects are not married to monads and approachable directly. The profound insight behind monads is the structuring, the separation of pure' (context-independent) and effectful computations. The structuring can be done without explicating mathematical monads, and especially without resorting to vernacular monads such as State, etc. This article gives an example: a simple, effectful, domain-specific sublanguage embedded into an expressive macro' metalanguage. Abstraction facilities of the metalanguage such higher-order functions and modules help keep the DSL to the bare minimum, often to the first order, easier to reason about and implement.

The key insight predates monads and goes all the way back to the origins of ML, as a scripting language for the Edinburgh LCF theorem prover. What has not been clear is how simple an effectful DSL may be while remaining useful. How convenient it is, especially compared to the monadic encodings. How viable it is to forsake the generality of first-class functions and monads and what benefits it may bring. We report on an experiment set out to explore these questions.

We pick a rather complex effect -- non-determinism -- and use it in OCaml, which at first blush seems unsuitable since it is call-by-value and has no monadic sugar. And yet, we can write non-deterministic programs just as naturally and elegantly as in Haskell or Curry.

The running tutorial example is computing all permutations of a given list of integers. The reader may want to try doing that in their favorite language or plain OCaml. Albeit a simple exercise, the code is often rather messy and not obviously correct. In the functional-logic language Curry, it is strikingly elegant: mere foldr insert []. It is the re-statement of the specification: a permutation is moving the elements of the source list one-by-one into some position in the initially empty list. The code immediately tells that the number of possible permutations of n elements is n!. From its very conception in the 1959 Rabin and Scott's paper, non-determinism was called for to write clear specifications -- and then to make them executable. That is what will shall do.