When composing a monad from a stack of monad transformers, it's important to be aware that the order in which the monad transformers are layered matters. Different orderings of the same set of transformers result in different monads.

This version of countLetters is just like the previous version, except it uses type classes to describe the set of available effects instead of providing a concrete monad:

def countLetters [Monad m] [MonadState LetterCounts m] [MonadExcept Err m] (str : String) : m Unit :=
let rec loop (chars : List Char) := do
match chars with
| [] => pure ()
| c :: cs =>
if c.isAlpha then
if vowels.contains c then
modify fun st => {st with vowels := st.vowels + 1}
else if consonants.contains c then
modify fun st => {st with consonants := st.consonants + 1}
else -- modified or non-English letter
pure ()
else throw (.notALetter c)
loop cs
loop str.toList


The state and exception monad transformers can be combined in two different orders, each resulting in a monad that has instances of both type classes:

abbrev M1 := StateT LetterCounts (ExceptT Err Id)
abbrev M2 := ExceptT Err (StateT LetterCounts Id)


When run on input for which the program does not throw an exception, both monads yield similar results:

#eval countLetters (m := M1) "hello" ⟨0, 0⟩

Except.ok ((), { vowels := 2, consonants := 3 })

#eval countLetters (m := M2) "hello" ⟨0, 0⟩

(Except.ok (), { vowels := 2, consonants := 3 })


However, there is a subtle difference between these return values. In the case of M1, the outermost constructor is Except.ok, and it contains a pair of the unit constructor with the final state. In the case of M2, the outermost constructor is the pair, which contains Except.ok applied only to the unit constructor. The final state is outside of Except.ok. In both cases, the program returns the counts of vowels and consonants.

On the other hand, only one monad yields a count of vowels and consonants when the string causes an exception to be thrown. Using M1, only an exception value is returned:

#eval countLetters (m := M1) "hello!" ⟨0, 0⟩

Except.error (StEx.Err.notALetter '!')


Using M2, the exception value is paired with the state as it was at the time that the exception was thrown:

#eval countLetters (m := M2) "hello!" ⟨0, 0⟩

(Except.error (StEx.Err.notALetter '!'), { vowels := 2, consonants := 3 })


It might be tempting to think that M2 is superior to M1 because it provides more information that might be useful when debugging. The same program might compute different answers in M1 than it does in M2, and there's no principled reason to say that one of these answers is necessarily better than the other. This can be seen by adding a step to the program that handles exceptions:

def countWithFallback
(str : String) : m Unit :=
try
countLetters str
catch _ =>
countLetters "Fallback"


This program always succeeds, but it might succeed with different results. If no exception is thrown, then the results are the same as countLetters:

#eval countWithFallback (m := M1) "hello" ⟨0, 0⟩

Except.ok ((), { vowels := 2, consonants := 3 })

#eval countWithFallback (m := M2) "hello" ⟨0, 0⟩

(Except.ok (), { vowels := 2, consonants := 3 })


However, if the exception is thrown and caught, then the final states are very different. With M1, the final state contains only the letter counts from "Fallback":

#eval countWithFallback (m := M1) "hello!" ⟨0, 0⟩

Except.ok ((), { vowels := 2, consonants := 6 })


With M2, the final state contains letter counts from both "hello" and from "Fallback", as one would expect in an imperative language:

#eval countWithFallback (m := M2) "hello!" ⟨0, 0⟩

(Except.ok (), { vowels := 4, consonants := 9 })


In M1, throwing an exception "rolls back" the state to where the exception was caught. In M2, modifications to the state persist across the throwing and catching of exceptions. This difference can be seen by unfolding the definitions of M1 and M2. M1 α unfolds to LetterCounts → Except Err (α × LetterCounts), and M2 α unfolds to LetterCounts → Except Err α × LetterCounts. That is to say, M1 α describes functions that take an initial letter count, returning either an error or an α paired with updated counts. When an exception is thrown in M1, there is no final state. M2 α describes functions that take an initial letter count and return a new letter count paired with either an error or an α. When an exception is thrown in M2, it is accompanied by a state.

In the jargon of functional programming, two monad transformers are said to commute if they can be re-ordered without the meaning of the program changing. The fact that the result of the program can differ when StateT and ExceptT are reordered means that state and exceptions do not commute. In general, monad transformers should not be expected to commute.
Even though not all monad transformers commute, some do. For example, two uses of StateT can be re-ordered. Expanding the definitions in StateT σ (StateT σ' Id) α yields the type σ → σ' → ((α × σ) × σ'), and StateT σ' (StateT σ Id) α yields σ' → σ → ((α × σ') × σ). In other words, the differences between them are that they nest the σ and σ' types in different places in the return type, and they accept their arguments in a different order. Any client code will still need to provide the same inputs, and it will still receive the same outputs.
Most programming languages that have both mutable state and exceptions work like M2. In those languages, state that should be rolled back when an exception is thrown is difficult to express, and it usually needs to be simulated in a manner that looks much like the passing of explicit state values in M1. Monad transformers grant the freedom to choose an interpretation of effect ordering that works for the problem at hand, with both choices being equally easy to program with. However, they also require care to be taken in the choice of ordering of transformers. With great expressive power comes the responsibility to check that what's being expressed is what is intended, and the type signature of countWithFallback is probably more polymorphic than it should be.
• Check that ReaderT and StateT commute by expanding their definitions and reasoning about the resulting types.
• Do ReaderT and ExceptT commute? Check your answer by expanding their definitions and reasoning about the resulting types.
• Construct a monad transformer ManyT based on the definition of Many, with a suitable Alternative instance. Check that it satisfies the Monad contract.
• Does ManyT commute with StateT? If so, check your answer by expanding definitions and reasoning about the resulting types. If not, write a program in ManyT (StateT σ Id) and a program in StateT σ (ManyT Id). Each program should be one that makes more sense for the given ordering of monad transformers.