Yeah, the free monad structures involve lots of lambdas. It's not so much that information is "lost", more that you cannot see beyond the next lambda abstraction.
From a DSL perspective, it's like you can only inspect the program so far as to know the next statement in the "do" block. To see what the next statement will be, you need to actually evaluate the current one.
Instead of free monads, if you want very analyzable structures, look at free applicative functors.
"Applicative functors are a generalisation of monads. Both allow the expression of effectful computations into an otherwise pure language, like Haskell. Applicative functors are to be preferred to monads when the structure of a computation is fixed a priori. That makes it possible to perform certain kinds of static analysis on applicative values. We define a notion of free applicative functor, prove that it satisfies the appropriate laws, and that the construction is left adjoint to a suitable forgetful functor. We show how free applicative functors can be used to implement embedded DSLs which can be statically analysed."
The problem is that applicative functors aren't powerful enough to allow computations that depend on previous results. I suspect what we want is something like free ArrowChoice, but I'm not aware of any work in that direction.
I might be completely missing your point (my apologies if i am).
applicative is for specific elements of the structure. It's totally reasonable to want access to nearby values, but that requires being a little bit tricky. You could do something like a window of averages
windows = map (take 5) tails $ [1,2,3,4,5,6]
and then do your fmap across that
fmap sum windows
For access to prior values, you need something that looks a lot more like a fold. In that kind of case, i'd point at traversable.
mapAccumL (\val accumulator -> val + accumulator) 0 [1,2,3,4,5]
which would gather up the sums, [1, 3, 6, 10, 15]
of course, accumulator can be as fancy as you want, hashmap, set, tree, or whatever. If you can formulate a dynamic programming solution, there's generally a way to stuff that into traversable.
This is sort of off the top of my head and my haskell is a bit rusty, so this probably won't compile. But i think it captures the essence.
The trouble is that applicative doesn't let you use the result of one effect to compute the next effect. (Indeed sometimes there is no there there: Const is a valid applicative). Monads have bind, but that's a little too powerful to implement efficiently. Like I said, I suspect ArrowChoice or the like might be the missing intermediate construct.
ApplicativeDo might be useful. You would be able to see the static applicative structure as far as the next monadic bind, which lets you do interesting types of optimization, e.g. Haxl.
The thing is we're used to thinking in monads, and used to thinking of certain constructs as equivalent, which ApplicativeDo would break. There's a real tension between being explicit about the performance - in which case many identities no longer hold - and not. I don't know what Haxl itself does, but Fetch (which claims to be a Scala port) is kind of unsatisfactory in that regard.
AplicativeDo solves this a little bit through using Applicative until the results are necessary for the next instruction, then opting for Monad instances
Right, but at that point you're no longer Free. (Unless you declare the inefficient implementation and the efficient implementation to be equivalent, but if you do that then every refactor risks destroying your performance).
Monad linearizes the call graph; the effects of f(g, h) have to look the same as those of f(g(h)). In particular your interpreter can't parallelize, because it can't tell whether there's a data dependency between one effect and the next or not.
From a DSL perspective, it's like you can only inspect the program so far as to know the next statement in the "do" block. To see what the next statement will be, you need to actually evaluate the current one.
Instead of free monads, if you want very analyzable structures, look at free applicative functors.
"Applicative functors are a generalisation of monads. Both allow the expression of effectful computations into an otherwise pure language, like Haskell. Applicative functors are to be preferred to monads when the structure of a computation is fixed a priori. That makes it possible to perform certain kinds of static analysis on applicative values. We define a notion of free applicative functor, prove that it satisfies the appropriate laws, and that the construction is left adjoint to a suitable forgetful functor. We show how free applicative functors can be used to implement embedded DSLs which can be statically analysed."
http://arxiv.org/abs/1403.0749