> Small bug fix to Juli8 on Mon 6 April.
-> Topics: [[Using the OCaml Monad library|/topics/week9_using_the_monad_library]]; the State monad; Programming with mutable state; Using multiple monads together; [[Homework for weeks 8-9|/exercises/assignment8-9]]
+> Topics: [[Using the OCaml Monad library|/topics/week9_using_the_monad_library]]; [[Programming with mutable state|/topics/week9_mutable_state]]; [[A State Monad Tutorial|/topics/week9_state_monad_tutorial]]; Using multiple monads together; [[Homework for weeks 8-9|/exercises/assignment8-9]]
> Reading for Week 10: Groenendijk, Stokhof, and Veltman, "[[Coreference and Modality|/readings/coreference-and-modality.pdf]]" (1996)
*Will explain later, and add more material.*
+Until we do, have a look at our [page on translating between OCaml Scheme and Haskell](http://lambda1.jimpryor.net/translating_between_OCaml_Scheme_and_Haskell) from the first time we offered this seminar.
+++ /dev/null
-[[!toc]]
-
-The seminar is now going to begin talking about more **imperatival** or **effect**-like elements in programming languages. The only effect-like element we've encountered so far is the possibility of divergence, in languages that permit fixed point combinators and so have the full power of recursion. What it means for something to be effect-like, and why this counts as an example of such, will emerge.
-
-Other effect-like elements in a language include: printing (recall the [[damn]] example at the start of term); continuations (also foreshadowed in the [[damn]] example) and exceptions (foreshadowed in our discussion of abortable list traversals in [[week4]]); and **mutation**. This last notion is our topic this week.
-
-
-## Mutation##
-
-What is mutation? It's helpful to build up to this in a series of fragments. For pedagogical purposes, we'll be using a made-up language that's syntactically similar to, but not quite the same as, OCaml.
-
-Recall from earlier discussions that the following two forms are equivalent:
-
- [A] let x be EXPRESSION in
- BODY
-
- (lambda (x) -> BODY) (EXPRESSION)
-
-This should seem entirely familiar:
-
- [B] let x be 1 + 2 in
- let y be 10 in
- (x + y, x + 20)
- ; evaluates to (13, 23)
-
-In fragment [B], we bound the variables `x` and `y` to `int`s. We can also bind variables to function values, as here:
-
- [C] let f be (lambda (x, y) -> x + y + 1) in
- (f (10, 2), f (20, 2))
- ; evaluates to (13, 23)
-
-If the expression that evaluates to a function value has a free variable in it, like `y` in the next fragment, it's interpreted as bound to whatever value `y` has in the surrounding lexical context:
-
- [D] let y be 3 in
- let f be (lambda (x) -> x + y) in
- (f (10), f (20))
- ; evaluates to (13, 23)
-
-Other choices about how to interpret free variables are also possible (you can read about "lexical scope" versus "dynamic scope"), but what we do here is the norm in functional programming languages, and seems to be easiest for programmers to reason about.
-
-In our next fragment, we re-use a variable that had been bound to another value in a wider context:
-
- [E] let y be 2 in
- let y be 3 in
- (y + 10, y + 20)
- ; evaluates to (13, 23)
-
-As you can see, the narrowest assignment is what's effective. This is just like in predicate logic: consider <code>∃y (Fy and ∃y ~Fy)</code>. The computer-science terminology to describe this is that the narrower assignment of `y` to the value 3 **shadows** the wider assignment to 2.
-
-I call attention to this because you might casually describe it as "changing the value that y is assigned to." What we'll go on to see is a more exotic phenomenon that merits that description better.
-
-Sometimes the shadowing is merely temporary, as here:
-
- [F] let y be 2 in
- let f be (lambda (x) ->
- let y be 3 in
- ; here the most local assignment to y applies
- x + y
- ) in
- ; here the assignment of 3 to y has expired
- (f (10), y, f (20))
- ; evaluates to (13, 2, 23)
-
-OK, now we're ready for our main event, **mutable variables.** We'll introduce new syntax to express an operation where we're not shadowing a wider assignment, but *changing* the original assignment:
-
- [G] let y be 2 in
- let f be (lambda (x) ->
- change y to 3 then
- x + y
- ) in
- ; here the change in what value y was assigned *sticks*
- ; because we *updated* the value of the original variable y
- ; instead of introducing a new y with a narrower scope
- (f (10), y, f (19))
- ; evaluates to (13, 3, 23)
-
-In languages that have native syntax for this, there are two styles in which it can be expressed. The *implicit style* is exemplified in fragment [G] above, and also in languages like C:
-
- {
- int y = 2; // this is like "let y be 2 in ..."
- ...
- y = 3; // this is like "change y to 3 then ..."
- return x + y; // this is like "x + y"
- }
-
-A different possibility is the *explicit style* for handling mutation. Here we explicitly create and refer to new "reference cells" to hold our values. When we change a variable's value, the variable stays associated with the same reference cell, but that reference cell's contents get modified. The same thing happens in the semantic machinery underlying implicit-style mutable variables, but there it's implicit---the reference cells aren't themselves expressed by any term in the object language. In explicit-style mutation, they are. OCaml has explicit-style mutation. It looks like this:
-
- let ycell = ref 2 (* this creates a new reference cell *)
- ...
- in let () = ycell := 3 (* this changes the contents of that cell to 3 *)
- (* the return value of doing so is () *)
- (* other return values could also be reasonable: *)
- (* such as the old value of ycell, the new value, an arbitrary int, and so on *)
- in x + !ycell;; (* the !ycell operation "dereferences" the cell---it retrieves the value it contains *)
-
-Scheme is similar. There are various sorts of reference cells available in Scheme. The one most like OCaml's `ref` is a `box`. Here's how we'd write the same fragment in Scheme:
-
- (let ([ycell (box 2)])
- ...
- (set-box! ycell 3)
- (+ x (unbox ycell)))
-
-C has explicit-style mutable variables, too, which it calls *pointers*. But simple variables in C are already mutable, in the implicit style. Scheme also has both styles of mutation. In addition to the explicit boxes, Scheme also lets you mutate unboxed variables:
-
- (begin
- (define x 1)
- (set! x 2)
- x)
- ; evaluates to 2
-
-When dealing with explicit-style mutation, there's a difference between the types and values of `ycell` and `!ycell` (or in Scheme, `(unbox ycell)`). The former has the type `int ref`: the variable `ycell` is assigned a reference cell that contains an `int`. The latter has the type `int`, and has whatever value is now stored in the relevant reference cell. In an implicit-style framework though, we only have the resources to refer to the contents of the relevant reference cell. `y` in fragment [G] or the C snippet above has the type `int`, and only ever evaluates to `int` values.
-
-
-##Controlling order##
-
-When we're dealing with mutable variables (or any other kind of effect), order matters. For example, it would make a big difference whether I evaluated `let z = !ycell` before or after evaluating `ycell := !ycell + 1`. Before this point, order never mattered except sometimes it played a role in avoiding divergence.
-
-OCaml does *not* guarantee what order expressions will be evaluated in arbitrary contexts. For example, in the following fragment, you cannot rely on `expression_a` being evaluated before `expression_b` before `expression_c`:
-
- let triple = (expression_a, expression_b, expression_c)
-
-OCaml does however guarantee that different let-expressions are evaluated in the order they lexically appear. So in the following fragment, `expression_a` *will* be evaluated before `expression_b` and that before `expression_c`:
-
- let a = expression_a
- in let b = expression_b
- in expression_c
-
-Scheme does the same. (*If* you use Scheme's `let*`, but not if you use its `let`. I agree this is annoying.)
-
-If `expression_a` and `expression_b` evaluate to (), for instance if they're something like `ycell := !ycell + 1`, that can also be expressed in OCaml as:
-
- let () = expression_a
- in let () = expression_b
- in expression_c
-
-And OCaml has a syntactic shorthand for this form, namely to use semi-colons:
-
- expression_a; expression_b; expression_c
-
-This is not the same role that semi-colons play in list expressions, like `[1; 2; 3]`. To be parsed correctly, these semi-colon'ed complexes sometimes need to be enclosed in parentheses or a `begin ... end` construction:
-
- (expression_a; expression_b; expression_c)
-
- begin expression_a; expression_b; expression_c end
-
-Scheme has a construction similar to the latter:
-
- (begin (expression_a) (expression_b) (expression_c))
-
-Though often in Scheme, the `(begin ...)` is implicit and doesn't need to be explicitly inserted, as here:
-
- (lambda (x) (expression_a) (expression_b) (expression_c))
-
-Another way to control evaluation order, you'll recall from [[week6]], is to use **thunks**. These are functions that only take the uninformative `()` as an argument, such as this:
-
- let f () = ...
-
-or this:
-
- let f = fun () -> ...
-
-In Scheme these are written as functions that take 0 arguments:
-
- (lambda () ...)
-
-or:
-
- (define (f) ...)
-
-How could such functions be useful? Well, as always, the context in which you build a function need not be the same as the one in which you apply it to some arguments. So for example:
-
- let ycell = ref 1
- in let f () = ycell := !ycell + 1
- in let z = !ycell
- in f ()
- in z;;
-
-We don't apply (or call or execute or however you want to say it) the function `f` until after we've extracted `ycell`'s value and assigned it to `z`. So `z` will get assigned 1. If on the other hand we called `f ()` before evaluating `let z = !ycell`, then `z` would have gotten assigned a different value.
-
-In languages with mutable variables, the free variables in a function definition are usually taken to refer back to the same *reference cells* they had in their lexical contexts, and not just their original value. So if we do this for instance:
-
- let factory (starting_value : int) =
- let free_var = ref starting_value
- in let getter () =
- !free_var
- in let setter (new_value : int) =
- free_var := new_value
- in (getter, setter)
- in let (getter, setter) = factory 1
- in let first = getter ()
- in let () = setter 2
- in let second = getter ()
- in let () = setter 3
- in let third = getter ()
- in (first, second, third)
-
-At the end, we'll get `(1, 2, 3)`. The reference cell that gets updated when we call `setter` is the same one that gets fetched from when we call `getter`. This should seem very intuitive here, since we're working with explicit-style mutation. When working with a language with implicit-style mutation, it can be more surprising. For instance, here's the same fragment in Python, which has implicit-style mutation:
-
- def factory (starting_value):
- free_var = starting_value
- def getter ():
- return free_var
- def setter (new_value):
- # the next line indicates that we're using the
- # free_var from the surrounding function, not
- # introducing a new local variable with the same name
- nonlocal free_var
- free_var = new_value
- return getter, setter
- getter, setter = factory (1)
- first = getter ()
- setter (2)
- second = getter ()
- setter (3)
- third = getter ()
- (first, second, third)
-
-Here, too, just as in the OCaml fragment, all the calls to getter and setter are working with a single mutable variable `free_var`.
-
-If you've got a copy of *The Seasoned Schemer*, which we recommended for the seminar, see the discussion at pp. 91-118 and 127-137.
-
-If however you called `factory` twice, you'd have different `getter`/`setter` pairs, each of which had their own, independent `free_var`. In OCaml:
-
- let factory (starting_val : int) =
- ... (* as above *)
- in let (getter, setter) = factory 1
- in let (getter', setter') = factory 1
- in let () = setter 2
- in getter' ()
-
-Here, the call to `setter` only mutated the reference cell associated with the `getter`/`setter` pair. The reference cell associated with `getter'` hasn't changed, and so `getter' ()` will still evaluate to 1.
-
-Notice in these fragments that once we return from inside the call to `factory`, the `free_var` mutable variable is no longer accessible, except through the helper functions `getter` and `setter` that we've provided. This is another way in which a thunk like `getter` can be useful: it still has access to the `free_var` reference cell that was created when it was, because its free variables are interpreted relative to the context in which `getter` was built, even if that context is otherwise no longer accessible. What `getter ()` evaluates to, however, will very much depend on *when* we evaluate it---in particular, it will depend on which calls to the corresponding `setter` were evaluated first.
-
-##Referential opacity##
-
-In addition to order-sensitivity, when you're dealing with mutable variables you also give up a property that computer scientists call "referential transparency." It's not obvious whether they mean exactly the same by that as philosophers and linguists do, or only something approximately the same.
-
-The core idea to referential transparency is that when the same value is supplied to a context, the whole should always evaluate the same way. Mutation makes it possible to violate this. Consider:
-
- let ycell = ref 1
- in let f x = x + !ycell
- in let first = f 1 (* first is assigned the value 2 *)
- in ycell := 2; let second = f 1 (* second is assigned the value 3 *)
- in first = second;; (* not true! *)
-
-Notice that the two invocations of `f 1` yield different results, even though the same value is being supplied as an argument to the same function.
-
-Similarly, functions like these:
-
- let f cell = !cell;;
-
- let g cell = cell := !cell + 1; !cell;;
-
-may return different results each time they're invoked, even if they're always supplied one and the same reference cell as argument.
-
-Computer scientists also associate referential transparency with a kind of substitution principle, illustrated here:
-
- let x = 1
- in (x, x)
-
-should evaluate the same as:
-
- let x = 1
- in (x, 1)
-
-or:
-
- (1, 1)
-
-Notice, however, that when mutable variables are present, the same substitution patterns can't always be relied on:
-
- let ycell = ref 1
- in ycell := 2; !ycell
- (* evaluates to 2 *)
-
- (ref 1) := 2; !(ref 1)
- (* creates a ref 1 cell and changes its contents *)
- (* then creates a *new* ref 1 cell and returns *its* contents *)
-
-
-
-
-##How to implement explicit-style mutable variables##
-
-We'll think about how to implement explicit-style mutation first. We suppose that we add some new syntactic forms to a language, let's call them `newref`, `deref`, and `setref`. And now we want to expand the semantics for the language so as to interpret these new forms.
-
-Well, part of our semantic machinery will be an assignment function, call it `g`. Somehow we should keep track of the types of the variables and values we're working with, but we won't pay much attention to that now. In fact, we won't even bother much at this point with the assignment function. Below we'll pay more attention to it.
-
-In addition to the assignment function, we'll also need a way to keep track of how many reference cells have been "allocated" (using `newref`), and what their current values are. We'll suppose all the reference cells are organized in a single data structure we'll call a **store**. This might be a big heap of memory. For our purposes, we'll suppose that reference cells only ever contain `int`s, and we'll let the store be a list of `int`s.
-
-In many languages, including OCaml, the first position in a list is indexed `0`, the second is indexed `1` and so on. If a list has length 2, then there won't be any value at index `2`; that will be the "next free location" in the list.
-
-Before we brought mutation on the scene, our language's semantics will have looked something like this:
-
-> \[[expression]]<sub>g</sub> = value
-
-Now we're going to relativize our interpretations not only to the assignment function `g`, but also to the current store, which I'll label `s`. Additionally, we're going to want to allow that evaluating some functions might *change* the store, perhaps by allocating new reference cells or perhaps by updating the contents of some existing cells. So the interpretation of an expression won't just return a value; it will also return a possibly updated store. We'll suppose that our interpretation function does this quite generally, even though for many expressions in the language, the store that's returned will be the same one that the interpretation function started with:
-
-> \[[expression]]<sub>g s</sub> = (value, s')
-
-For expressions we already know how to interpret, expect `s'` to just be `s`.
-An exception is complex expressions like `let var = expr1 in expr2`. Part of
-interpreting this will be to interpret the sub-expression `expr1`, and we have
-to allow that in doing that, the store may have already been updated. We want
-to use that possibly updated store when interpreting `expr2`. Like this:
-
- let rec eval expression g s =
- match expression with
- ...
- | Let (c, expr1, expr2) ->
- let (value, s') = eval expr1 g s
- (* s' may be different from s *)
- (* now we evaluate expr2 in a new environment where c has been associated
- with the result of evaluating expr1 in the current environment *)
- eval expr2 ((c, value) :: g) s'
- ...
-
-Similarly:
-
- ...
- | Addition (expr1, expr2) ->
- let (value1, s') = eval expr1 g s
- in let (value2, s'') = eval expr2 g s'
- in (value1 + value2, s'')
- ...
-
-Let's consider how to interpet our new syntactic forms `newref`, `deref`, and `setref`:
-
-
-1. When `expr` evaluates to `starting_val`, **newref expr** should allocate a new reference cell in the store and insert `starting_val` into that cell. It should return some "key" or "index" or "pointer" to the newly created reference cell, so that we can do things like:
-
- let ycell = newref 1
- in ...
-
- and be able to refer back to that cell later by using the value that we assigned to the variable `ycell`. In our simple implementation, we're letting the store just be an `int list`, and we can let the "keys" be indexes in that list, which are (also) just `int`s. Somehow we should keep track of which variables are assigned `int`s as `int`s and which are assigned `int`s as indexes into the store. So we'll create a special type to wrap the latter:
-
- type store_index = Index of int;;
-
- Our interpretation function will look something like this:
-
- let rec eval expression g s =
- match expression with
- ...
- | Newref (expr) ->
- let (starting_val, s') = eval expr g s
- (* note that s' may be different from s, if expr itself contained any mutation operations *)
- (* now we want to retrieve the next free index in s' *)
- in let new_index = List.length s'
- (* now we want to insert starting_val there; the following is an easy but inefficient way to do it *)
- in let s'' = List.append s' [starting_val]
- (* now we return a pair of a wrapped new_index, and the new store *)
- in (Index new_index, s'')
- ...
-
-2. When `expr` evaluates to a `store_index`, then **deref expr** should evaluate to whatever value is at that index in the current store. (If `expr` evaluates to a value of another type, `deref expr` is undefined.) In this operation, we don't change the store at all; we're just reading from it. So we'll return the same store back unchanged (assuming it wasn't changed during the evaluation of `expr`).
-
- let rec eval expression g s =
- match expression with
- ...
- | Deref (expr) ->
- let (Index n, s') = eval expr g s
- (* note that s' may be different from s, if expr itself contained any mutation operations *)
- in (List.nth s' n, s')
- ...
-
-3. When `expr1` evaluates to a `store_index` and `expr2` evaluates to an `int`, then **setref expr1 expr2** should have the effect of changing the store so that the reference cell at that index now contains that `int`. We have to make a decision about what value the `setref ...` call should itself evaluate to; OCaml makes this `()` but other choices are also possible. Here I'll just suppose we've got some appropriate value in the variable `dummy`.
-
- let rec eval expression g s =
- match expression with
- ...
- | Setref (expr1, expr2) ->
- let (Index n, s') = eval expr1 g s
- (* note that s' may be different from s, if expr1 itself contained any mutation operations *)
- in let (new_value, s'') = eval expr2 g s'
- (* now we create a list which is just like s'' except it has new_value in index n *)
- in let rec replace_nth lst m =
- match lst with
- | [] -> failwith "list too short"
- | x::xs when m = 0 -> new_value :: xs
- | x::xs -> x :: replace_nth xs (m - 1)
- in let s''' = replace_nth s'' n
- in (dummy, s''')
- ...
-
-
-
-
-
-##How to implement implicit-style mutable variables##
-
-With implicit-style mutation, we don't have new syntactic forms like `newref` and `deref`. Instead, we just treat ordinary variables as being mutable. You could if you wanted to have some variables be mutable and others not; perhaps the first sort are written in Greek and the second in Latin. But we will suppose all variables in our language are mutable.
-
-We will still need a store to keep track of reference cells and their current values, just as in the explicit-style implementation. This time, every variable will be associated with an index into the store. So this is what we'll have our assignment function keep track of. The assignment function will bind variables to indexes into the store, rather than to the variables' current values. The variables will only indirectly be associated with "their values" by virtue of the joint work of the assignment function and the store.
-
-This brings up an interesting conceptual distinction. Formerly, we'd naturally think that a variable `x` is associated with only one type, and that that's the type that the expression `x` would *evaluate to*, and also the type of value that the assignment function *bound* `x` to. However, in the current framework these two types come apart. The assignment function binds `x` to an index into the store, and what the expression `x` evaluates to will be the value at that location in the store, which will usually be some type other than an index into a store, such as a `bool` or a `string`.
-
-To handle implicit-style mutation, we'll need to re-implement the way we interpret expressions like `x` and `let x = expr1 in expr2`. We will also have just one new syntactic form, `change x to expr1 then expr2`.
-
-Here's how to implement these. We'll suppose that our assignment function is list of pairs, as above and as in [week7](/reader_monad_for_variable_binding).
-
- let rec eval expression g s =
- match expression with
- ...
- | Var (c : char) ->
- let index = List.assoc c g
- (* retrieve the value at that index in the current store *)
- in let value = List.nth s index
- in (value, s)
-
- | Let ((c : char), expr1, expr2) ->
- let (starting_val, s') = eval expr1 g s
- (* get next free index in s' *)
- in let new_index = List.length s'
- (* insert starting_val there *)
- in let s'' = List.append s' [starting_val]
- (* evaluate expr2 using a new assignment function and store *)
- in eval expr2 ((c, new_index) :: g) s''
-
- | Change ((c : char), expr1, expr2) ->
- let (new_value, s') = eval expr1 g s
- (* lookup which index is associated with Var c *)
- in let index = List.assoc c g
- (* now we create a list which is just like s' except it has new_value at index *)
- in let rec replace_nth lst m =
- match lst with
- | [] -> failwith "list too short"
- | x::xs when m = 0 -> new_value :: xs
- | x::xs -> x :: replace_nth xs (m - 1)
- in let s'' = replace_nth s' index
- (* evaluate expr2 using original assignment function and new store *)
- in eval expr2 g s''
-
-Note: Chris uses this kind of machinery on the third page of the Nov 22 handout. Except he implements `Let` the way we here implement `Change`. And he adds an implementation of `Alias` (see below). Some minor differences: on his handout (and following Groenendijk, Stokhof and Veltman), he uses `r` and `g` where we use `g` and `s` respectively. Also, he implements his `r` with a function from `char` to `int`, instead of a `(char * int) list`, as we do here. It should be obvious how to translate between these. His implementation requires that variables always already have an associated peg. So that when we call `Let(c, expr1, expr2)` for the first time with `c`, there's a peg whose value is to be updated. That's easier to ensure when you implement the assignment as a function than as a `(char * int) list`.
-
-
-##How to implement mutation with a State monad##
-
-It's possible to do all of this monadically, and so using a language's existing resources, instead of adding new syntactic forms and new interpretation rules to the semantics. The patterns we use to do this in fact closely mirror the machinery described above.
-
-We call this a State monad. It's a lot like the Reader monad, except that with the Reader monad, we could only read from the environment. We did have the possibility of interpreting sub-expressions inside a "shifted" environment, but as you'll see, that corresponds to the "shadowing" behavior described before, not to the mutation behavior that we're trying to implement now.
-
-With a State monad, we call our book-keeping apparatus a "state" or "store" instead of an environment, and this time we are able to both read from it and write to it. To keep things simple, we'll work here with the simplest possible kind of store, which only holds a single value. One could also have stores that were composed of a list of values, of a length that could expand or shrink, or even more complex structures.
-
-Here's the implementation of the State monad, together with an implementation of the Reader monad for comparison:
-
- type env = (char * int) list;;
- (* alternatively, an env could be implemented as type char -> int *)
-
- type 'a reader = env -> 'a;;
- let reader_unit (value : 'a) : 'a reader =
- fun e -> value;;
- let reader_bind (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
- fun e -> let a = u e
- in let u' = f a
- in u' e;;
-
- type store = int;;
- (* very simple store, holds only a single int *)
- (* this corresponds to having only a single mutable variable *)
-
- type 'a state = store -> ('a, store);;
- let state_unit (value : 'a) : 'a state =
- fun s -> (value, s);;
- let state_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
- fun s -> let (a, s') = u s
- in let u' = f a
- in u' s';;
-
-Notice the similarities (and differences) between the implementation of these two monads.
-
-With the Reader monad, we also had some special-purpose operations, beyond its general monadic operations. These were `lookup` and `shift`. With the State monad, we'll also have some special-purpose operations. We'll consider two basic ones here. One will be to retrieve what is the current store. This is like the Reader monad's `lookup`, except in this simple implementation there's only a single location for a value to be looked up from. Here's how we'll do it:
-
- let state_get : store state =
- fun s -> (s, s);;
-
-This passes through the current store unaltered, and also returns a copy of the store as its value. We can use this operation like this:
-
- some_existing_state_monad_box >>= fun _ -> state_get >>= (fun cur_store -> ...)
-
-The `fun _ ->` part here discards the value wrapped by `some_existing_state_monad_box`. We're only going to pass through, unaltered, whatever *store* is generated by that monadic box. We also wrap that store as *our own value*, which can be retrieved by further operations in the `... >>= ...` chain, such as `(fun cur_store -> ...)`.
-
-The other operation for the State monad will be to update the existing store to a new one. This operation looks like this:
-
- let state_put (new_store : int) : dummy state =
- fun s -> (dummy, new_store);;
-
-If we want to stick this in a `... >>= ...` chain, we'll need to prefix it with `fun _ ->` too, like this:
-
- some_existing_state_monad_box >>= fun _ -> state_put 100 >>= ...
-
-In this usage, we don't care what value is wrapped by `some_existing_state_monad_box`. We don't even care what store it generates, since we're going to replace that store with our own new store. A more complex kind of `state_put` operation might insert not just some constant value as the new store, but rather the result of applying some function to the existing store. For example, we might want to increment the current store. Here's how we could do that:
-
- some_existing_state_monad_box >>= fun _ -> state_get >>= (fun cur_store -> state_put (cur_store + 1) >>= ...
-
-We can of course define more complex functions that perform the `state_get >>= (fun cur_store -> state_put (cur_store + 1)` as a single operation.
-
-In general, a State monadic **box** (type `'a state`, what appears at the start of a `... >>= ... >>= ...` chain) is an operation that accepts some starting store as input---where the store might be simple as it is here, or much more complex---and returns a value plus a possibly modified store. This can be thought of as a static encoding of some computation on a store, which encoding is used as a box wrapped around a value of type `'a`. (And also it's a burrito.)
-
-State monadic **operations** (type `'a -> 'b state`, what appears anywhere in the middle or end of a `... >>= ... >>= ...` chain) are operations that generate new State monad boxes, based on what value was wrapped by the preceding elements in the `... >>= ... >>= ...` chain. The computations on a store that these encode (which their values may or may not be sensitive to) will be chained in the order given by their position in the `... >>= ... >>= ...` chain. That is, the computation encoded by the first element in the chain will accept a starting store s0 as input, and will return (a value and) a new store s1 as output, the next computation will get s1 as input and will return s2 as output, the next computation will get s2 as input, ... and so on.
-
-To get the whole process started, the complex computation so defined will need to be given a starting store. So we'd need to do something like this:
-
- let computation = some_state_monadic_box >>= operation >>= operation
- in computation initial_store;;
-
-
-* See also our [[State Monad Tutorial]].
-
-
-##Aliasing or Passing by reference##
-
--- FIXME --
-
- [H] ; *** aliasing ***
- let y be 2 in
- let x be y in
- let w alias y in
- (y, x, w)
- ; evaluates to (2, 2, 2)
-
- [I] ; mutation plus aliasing
- let y be 2 in
- let x be y in
- let w alias y in
- change y to 3 then
- (y, x, w)
- ; evaluates to (3, 2, 3)
-
- [J] ; as we already know, these are all equivalent:
-
- let f be (lambda (y) -> BODY) in ; #1
- ... f (EXPRESSION) ...
-
- (lambda (y) -> BODY) EXPRESSION ; #2
-
- let y be EXPRESSION in ; #3
- ... BODY ...
-
- [K] ; *** passing by reference ***
- ; now think: "[J#1] is to [J#3] as [K#1] is to [K#2]"
-
- ? ; #1
-
- let w alias y in ; #2
- ... BODY ...
-
- ; We introduce a special syntactic form to supply
- ; the missing ?
-
- let f be (lambda (alias w) -> ; #1
- BODY
- ) in
- ... f (y) ...
-
- [L] let f be (lambda (alias w) ->
- change w to 2 then
- w + 2
- ) in
- let y be 1 in
- let z be f (y) in
- ; y is now 2, not 1
- (z, y)
- ; evaluates to (4, 2)
-
- [M] ; hyper-evaluativity
- let h be 1 in
- let p be 1 in
- let f be (lambda (alias x, alias y) ->
- ; contrast here: "let z be x + y + 1"
- change y to y + 1 then
- let z be x + y in
- change y to y - 1 then
- z
- ) in
- (f (h, p), f (h, h))
- ; evaluates to (3, 4)
-
-Notice: in [M], `h` and `p` have same value (1), but `f (h, p)` and `f (h, h)` differ.
-
-See Pryor's "[Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt)".
-
-
-##Four grades of mutation involvement##
-
-Programming languages tend to provide a bunch of mutation-related capabilities at once, if they provide any. For conceptual clarity, however, it's helped me to distill these into several small increments.
-
-* At the first stage, we have a purely functional language, like we've been working with up until this week.
-
-
-* One increment would be to add aliasing or passing by reference, as illustrated above. In the illustration, we relied on the combination of passing by reference and mutation to demonstrate how you could get different behavior depending on whether an argument was passed to a function by reference or instead passed in the more familiar way (called "passing by value"). However, it would be possible to have passing by reference in a language without having mutation. For it to make any difference whether an argument is passed by reference or by value, such a language would have to have some primitive predicates which are sensitive to whether their arguments are aliased or not. In Jim's paper linked above, he calls such predicates "hyper-evaluative."
-
- The simplest such predicate we might call "hyperequals": `y hyperequals w` should evaluate to true when and only when the arguments `y` and `w` are aliased.
-
-
-* Another increment would be to add implicit-style mutable variables, as we explained above. You could do this with or without also adding passing-by-reference.
-
- The semantic machinery for implicit-style mutable variables will have something playing the role of a reference cell. However these won't be **first-class values** in the language. For something to be a first-class value, it has to be possible to assign that value to variables, to pass it as an argument to functions, and to return it as the result of a function call. Now for some of these criteria it's debatable that they are already here satisfied. For example, in some sense the introduction of a new implicitly mutable variable (`let x = 1 in ...`) will associate a reference cell with `x`. That won't be what `x` evaluates to, but it will be what the assignment function *binds* `x` to, behind the scenes. Similarly, if we bring in passing by reference, then again in some sense we are passing reference cells as arguments to functions. Not explicitly---in a context like:
-
- let f = (lambda (alias w) -> ...)
- in let x = 1
- in f (x)
-
- the expression `w` won't evaluate to a reference cell anywhere inside the `...`. But it will be associated with a reference cell, in the same way that `x` is (and indeed, with the same reference cell).
-
- However, in language with implicit-style mutation, even when combined with passing by reference, what you're clearly not able to do is to return a reference cell as the result of a function call, or indeed of any expression. This is connected to---perhaps it's the same point as---the fact that `x` and `w` don't evalute to reference cells, but rather to the values that the reference cell they're implicitly associated with contains, at that stage in the computation.
-
-* A third grade of mutation involvement is to have explicit-style mutation. Here we might say we have not just mutable variables but also first-class values whose contents can be altered. That is, we have not just mutable variables but **mutable values**.
-
- This introduces some interesting new conceptual possibilities. For example, what should be the result of the following fragment?
-
- let ycell = ref 1
- in let xcell = ref 1
- in ycell = xcell
-
- Are the two reference cell values equal or aren't they? Well, at this stage in the computation, they're qualitatively indiscernible. They're both `int ref`s containing the same `int`. And that is in fact the relation that `=` expresses in OCaml. In Scheme the analogous relation is spelled `equal?` Computer scientists sometimes call this relation "structural equality."
-
- On the other hand, these are numerically *two* reference cells. If we mutate one of them, the other one doesn't change. For example:
-
- let ycell = ref 1
- in let xcell = ref 1
- in ycell := 2
- in !xcell;;
- (* evaluates to 1, not to 2 *)
-
- So we have here the basis for introducing a new kind of equality predicate into our language, which tests not for qualitative indiscernibility but for numerical equality. In OCaml this relation is expressed by the double equals `==`. In Scheme it's spelled `eq?` Computer scientists sometimes call this relation "physical equality". Using this equality predicate, our comparison of `ycell` and `xcell` will be `false`, even if they then happen to contain the same `int`.
-
- Isn't this interesting? Intuitively, elsewhere in math, you might think that qualitative indicernibility always suffices for numerical identity. Well, perhaps this needs discussion. In some sense the imaginary numbers ι and -ι are qualitatively indiscernible, but numerically distinct. However, arguably they're not *fully* qualitatively indiscernible. They don't both bear all the same relations to ι for instance. But then, if we include numerical identity as a relation, then `ycell` and `xcell` don't both bear all the same relations to `ycell`, either. Yet there is still a useful sense in which they can be understood to be qualitatively equal---at least, at a given stage in a computation.
-
- Terminological note: in OCaml, `=` and `<>` express the qualitative (in)discernibility relations, also expressed in Scheme with `equal?`. In OCaml, `==` and `!=` express the numerical (non)identity relations, also expressed in Scheme with `eq?`. `=` also has other syntactic roles in OCaml, such as in the form `let x = value in ...`. In other languages, like C and Python, `=` is commonly used just for assignment (of either of the sorts we've now seen: `let x = value in ...` or `change x to value in ...`). The symbols `==` and `!=` are commonly used to express qualitative (in)discernibility in these languages. Python expresses numerical (non)identity with `is` and `is not`. What an unattractive mess. Don't get me started on Haskell (qualitative discernibility is `/=`) and Lua (physical (non)identity is `==` and `~=`).
-
- Because of the particular way the numerical identity predicates are implemented in all of these languages, it doesn't quite match our conceptual expectations. For instance, For instance, if `ycell` is a reference cell, then `ref !ycell` will always be a numerically distinct reference cell containing the same value. We get this pattern of comparisons in OCaml:
-
- ycell == ycell
- ycell != ref !ycell (* true, these aren't numerically identical *)
-
- ycell = ycell
- ycell = ref !ycell (* true, they are qualitatively indiscernible *)
-
- But now what about?
-
- (0, 1, ycell) ? (0, 1, ycell)
- (0, 1. ycell) ? (0, 1. ref !ycell)
-
- You might expect the first pair to be numerically identical too---after all, they involve the same structure (an immutable triple) each of whose components is numerically identical. But OCaml's "physical identity" predicate `==` does not detect that identity. It counts both of these comparisons as false. OCaml's `=` predicate does count the first pair as equal, but only because it's insensitive to numerical identity; it also counts the second pair as equal. This shows up in all the other languages I know, as well. In Python, `y = []; (0, 1, y) is (0, 1, y)` evaluates to false. In Racket, `(define y (box 1)) (eq? (cons 0 y) (cons 0 y))` also evaluates to false (and in Racket, unlike traditional Schemes, `cons` is creating immutable pairs). They chose an implementation for their numerical identity predicates that is especially efficient and does the right thing in the common cases, but doesn't quite match our mathematical expectations.
-
- Additionally, note that none of the equality predicates so far considered is the same as the "hyperequals" predicate mentioned above. For example, in the following (fictional) language:
-
- let ycell = ref 1
- in let xcell = ref 1
- in let wcell alias ycell
- in let zcell = ycell
- in ...
-
- at the end, `hyperequals ycell wcell` (and the converse) would be true, but no other non-reflexive hyperequality would be true. `hyperequals ycell zcell` for instance would be false. If we express numerical identity using `==`, as OCaml does, then both of these (and their converses) would be true:
-
- ycell == wcell
- ycell == zcell
-
- but these would be false:
-
- xcell == ycell
- xcell == wcell
- xcell == zcell
-
- If we express qualitative indiscernibility using `=`, as OCaml does, then all of the salient comparisons would be true:
-
- ycell = wcell
- ycell = zcell
- xcell = ycell
- ...
-
- Another interesting example of "mutable values" that illustrate the coming apart of qualitative indiscernibility and numerical identity are the `getter`/`setter` pairs we discussed earlier. Recall:
-
- let factory (starting_val : int) =
- let free_var = ref starting_value
- in let getter () =
- !free_var
- in let setter (new_value : int) =
- free_var := new_value
- in (getter, setter)
- in let (getter, setter) = factory 1
- in let (getter', setter') = factory 1
- in ...
-
- After this, `getter` and `getter'` would (at least, temporarily) be qualitatively indiscernible. They'd return the same value whenever called with the same argument (`()`). So too would `adder` and `adder'` in the following example:
-
- let factory (starting_val : int) =
- let free_var = ref starting_value
- in let adder x =
- x + !free_var
- in let setter (new_value : int) =
- free_var := new_value
- in (adder, setter)
- in let (adder, setter) = factory 1
- in let (adder', setter') = factory 1
- in ...
-
- Of course, in most languages you wouldn't be able to evaluate a comparison like `getter = getter'`, because in general the question whether two functions always return the same values for the same arguments is not decidable. So typically languages don't even try to answer that question. However, it would still be true that `getter` and `getter'` (and `adder` and `adder'`) were extensionally equivalent; you just wouldn't be able to establish so.
-
- However, they're not numerically identical, because by calling `setter 2` (but not calling `setter' 2`) we can mutate the function value `getter` (and `adder`) so that it's *no longer* qualitatively indiscernible from `getter'` (or `adder'`).
-
-
-
-* A fourth grade of mutation involvement: (--- FIXME ---)
-
- structured references
- (a) if `a` and `b` are mutable variables that uncoordinatedly refer to numerically the same value
- then mutating `b` won't affect `a` or its value
- (b) if however their value has a mutable field `f`, then mutating `b.f` does
- affect their shared value; will see a difference in what `a.f` now evaluates to
- (c) examples: Scheme mutable pairs, OCaml mutable arrays or records
-
-
-
-##Miscellany##
-
-* When using mutable variables, programmers will sometimes write using *loops* that repeatedly mutate a variable, rather than the recursive techniques we've been using so far. For example, we'd define the factorial function like this:
-
- let rec factorial n =
- if n = 0 then 1 else n * factorial (n - 1)
-
- or like this:
-
- let factorial n =
- let rec helper n sofar =
- if n = 0 then sofar else helper (n - 1) (n * sofar)
- in helper n 1
-
- (The second version is more efficient than the first; so you may sometimes see this programming style. But for our purposes, these can be regarded as equivalent.)
-
- When using mutable variables, on the other hand, this may be written as:
-
- let factorial n =
- let current = ref n
- in let total = ref 1
- in while !current > 0 do
- total := !total * !current; current := !current - 1
- done; !total
-
-
-* Mutable variables also give us a way to achieve recursion, in a language that doesn't already have it. For example:
-
- let fact_cell = ref None
- in let factorial n =
- if n = 0 then 1 else match !fact_cell with
- | Some fact -> n * fact (n - 1)
- | None -> failwith "can't happen"
- in let () = fact_cell := Some factorial
- in ...
-
- We use the `None`/`Some factorial` option type here just as a way to ensure that the contents of `fact_cell` are of the same type both at the start and the end of the block.
-
- If you've got a copy of *The Seasoned Schemer*, which we recommended for the seminar, see the discussion at pp. 118-125.
-
-* Now would be a good time to go back and review some material from [[week1]], and seeing how much we've learned. There's discussion back then of declarative or functional languages versus languages using imperatival features, like mutation. Mutation is distinguished from shadowing. There's discussion of sequencing, and of what we mean by saying "order matters."
-
- In point 7 of the Rosetta Stone discussion, the contrast between call-by-name and call-by-value evaluation order appears (though we don't yet call it that). We'll be discussing that more in coming weeks. In the [[damn]] example, continuations and other kinds of side-effects (namely, printing) make an appearance. These too will be center-stage in coming weeks.
-
-* Now would also be a good time to read [Calculator Improvements](/week10). This reviews the different systems discussed above, as well as other capabilities we can add to the calculators introduced in [week7](/reader_monad_for_variable_binding). We will be building off of that in coming weeks.
-
-
-##Offsite Reading##
-
-* [[!wikipedia Declarative programming]]
-* [[!wikipedia Functional programming]]
-* [[!wikipedia Purely functional]]
-* [[!wikipedia Side effect (computer science) desc="Side effects"]]
-* [[!wikipedia Referential transparency (computer science)]]
-* [[!wikipedia Imperative programming]]
-* [[!wikipedia Reference (computer science) desc="References"]]
-* [[!wikipedia Pointer (computing) desc="Pointers"]]
-* [Pointers in OCaml](http://caml.inria.fr/resources/doc/guides/pointers.html)
-
-<!--
-# General issues about variables and scope in programming languages #
-
-* [[!wikipedia Variable (programming) desc="Variables"]]
-* [[!wikipedia Free variables and bound variables]]
-* [[!wikipedia Variable shadowing]]
-* [[!wikipedia Name binding]]
-* [[!wikipedia Name resolution]]
-* [[!wikipedia Parameter (computer science) desc="Function parameters"]]
-* [[!wikipedia Scope (programming) desc="Variable scope"]]
-* [[!wikipedia Closure (computer science) desc="Closures"]]
-
--->
-
+++ /dev/null
-This walks through most of [A State Monad Tutorial](http://strabismicgobbledygook.wordpress.com/2010/03/06/a-state-monad-tutorial/), which is addressed to a Haskell-using audience. But we convert it to OCaml. See our page on
-[[Translating between OCaml Scheme and Haskell]].
-
-Some of what we do here will make use of our [[monad library]] for OCaml.
-
-As we discussed in [[week9]], a State monad is implemented with the type:
-
- store -> ('a * store)
-
-It's common practice to encapsulate this in some way, so that the interpreter knows the difference between arbitrary functions from a `blah` to a pair of something and a `blah` and the values that you've specially designated as being State monadic values.
-
-The most lightweight way encapsulate it would be just to add a data constructor to the type. In the same way that the `'a option` type has the `None` and `Some` data constructors, we give our `'a state` type a `State` data constructor:
-
- (* we assume that the store type has already been declared *)
- type 'a state = State of (store -> ('a * store))
-
-Then a function expecting an `'a store` will look for a value with the structure `State ...` rather than just one with the structure `...`.
-
-To take a `State (s -> (a,s))` and get at the `s -> (a,s)` it wraps, you use the same techniques you use to take an `Some int` and get at the `int` it wraps:
-
- let u = State (fun s -> (1, s))
- in let State unwrapped_state = u
- in ...
-
-or:
-
- let u = State (fun s -> (1, s))
- in match u with
- | State unwrapped_state -> ...
-
-There are two heavierweight ways to encapsulate the type of the State monad. One is used by our [[monad library]]---the type is hidden from the outside user, and only gets exposed by the `run` function. But the result of `run u` is not itself recognized as a monadic value any longer. You can't replace `u` in:
-
- Monad.(u >>= ...)
-
-with `run u`. So you should only apply `run` when you've finished building up a monadic value. (That's why it's called `run`.)
-
-Of course you can do this:
-
- let u = Monad.(...)
- in let intermediate_result = Monad.run u 0
- in let v = Monad.(u >>= ...)
- in let final_result = Monad.run v 0
- in ...
-
-The other heavyweight way to encapsulate the type of a monad is to use records. See [here](/translating_between_OCaml_Scheme_and_Haskell) and [here](/coroutines_and_aborts/) for some introduction to these. We don't use this design in our OCaml monad library, but the Haskell monad libraries do, and it would be good for you to get acquainted with it so that you can see how to ignore it when you come across it in Haskell-based literature. (Or you might want to learn Haskell, who knows?)
-
-We'll illustrate this technique in OCaml code, for uniformity. See the [translation page](/translating_between_OCaml_Scheme_and_Haskell) about how this looks in Haskell.
-
-To use the record technique, instead of saying;
-
- type 'a state = State of (store -> ('a * store))
-
-you'd say:
-
- type 'a state = { state : store -> ('a * store) }
-
-and instead of saying:
-
- let u = State (fun s -> (1, s))
- in let State unwrapped_state = u
- in ...
-
-you'd say:
-
- let u = { state = fun s -> (1, s) }
- in let unwrapped_state = u.state
- in ...
-
-That's basically it. As with the other two techniques, the type of `u` is not the same as `store -> ('a * store)`, but the relevant code will know how to convert between them.
-
-The main benefit of these techniques is that it gives you better type-checking: it makes sure that you're only using your monadic values in the hygenic ways you're supposed to. Perhaps you don't care about that. Well, then, if you want to write all your own monadic code, you can proceed as you like. If you ever want to use other people's code, though, or read papers or web posts about monads, you will encounter one or more of these techniques, and so you need to get comfortable enough with them not to let them confuse you.
-
-OK, back to our walk-through of "A State Monad Tutorial". What shall we use for a store? Instead of a plain `int`, let's suppose our store is a structure of two values: a running total, and a count of how many times the store has been modified. We'll implement this with a record. Hence:
-
- type store' = { total : int; modifications: int };;
-
-State monads employing this store will then have *three* salient values at any point in the computation: the `total` and `modifications` field in the store, and also the `'a` value that is then wrapped in the monadic box.
-
-Here's a monadic box that encodes the operation of incrementing the store's `total` and wrapping the value that was the former `total`:
-
- let increment_store : store' -> (int * store') =
- fun s ->
- let value = s.total
- in let s' = { total = succ s.total; modifications = succ s.modifications }
- in (value, s')
-
-If we wanted to work with one of the encapsulation techniques described above, we'd have to proceed a bit differently. Here is how to do it with the first, lightweight technique:
-
- let increment_store' : 'a state =
- State (fun s ->
- let value = s.total
- in let s' = { total = succ s.total; modifications = succ s.modifications }
- in (value, s'))
-
-
-Here is how you'd have to do it using our OCaml monad library:
-
- # #use "path/to/monads.ml";;
- # module S = State_monad(struct type store = store' end);;
- # let increment_store'' : ('x,'a) S.m =
- S.(get >>= fun cur ->
- let value = cur.total
- in let s' = { total = succ cur.total; modifications = succ cur.modifications }
- in put s' >> unit value);;
-
-Let's try it out:
-
- # let s0 = { total = 42; modifications = 3 };;
- # increment_store s0;;
- - : int * store' = (42, {total = 43; modifications = 4})
-
-Or if you used the OCaml monad library:
-
- # S.(run(increment_store'')) s0;;
- - : int * S.store = (42, {total = 43; modifications = 4})
-
-Great!
-
-Can you write a monadic value that instead of incrementing each of the `total` and `modifications` fields in the store, doubles the `total` field and increments the `modifications` field?
-
-What about a value that increments each of `total` and `modifications` twice? Well, you could custom-write that, as with the previous question. But we already have the tools to express it easily, using our existing `increment_store` value:
-
- increment_store >>= fun value -> increment_store >> unit value
-
-That ensures that the value we get at the end is the value returned by the first application of `increment_store`, that is, the contents of the `total` field in the store before we started modifying the store at all.
-
-You should start to see here how chaining monadic values together gives us a kind of programming language. Of course, it's a cumbersome programming language. It'd be much easier to write, directly in OCaml:
-
- let value = s0.total
- in (value, { total = s0.total + 2; modifications = s0.modifications + 2};;
-
-or, using pattern-matching on the record (you don't have to specify every field in the record):
-
- let { total = value; _ } = s0
- in (value, { total = s0.total + 2; modifications = s0.modifications + 2};;
-
-But **the point of learning how to do this monadically** is that (1) monads show us how to embed more sophisticated programming techniques, such as imperative state and continuations, into frameworks that don't natively possess them (such as the set-theoretic metalanguage of Groenendijk, Stokhof and Veltman's paper); (2) becoming familiar with monads will enable you to see patterns you'd otherwise miss, and implement some seemingly complex computations using the same simple patterns (same-fringe is an example); and finally, of course (3) monads are delicious.
-
-Keep in mind that the final result of a bind chain doesn't have to be the same type as the starting value:
-
- increment_store >>= fun value -> increment_store >> unit (string_of_int value)
-
-Or:
-
- unit 1 >> unit "blah"
-
-The store keeps the same type throughout the computation, but the type of the wrapped value can change.
-
-What are the special-purpose operations that the `State_monad` module defines for us?
-
-* `get` is a monadic value that passes through the existing store unchanged, and also wraps that same store as its boxed value. You use it like this:
-
- ... >>= fun _ -> get >>= fun cur -> ... here we can use cur ...
-
- As we've said, that's equivalent to:
-
- ... >> get >>= fun cur -> ...
-
- You can also get the current store at the start of the computation:
-
- get >> = fun cur -> ...
-
-* `gets selector` is like `get`, but it additionally applies the `selector` function to the store before depositing it in the box. If your store is structured, you can use this to only extract a piece of the structure:
-
- ... >> gets (fun cur -> cur.total) >>= fun total -> ...
-
- For more complex structured stores, consider using the `Ref_monad` version of the State monad in the OCaml library.
-
-* `put new_store` replaces the existing store with `new_store`. Use it like this:
-
- ... >> put new_store >> fun () -> ...
-
- As that code snippet suggests, the boxed value after the application of `puts new_store` is just `()`. If you want to preserve the existing boxed value but replace the store, do this:
-
- ... >>= fun value -> put new_store >> unit value >>= ...
-
-* Finally, `puts modifier` applies `modifier` to whatever the existing store is, and substitutes that as the new store. As with `put`, the boxed value afterwards is `()`.
-
- Haskell calls this operation `modify`. We've called it `puts` because it seems to fit naturally with the convention of `get` vs `gets`. (See also `ask` vs `asks` in `Reader_monad`, which are also the names used in Haskell.)
-
-
-Here's an example from "A State Monad Tutorial":
-
- increment_store >> get >>= fun cur ->
- State (fun s -> ((), { total = s.total / 2; modifications = succ s.modifications })) >>
- increment_store >> unit cur.total
-
-Or, as you'd have to write it using our OCaml monad library:
-
- increment_store'' >> get >>= fun cur ->
- put { total = cur.total / 2; modifications = succ cur.modifications } >>
- increment_store'' >> unit cur.total
-
-
-The last topic covered in "A State Monad Tutorial" is the use of do-notation to work with monads in Haskell. We discuss that on our [translation page](/translating_between_OCaml_Scheme_and_Haskell).
-
-
-[[!toc]]
-
-Multi-layered monadic boxes
-===========================
-
-So far, we've defined monads as single-layered boxes. Though in the Groenendijk, Stokhof, and Veltman homework, we had to figure out how to combine Reader, State, and Set monads in an ad-hoc way. In practice, one often wants to combine the abilities of several monads. Corresponding to each monad like Reader, there's a corresponding ReaderT **monad transformer**. That takes an existing monad M and wraps Readerish monad packaging around it. The way these are defined parallels the way the single-layer versions are defined. For example, here's the Reader monad:
+So far, we've defined monads as single-layered boxes. In practice, one often wants to combine the abilities of several monads. Corresponding to each monad like Reader, there's a corresponding ReaderT **monad transformer**. That takes an existing monad M and wraps Readerish monad packaging around it. The way these are defined parallels the way the single-layer versions are defined. For example, here's the Reader monad:
(* monadic operations for the Reader monad *)
type 'a reader =
env -> 'a;;
- let unit (a : 'a) : 'a reader =
- fun e -> a;;
- let bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader =
- fun e -> (fun a -> f a e) (u e);;
+ let mid (x : 'a) : 'a reader =
+ fun e -> x;;
+ let mbind (xx: 'a reader) (k : 'a -> 'b reader) : 'b reader =
+ fun e -> (fun x -> k x e) (xx e);;
-We've just beta-expanded the familiar `f (u e) e` into `(fun a -> f a
-e) (u e)`. We did that so as to factor out the parts where any Reader monad is
-being supplied as an argument to another function. That will help make some patterns that are coming up more salient.
+We've just beta-expanded the familiar `k (xx e) e` into `(fun x -> k x e) (xx e)`. We did that so as to factor out the parts where the payload of any Reader value is being supplied as an argument to another function. That will help make some patterns that are coming up more salient.
That was the plain Reader monad. Now if we want instead to wrap some other monad M inside Readerish packaging. How could we do it?
-Well, one way to proceed would be to just let values of the other monad M be the `'a` in your `'a reader`. Then you could apply `reader_bind` to get at the wrapped `'a M`, and then apply `M.bind` to get at *its* wrapped `'a`. This sometimes works. It's what we did in the hints to GSV assignment, where as we said, we "combined State and Set in an ad hoc way."
+Well, one way to proceed would be to just let values of the other monad M be the `'a` in your `'a reader`. Then you could apply `reader_mbind` to get at the wrapped `'a M`, and then apply `M.mbind` to get at *its* wrapped `'a`. This sometimes works.
-But there are two problems: (1) It's cumbersome having to work with *both* `reader_bind` and `M.bind`. It'd be nice to figure out some systematic way to connect the plumbing of the different monadic layers, so that we could have a *single* `bind` that took our `'a M_inside_Reader`, and sequenced it with a single `'a -> 'b M_inside_Reader` function. Similarly for `unit`. This is what the ReaderT monad transformer will let us do.
+But there are two problems: (1) It's cumbersome having to work with *both* `reader_mbind` and `M.mbind`. It'd be nice to figure out some systematic way to connect the plumbing of the different monadic layers, so that we could have a *single* `mbind` that took our `'a Reader_around_M`, and sequenced it with a single `'a -> 'b Reader_around_M` function. Similarly for `mid`. This is what the ReaderT monad transformer will let us do.
(2) For some combinations of monads, the best way to implement a Tish monadic wrapper around an inner M monad won't be equivalent to either an `('a m) t` or an `('a t) m`. It will be a tighter intermingling of the two. So some natural activities will remain out of reach until we equip ourselves to go beyond `('a m) t`s and so on.
-What we want in general are monadic transformers. For example, a ReaderT transformer will be parameterized not just on the type of its innermost contents `'a`, but also on the monadic box `M` that wraps `'a`. It will make use of monad `M`'s existing operations `M.unit` and `M.bind`, together with the Reader patterns for unit and bind, to define a new monad ReaderT(M) that fuses the behavior of Reader and M.
+What we want in general are monadic transformers. For example, a ReaderT transformer will be parameterized not just on the type of its innermost payload `'a`, but also on the monadic box type `M` that wraps `'a`. It will make use of monad `M`'s existing operations `M.mid` and `M.mbind`, together with the Reader patterns for `mid` and `mbind`, to define a new monad ReaderT(M) that fuses the behavior of Reader and M.
To be clear: ReaderT isn't itself a monad. It's a kind of Readerish packaging (wrapping paper) that converts one monadic box M into another monadic box ReaderT(M).
(* We're not giving valid OCaml code, but rather something
* that's conceptually easier to digest.
- * How you really need to write this in OCaml is more circuitous...
- * see http://lambda.jimpryor.net/code/tree_monadize.ml
- * and http://lambda.jimpryor.net/code/monads.ml
- * for some details. *)
+ * How you really need to write this in OCaml is more circuitous... *)
type 'a readerT(M) =
env -> 'a M;;
(* this _happens_ also to be the type of an ('a M) reader
* but as we noted, you can't rely on that pattern always to hold *)
- let unit (a : 'a) : 'a readerT(M) =
- fun e -> M.unit a;;
+ let mid (x : 'a) : 'a readerT(M) =
+ fun e -> M.mid x;;
- let bind (u : 'a readerT(M)) (f : 'a -> 'b readerT(M)) : 'b readerT(M) =
- fun e -> M.bind (u e) (fun a -> f a e);;
+ let mbind (xx : 'a readerT(M)) (k : 'a -> 'b readerT(M)) : 'b readerT(M) =
+ fun e -> M.mbind (xx e) (fun x -> k x e);;
-Notice the key differences: where before `unit` was implemented by a function that just returned `a`, now we
-instead return `M.unit a`. Where before `bind` just supplied value `u e`
-of type `'a reader` as an argument to a function, now we instead
-`M.bind` the corresponding value to the function. Notice also the differences
+Notice the key differences: where before `mid x` was implemented by a function that just returned `x`, now we
+instead return `M.mid x`. Where before `mbind` just supplied the payload `xx e`
+of an `'a reader` as an argument to a function, now we instead
+`M.mbind` the corresponding value to the function. Notice also the differences
in the types.
What is the relation between Reader and ReaderT? Well, suppose you started with the Identity monad:
type 'a identity = 'a;;
- let unit (a : 'a) : 'a = a;;
- let bind (u : 'a) (f : 'a -> 'b) : 'b = f u;;
+ let mid (x : 'a) : 'a = x;;
+ let mbind (xx : 'a) (k : 'a -> 'b) : 'b = k xx;;
and you used the ReaderT transformer to wrap the Identity monad inside Readerish packaging. What do you suppose you would get?
type 'a state =
store -> ('a * store);;
- let unit (a : 'a) : 'a state =
- fun s -> (a, s);;
+ let mid (x : 'a) : 'a state =
+ fun s -> (x, s);;
- let bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
- fun s -> (fun (a, s') -> f a s') (u s);;
+ let mbind (xx : 'a state) (k : 'a -> 'b state) : 'b state =
+ fun s -> (fun (x, s') -> k x s') (xx s);;
-We've used `(fun (a, s') -> f a s') (u s)` instead of the more familiar `let (a, s') = u s in f a s'` in order to factor out the part where a value of type `'a state` is supplied as an argument to a function. Now StateT will be:
+We've used `(fun (x, s') -> k x s') (xx s)` instead of the more familiar `let (x, s') = xx s in k x s'` in order to factor out the part where the payload of an `'a state` value is supplied as an argument to a function. Now StateT will be:
(* monadic operations for the StateT monadic transformer *)
store -> ('a * store) M;;
(* notice this is not an ('a M) state *)
- let unit (a : 'a) : 'a stateT(M) =
- fun s -> M.unit (a, s);;
+ let mid (x : 'a) : 'a stateT(M) =
+ fun s -> M.mid (x, s);;
- let bind (u : 'a stateT(M)) (f : 'a -> 'b stateT(M)) : 'b stateT(M) =
- fun s -> M.bind (u s) (fun (a, s') -> f a s');;
+ let mbind (xx : 'a stateT(M)) (k : 'a -> 'b stateT(M)) : 'b stateT(M) =
+ fun s -> M.mbind (xx s) (fun (x, s') -> k x s');;
-Do you see the pattern? Where before `unit` was implemented by a function that returned an `'a * store` value, now we instead use `M.unit` to return an `('a * store) M` value. Where before `bind` supplied an `'a state` value `(u s)` as an argument to a function, now we instead `M.bind` it to that function.
+Do you see the pattern? Where before `mid` was implemented by a function that returned an `'a * store` value, now we instead use `M.mid` to return an `('a * store) M` value. Where before `mbind` supplied an `'a state` payload `(xx s)` as an argument to a function, now we instead `M.mbind` it to that function.
Once again, what do you think you'd get if you wrapped StateT monadic packaging around an Identity monad?
-We spell out all the common monads, their common dedicated operations (such as `lookup`- and `shift`-like operations for the Reader monad), and monad transformer cousins of all of these, in an OCaml [[monad library]]. Read the linked page for details about how to use the library, and some design choices we made. Our [[State Monad Tutorial]] gives some more examples of using the library.
+We spell out all the common monads, their common dedicated operations (such as `asks`- and `shift`-like operations for the Reader monad), and monad transformer cousins of all of these, in an OCaml [[monad library]]. Read the linked page for details about how to use the library, and some design choices we made. Our [[State Monad Tutorial]] gives some more examples of using the library. LINK TODO
-When a T monadic layer encloses an inner M monad, the T's interface is the most exposed one. To use operations defined in the inner M monad, you'll have to "elevate" them into the outer T packaging. Haskell calls this operation `lift`, but we call it `elevate` because the term "lift" is already now too overloaded. In our usage, `lift` (and `lift2`) are functions that bring non-monadic operations into a monad; `elevate` brings monadic operations from a wrapped monad out into the wrapping.
+When a T monadic layer encloses an inner M monad, the T's interface is the most exposed one. To use operations defined in the inner M monad, you'll have to `hoist` them into the outer T packaging. Haskell calls this operation `lift`, but we call it `hoist` because the term "lift" is already now too overloaded. The `hoist` operation brings monadic values or functions from a wrapped monad out into the wrapping.
Here's an example. Suppose `S` is an instance of a State monad:
- # #use "path/to/monads.ml";;
- # module S = State_monad(struct type store = int end);;
+ # module S = Monad.State(struct type store = int end);;
+
+and `OS` is a OptionT wrapped around `S`:
-and `MS` is a MaybeT wrapped around `S`:
+ # module OS = Monad.Option.T(S);;
- # module MS = Maybe_monad.T(S);;
+Then if you want to use an `S`-specific monad like `modify succ` inside `OS`, you'll have to use `OS`'s `hoist` function, like this:
-Then if you want to use an `S`-specific monad like `puts succ` inside `MS`, you'll have to use `MS`'s `elevate` function, like this:
+ # OS.(...hoist (S.modify succ) ...)
- # MS.(...elevate (S.puts succ) ...)
+Each monad transformer's `hoist` function will be defined differently. They have to obey the following laws:
-Each monad transformer's `elevate` function will be defined differently. They have to obey the following laws:
+* `Outer.hoist (Inner.mid a) <~~> Outer.mid a`
+* `Outer.hoist (Inner.mbind xx k) <~~> Outer.mbind (Outer.hoist xx) (fun x -> Outer.hoist (k x))`
-* `Outer.elevate (Inner.unit a) <~~> Outer.unit a`
-* `Outer.elevate (Inner.bind u f) <~~> Outer.bind (Outer.elevate u) (fun a -> Outer.elevate (f a))`
+We said that when T encloses M, you can rely on T's interface to be most exposed. That is intuitive. What you cannot also assume is that the implementing type has a Tish structure surrounding an Mish structure. Often it will be reverse: a ListT(Option) is implemented by a `'a list option`, not by an `'a option list`. Until you've tried to write the code to a monadic transformer library yourself, this will probably remain counter-intuitive. But you don't need to concern yourself with it in practice. Think of what you have as a ListT(Option); don't worry about whether the underlying implementation is as an `'a list option` or an `'a option list` or as something more complicated.
-We said that when T encloses M, you can rely on T's interface to be most exposed. That is intuitive. What you cannot also assume is that the implementing type has a Tish structure surrounding an Mish structure. Often it will be reverse: a ListT(Maybe) is implemented by a `'a list option`, not by an `'a option list`. Until you've tried to write the code to a monadic transformer library yourself, this will probably remain counter-intuitive. But you don't need to concern yourself with it in practise. Think of what you have as a ListT(Maybe); don't worry about whether the underlying implementation is as an `'a list option` or an `'a option list` or as something more complicated.
+(Except for the homework. There we do prompt you to experiment and figure some of these out. But you won't remember them.)
Notice from the code for StateT, above, that an `'a stateT(M)` is not an `('a M) state`; neither is it an `('a state) M`. The pattern by which we transform the types from a Blah monad to a BlahT monad transformer is:
As best we know, figuring out how a monad transformer should be defined is still something of an art, not something that can be done mechanically. However, you can think that all of the art goes into deciding what StateT and so on should be; having figured that out, plain State would follow as the simple case where StateT is parameterized on the Identity monad.
-Apart from whose interface is outermost, the behavior of a StateT(Maybe) and a MaybeT(State) will partly coincide. But in certain crucial respects they will diverge, and you need to think carefully about which behavior you want and what the appropriate layering is for your needs. Consider these examples:
+Apart from whose interface is outermost, the behavior of a StateT(Option) and a OptionT(State) will partly coincide. But in certain crucial respects they will diverge, and you need to think carefully about which behavior you want and what the appropriate layering is for your needs. Consider these examples:
- # module MS = Maybe_monad.T(S);;
- # module SM = S.T(Maybe_monad);;
- # MS.(run (elevate (S.puts succ) >> zero () >> elevate S.get >>= fun cur -> unit (cur+10) )) 0;;
+ # module OS = Monad.Option.T(S);;
+ # module SO = S.Z(Monad.Option);; (* we use S.Z instead of S.T because Monad.Option has a mzero *)
+ # OS.(run (hoist (S.modify succ) >> mzero >> hoist S.get >>= fun cur -> mid (cur+10) )) 0;;
- : int option * S.store = (None, 1)
- # MS.(run (elevate (S.puts succ) >> zero () >> elevate (S.put 5) )) 0;;
+ # OS.(run (hoist (S.modify succ) >> mzero >> hoist (S.put 5) )) 0;;
- : unit option * S.store = (None, 1)
-Although we have a wrapped `None`, notice that the store (as it was at the point of failure) is still retrievable.
+Although we have a wrapped `None`, notice that the store (as it was at the point of failure) was still retrievable.
- # SM.(run (puts succ >> elevate (Maybe_monad.zero ()) >> get >>= fun cur -> unit (cur+10) )) 0;;
- - : ('a, int * S.store) Maybe_monad.result = None
+ # SO.(run (modify succ >> mzero >> get >>= fun cur -> mid (cur+10) )) 0;;
+ - : (int * S.store) Monad.Option.result = None
-When Maybe is on the inside, on the other hand, a failure means the whole computation has failed, and even the store is no longer available.
+When Option is on the inside, on the other hand, as in SO, failure means the whole computation has failed, and even the store is no longer available.
<!--
- # ES.(run( elevate (S.puts succ) >> throw "bye" >> elevate S.get >>= fun i -> unit(i+10) )) 0;;
+ # ES.(run(hoist (S.modify succ) >> throw "bye" >> hoist S.get >>= fun i -> mid (i+10) )) 0;;
- : int Failure.error * S.store = (Failure.Error "bye", 1)
- # SE.(run( puts succ >> elevate (Failure.throw "bye") >> get >>= fun i -> unit(i+10) )) 0;;
+ # SE.(run(modify succ >> hoist (Failure.throw "bye") >> get >>= fun i -> mid (i+10) )) 0;;
- : (int * S.store) Failure.result = Failure.Error "bye"
- # ES.(run_exn( elevate (S.puts succ) >> throw "bye" >> elevate S.get >>= fun i -> unit(i+10) )) 0;;
+ # ES.(run_exn(hoist (S.modify succ) >> throw "bye" >> hoist S.get >>= fun i -> mid (i+10) )) 0;;
Exception: Failure "bye".
- # SE.(run_exn( puts succ >> elevate (Failure.throw "bye") >> get >>= fun i -> unit(i+10) )) 0;;
+ # SE.(run_exn(modify succ >> hoist (Failure.throw "bye") >> get >>= fun i -> mid (i+10) )) 0;;
Exception: Failure "bye".
-->
-Here's an example wrapping Maybe around List, and vice versa:
+<!--
+Here's an example wrapping Option around List, and vice versa:
- # module LM = List_monad.T(Maybe_monad);;
- # module ML = Maybe_monad.T(List_monad);;
- # ML.(run (plus (zero ()) (unit 20) >>= fun i -> unit (i+10)));;
- - : ('_a, int) ML.result = [Some 30]
+ # module LO = Monad.List.T(Monad.Option);;
+ # module OL = Monad.Option.T(Monad.List);;
+ # let plus = ???;;
+ # OL.(run (plus mzero mid 20 >>= fun i -> mid (i+10)));;
+ - : int OL.result = [Some 30]
-When List is on the inside, the failed results just get dropped and the computation proceeds without them.
+When List is on the inside, the failed results just got dropped and the computation proceeds without them.
- # LM.(run (plus (elevate (Maybe_monad.zero ())) (unit 20) >>= fun i -> unit (i+10)));;
- - : ('_a, int) LM.result = None
+ # LO.(run ((++) (hoist Monad.Option.mzero) (mid 20) >>= fun i -> mid (i+10)));;
+ - : int LO.result = None
-On the other hand, when Maybe is on the inside, failures abort the whole computation.
+On the other hand, when Option is on the inside, as in LO, failures (which we represent by `mzero`s from the Option monad, here `hoist Monad.Option.mzero`, not the List monad's own `mzero`) abort the whole computation.
+-->
<!--
- # EL.(run( plus (throw "bye") (unit 20) >>= fun i -> unit(i+10)));;
+ # EL.(run((++) (throw "bye") (mid 20) >>= fun i -> mid (i+10)));;
- : int EL.result = [Failure.Error "bye"; Failure.Success 30]
- # LE.(run( plus (elevate (Failure.throw "bye")) (unit 20) >>= fun i -> unit(i+10)));;
+ # LE.(run((++) (hoist (Failure.throw "bye")) (mid 20) >>= fun i -> mid (i+10)));;
- : int LE.result = Failure.Error "bye"
- # EL.(run_exn( plus (throw "bye") (unit 20) >>= fun i -> unit(i+10)));;
+ # EL.(run_exn((++) (throw "bye") (mid 20) >>= fun i -> mid (i+10)));;
Exception: Failure "bye".
- # LE.(run_exn( plus (elevate (Failure.throw "bye")) (unit 20) >>= fun i -> unit(i+10)));;
+ # LE.(run_exn((++) (hoist (Failure.throw "bye")) (mid 20) >>= fun i -> mid (i+10)));;
Exception: Failure "bye".
-->
-This is fun. Notice the difference it makes whether the second `plus` is native to the outer `List_monad`, or whether it's the inner `List_monad`'s `plus` elevated into the outer wrapper:
+This is fun. Notice the difference it makes whether the second `++` is native to the outer `Monad.List`, or whether it's the inner `Monad.List`'s `++` hoisted into the outer wrapper:
- # module LL = List_monad.T(List_monad);;
+ # module LL = Monad.List.T(Monad.List);;
- # LL.(run(plus (unit 1) (unit 2) >>= fun i -> plus (unit i) (unit(10*i)) ));;
- - : ('_a, int) LL.result = \[[1; 10; 2; 20]]
- # LL.(run(plus (unit 1) (unit 2) >>= fun i -> elevate L.(plus (unit i) (unit(10*i)) )));;
- - : ('_a, int) LL.result = [[1; 2]; [1; 20]; [10; 2]; [10; 20]]
+ # LL.(run((++) (mid 1) (mid 2) >>= fun i -> (++) (mid i) (mid (10*i)) ));;
+ - : int LL.result = \[[1; 10; 2; 20]]
+ # LL.(run((++) (mid 1) (mid 2) >>= fun i -> hoist L.((++) (mid i) (mid (10*i)) )));;
+ - : int LL.result = [[1; 2]; [1; 20]; [10; 2]; [10; 20]]
* Read Part III of [All About Monads](http://web.archive.org/web/20071106232016/haskell.org/all_about_monads/html/introIII.html). This link is to an archived version, the main link to haskell.org seems to be broken. Some but not all of this site has been [absorbed into the Haskell wikibook](http://en.wikibooks.org/wiki/Haskell/Monad_transformers).
-
-Tree Monads
-===========
-
-Our [[monad library]] includes a `Tree_monad`, for binary, leaf-labeled trees. There are other kinds of trees you might want to monadize, but we took the name `Tree_monad` for this one. Like the Haskell [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) monad, our `Tree_monad` also incorporates an Optionish layer. (See the comments in our library code about `plus` and `zero` for discussion of why.)
-
-So how does our `Tree_monad` behave? Simplified, its implementation looks something like this:
-
- (* monadic operations for the Tree monad *)
-
- type 'a tree =
- Leaf of 'a | Node of ('a tree) * ('a tree);;
-
- let unit (a: 'a) : 'a tree =
- Leaf a;;
-
- let rec bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
- match u with
- | Leaf a -> f a;;
- | Node (l, r) ->
- let l' = bind l f in
- let r' = bind r f in
- Node (l', r')
-
-Recall how `bind` works for the List monad. If you have a list:
-
- let u = [1; 2; 4; 8];;
-
-and a function `f` such that:
-
- f 1 ~~> []
- f 2 ~~> [2]
- f 4 ~~> [2; 4]
- f 8 ~~> [2; 4; 8]
-
-then `list_bind u f` would be `concat [[]; [2]; [2; 4]; [2; 4; 8]]`, that is `[2; 2; 4; 2; 4; 8]`. It splices the lists returned by `f` into the corresponding positions in the original list structure. The `tree_bind` operation works the same way. If `f'` maps `2` to the tree `Leaf 2` and `8` to the tree `Node (Leaf 2, Node (Leaf 4, Leaf 8))`, then binding the tree `u` to `f'` will splice the trees returned by `f'` to the corresponding positions in the original structure:
-
- u
- . .
- _|__ >>= f' ~~> _|__
- | | | |
- 2 8 2 .
- _|__
- | |
- 2 .
- _|__
- | |
- 4 8
-
-Except, as we mentioned, our implementation of the Tree monad incorporates an Optionish layer too. So `f' 2` should be not `Leaf 2` but `Some (Leaf 2)`. What if `f'` also mapped `1` to `None` and `4` to `Some (Node (Leaf 2, Leaf 4))`. Then binding the tree `Node (Leaf 1, Node (Leaf 2, Leaf 4))` (really the tree itself needs to be wrapped in a `Some`, too, but let me neglect that) to `f'` would delete the branch corresponding to the original `Leaf 1`, and would splice in the results for `f' 2` and `f' 4`, yielding:
-
- .
- _|__ >>= f' ~~>
- | |
- 1 . .
- _|__ _|__
- | | | |
- 2 4 2 .
- _|__
- | |
- 2 4
-
-As always, the functions you bind an `'a tree` to need not map `'a`s to `'a tree`s; they can map them to `'b tree`s instead. For instance, we could transform `Node (Leaf 1, Node (Leaf 2, Leaf 4))` instead into `Node (Leaf "two", Node (Leaf "two", Leaf "four"))`.
-
-As we [mention in the notes](/monad_library), our monad library encapsulates the implementation of its monadic types. So to work with it you have to use the primitives it provides. You can't say:
-
- # Tree_monad.(orig_tree >>= fun a -> match a with
- | 4 -> Some (Node (Leaf 2, Leaf 4))
- | _ -> None);;
- Error: This expression has type int Tree_monad.tree option
- but an expression was expected of type ('a, 'b) Tree_monad.m
-
-You have to instead say something like this:
-
- # Tree_monad.(orig_tree >>= fun a -> match a with
- | 4 -> plus (unit 2) (unit 4)
- | _ -> zero () );;
- - : ('_a, int) Tree_monad.m = <abstr>
-
-
-
-How is all this related to our tree\_monadize function?
--------------------------------------------------------
-
-Our Tree monad has a corresponding TreeT transformer. Simplified, its implementation looks something like this (we apply it to an inner Reader monad):
-
-
- type 'a tree_reader = 'a tree reader;;
- (* really it's an 'a tree option reader, but as I said we're simplifying *)
-
- let tree_reader_unit (a:'a) : 'a tree_reader = reader_unit (Leaf a);;
-
- let tree_reader_bind (u: 'a tree_reader) (f: 'a -> 'b tree_reader) : 'b tree_reader =
- reader_bind u (fun us ->
- let rec loop us = match us with
- | Leaf a ->
- f a
- | Node(l,r) ->
- reader_bind (loop l) (fun ls ->
- reader_bind (loop r) (fun rs ->
- reader_unit (Node(ls, rs))))
- in loop us);;
-
- let tree_reader_elevate (inner : 'a reader) : 'a tree_reader =
- reader_bind inner (fun a -> reader_unit (Leaf a))
-
-Recall our earlier definition of `tree_monadize`, specialized for the Reader monad:
-
- let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
- match t with
- | Leaf a ->
- (* the next line is equivalent to: tree_reader_elevate (f a) *)
- reader_bind (f a) (fun b -> reader_unit (Leaf b))
- | Node (l, r) ->
- reader_bind (tree_monadize f l) (fun ls ->
- reader_bind (tree_monadize f r) (fun rs ->
- reader_unit (Node (ls, rs))));;
-
-We rendered the result type here as `'b tree reader`, as we did in our earlier discussion, but as we can see from the above implementation of TreeT(Reader), that's the type of an `'b tree_reader`, that is, of a layered box consisting of TreeT packaging wrapped around an inner Reader box.
-
-The definitions of `tree_monadize` and `tree_reader_bind` should look very similar. They're not quite the same. There's the difference in the order of their function-like and tree-like arguments, but that's inconsequential. More important is that the types of their arguments differs. `tree_reader_bind` wants a tree that's already fused with a reader; `tree_monadize` instead just wants a plain tree. `tree_reader_bind` wants a function that takes the elements occupying its leaves into other `tree_reader`s; `tree_monadize` just wants it to take them into plain `reader`s. That's why the application of `f` to `a` has to be `elevate`d in the `tree_monadize` clause for `Leaf a -> ...`.
-
-But there is an obvious common structure to these two functions, and indeed in the [[monad library]] their more complicated cousins are defined in terms of common pieces. In the monad library, the `tree_monadize` function is called `distribute`; this is an operation living inside the TreeT packaging. There's an analogous `distribute` function living inside the ListT packaging. (Haskell has the second but not the first; it calls it `mapM` and it lives inside the wrapped base monad, instead of the List packaging.)
-
-We linked to [some code](/code/tree_monadize.ml) earlier that demonstrated all the `tree_monadize` examples in a compact way.
-
-Here's how to demonstrate the same examples, using the monad library. First, preliminaries:
-
- # #use "path/to/monads.ml";;
- # module T = Tree_monad;;
- # module R = Reader_monad(struct type env = int -> int end);;
- # module S = State_monad(struct type store = int end);;
- # module L = List_monad;;
- # module C = Continuation_monad;;
- # module TR = T.T(R);;
- # module TS = T.T(S);;
- # module TL = T.T(L);;
- # module TC = T.T(C);;
- # let t1 = Some (T.Node (T.Node (T.Leaf 2, T.Leaf 3), T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11))));;
-
-We can use TreeT(Reader) to modify leaves:
-
- # let tree_reader = TR.distribute (fun i -> R.asks (fun e -> e i)) t1;;
- # TR.run tree_reader (fun i -> i+i);;
- - : int T.tree option =
- Some
- (T.Node
- (T.Node (T.Leaf 4, T.Leaf 6),
- T.Node (T.Leaf 10, T.Node (T.Leaf 14, T.Leaf 22))))
-
-Here's a comparison of how distribute works for trees and how it works for lists:
-
- # module LR = L.T(R);;
- # let l1 = [2; 3; 5; 7; 11];;
- # LR.(run (distribute (fun i -> R.(asks (fun e -> e i))) l1)) (fun i -> i+i);;
- - : int list = [4; 6; 10; 14; 22]
-
-<!--
-More complex: here we use the monadic `list_reader` or `tree_reader` we got back from `distribute` and `bind` it to other operations:
-
- # let u = LR.distribute (fun i -> R.(asks (fun e -> e i))) l1 in
- LR.(run(u >>= fun i -> plus (unit i) (unit (10*i)))) (fun i -> i + i);;
- - : int list = [4; 40; 6; 60; 10; 100; 14; 140; 22; 220]
- # let v = TR.distribute (fun i -> R.(asks (fun e -> e i))) t1 in
- TR.(run(v >>= fun i -> plus (unit i) (unit (10*i)))) (fun i -> i + i);;
- - : int T.tree option =
- Some
- (T.Node
- (T.Node (T.Node (T.Leaf 4, T.Leaf 40), T.Node (T.Leaf 6, T.Leaf 60)),
- T.Node
- (T.Node (T.Leaf 10, T.Leaf 100),
- T.Node (T.Node (T.Leaf 14, T.Leaf 140), T.Node (T.Leaf 22, T.Leaf 220)))))
--->
-
-We can use TreeT(State) to count leaves:
-
- # let tree_counter = TS.distribute (fun i -> S.(puts succ >> unit i)) t1 in
- TS.run tree_counter 0;;
- - : int T.tree option * S.store =
- (Some
- (T.Node
- (T.Node (T.Leaf 2, T.Leaf 3),
- T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11)))),
- 5)
-
-or to annotate leaves:
-
- # let tree_annotater = TS.distribute (fun i -> S.(puts succ >> get >>= fun s -> unit (i,s))) t1 in
- TS.run tree_annotater 0;;
- - : (int * S.store) T.tree option * S.store =
- (Some
- (T.Node
- (T.Node (T.Leaf (2, 1), T.Leaf (3, 2)),
- T.Node (T.Leaf (5, 3), T.Node (T.Leaf (7, 4), T.Leaf (11, 5))))),
- 5)
-
-Here's a comparison of how distribute works for trees and how it works for lists:
-
- # module LS = L.T(S);;
-
- # let list_counter = LS.distribute (fun i -> S.(puts succ >> unit i)) l1 in
- LS.run list_counter 0;;
- - : int list * S.store = ([2; 3; 5; 7; 11], 5)
-
- # let list_annotater = LS.distribute (fun i -> S.(puts succ >> get >>= fun s -> unit (i,s) )) l1 in
- LS.run list_annotater 0;;
- - : (int * S.store) list * S.store =
- ([(2, 1); (3, 2); (5, 3); (7, 4); (11, 5)], 5)
-
-
-<!--
-# let u = LS.distribute (fun i -> if i = -1 then S.get else if i < 0 then S.(puts succ >> unit 0) else S.unit i) [10;-1;-2;-1;20] in
- LS.run u 0;;
-- : S.store list * S.store = ([10; 0; 0; 1; 20], 1)
--->
-
-
-We can use TreeT(List) to copy the tree with different choices for some of the leaves:
-
- # let tree_chooser = TL.distribute (fun i -> L.(if i = 2 then plus (unit 20) (unit 21) else unit i)) t1;;
- # TL.run tree_chooser;;
- - : ('_a, int) TL.result =
- [Some
- (T.Node
- (T.Node (T.Leaf 20, T.Leaf 3),
- T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11))));
- Some
- (T.Node
- (T.Node (T.Leaf 21, T.Leaf 3),
- T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11))))]
-
-
-Finally, we use TreeT(Continuation) to do various things. For reasons I won't explain here, the library currently requires you to run the Tree-plus-Continuation bundle using a different sequence of `run` commands:
-
-We can do nothing:
-<!--
-let initial_continuation = fun t -> t in
-TreeCont.monadize Continuation_monad.unit t1 initial_continuation;;
--->
-
- # C.run_exn TC.(run (distribute C.unit t1)) (fun t -> t);;
- - : int T.tree option =
- Some
- (T.Node
- (T.Node (T.Leaf 2, T.Leaf 3),
- T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11))))
-
-We can square each leaf:
-<!--
-let initial_continuation = fun t -> t in
-TreeCont.monadize (fun a k -> k (a*a)) t1 initial_continuation;;
--->
-
- # C.run_exn TC.(run (distribute C.(fun a -> shift (fun k -> k (a*a))) t1)) (fun t -> t);;
- - : int T.tree option =
- Some
- (T.Node
- (T.Node (T.Leaf 4, T.Leaf 9),
- T.Node (T.Leaf 25, T.Node (T.Leaf 49, T.Leaf 121))))
-
-The meaning of `shift` will be explained in [[CPS and Continuation Operators]]. Here you should just regard it as a primitive operation in our Continuation monad. In [this code](/code/tree_monadize.ml) you could simply write:
-
- TreeCont.monadize (fun a -> fun k -> k (a*a)) t1 (fun t -> t);;
-
-But because of the way our monad library hides the underlying machinery, here you can no longer just say `fun k -> k (a*a)`; you have to say `shift (fun k -> k (a*a))`.
-
-Moving on, we can count the leaves:
-<!--
-let initial_continuation = fun t -> 0 in
-TreeCont.monadize (fun a k -> 1 + k a) t1 initial_continuation;;
--->
-
- # C.run_exn TC.(run (distribute C.(fun a -> shift (fun k -> k a >>= fun v -> unit (1+v))) t1)) (fun t -> 0);;
- - : int = 5
-
-
-And we can convert the tree to a list of leaves:
-<!--
-let initial_continuation = fun t -> [] in
-TreeCont.monadize (fun a k -> a :: k a) t1 initial_continuation;;
--->
-
- # C.run_exn TC.(run (distribute C.(fun a -> shift (fun k -> k a >>= fun v -> unit (a::v))) t1)) (fun t -> []);;
- - : int list = [2; 3; 5; 7; 11]
-
-
--- /dev/null
+[[!toc]]
+
+The seminar is now going to begin talking about more **imperatival** or **effect**-like elements in programming languages. The only effect-like element we've encountered so far is the possibility of divergence, in languages that permit fixed point combinators and so have the full power of recursion. What it means for something to be effect-like, and why this counts as an example of such, will emerge.
+
+Other effect-like elements in a language include: printing; continuations (which we'll study more in the coming weeks) and exceptions (like OCaml's `failwith "message"` or `raise Not_found`); and **mutation**. This last notion is our topic this week.
+
+
+## Mutation##
+
+What is mutation? It's helpful to build up to this in a series of fragments. For present pedagogical purposes, we'll be using a made-up language that's syntactically similar to, but not quite the same as, OCaml. (It's not quite Kapulet either.)
+
+This should seem entirely familiar:
+
+ [A] let y = 1 + 2 in
+ let x = 10 in
+ (x + y, 20 + y)
+ ; evaluates to (13, 23)
+
+In our next fragment, we re-use a variable that had been bound to another value in a wider context:
+
+ [B] let y = 2 in ; will be shadowed by the binding on the next line
+ let y = 3 in
+ (10 + y, 20 + y)
+ ; evaluates to (13, 23)
+
+As you can see, the narrowest assignment is what's effective. This is just like in predicate logic: consider <code>∃y (Fy and ∃y ~Fy)</code>. The computer-science terminology to describe this is that the narrower assignment of `y` to the value 3 **shadows** the wider assignment to 2.
+
+I call attention to this because you might casually describe it as "changing the value that y is assigned to." But what we'll see below is a more exotic phenomenon that merits that description better.
+
+In the previous fragments, we bound the variables `x` and `y` to `int`s. We can also bind variables to function values, as here:
+
+ [C] let f = (\x y. x + y + 1) in
+ (f 10 2, f 20 2)
+ ; evaluates to (13, 23)
+
+If the expression that evaluates to a function value has a free variable in it, like `y` in the next fragment, it's interpreted as bound to whatever value `y` has in that expression's lexical context:
+
+ [D] let y = 3 in
+ let f = (\x. x + y) in
+ let y = 2 in
+ (f 10, y, f 20)
+ ; evaluates to (13, 2, 23)
+
+Other choices about how to interpret free variables are also possible (you can read about "lexical scope" versus "dynamic scope"), but what we do here is the contemporary norm in functional programming languages, and seems to be easiest for programmers to reason about.
+
+Sometimes bindings are shadowed merely in a temporary, local context, as here:
+
+ [E] let y = 3 in
+ let f = (\x. let y = 2 in
+ ; here the most local binding for y applies
+ x + y) in
+ ; here the binding of y to 2 has expired
+ (y, f 10, y, f 20)
+ ; evaluates to (3, 12, 3, 22)
+
+Notice that the `y`s in the tuple at the end use the outermost binding of `y` to `3`, but the `y` in `x + y` in the body of the `f` function uses the more local binding.
+
+OK, now we're ready for our main event, **mutable variables.** We'll introduce new syntax to express an operation where we're not merely *shadowing* a wider binding, but *changing* or *mutating* that binding. The new syntax will show up both when we introduce the variable, using `var y = ...` rather than `let y = ...`; and also when we change `y`'s value using `set`.
+
+ [F] var y = 3 in
+ let f = (\x. set y to 2 then
+ x + y) in
+ ; here the change in what value y is bound to *sticks*
+ ; because we *mutated* the value of the *original* variable y
+ ; instead of introducing a new y with a narrower scope
+ (y, f 10, y, f 20)
+ ; evaluates to (3, 12, 2, 22)
+
+Notice the difference in the how the second `y` is evaluated in the tuple at the end. By the way, I am assuming here that the tuple gets evaluated left-to-right. Other languages may or may not conform to that. OCaml doesn't always.
+
+In languages that have native syntax for mutation, there are two styles in which it can be expressed. The *implicit style* is exemplified in fragment [F] above, and also in languages like C:
+
+ {
+ int y = 3; // this is like "var y = 3 in ..."
+ ...
+ y = 2; // this is like "set y to 2 then ..."
+ return x + y; // this is like "x + y"
+ }
+
+A different possibility is the *explicit style* for handling mutation. Here we explicitly create and refer to new "reference cells" to hold our values. When we mutate a variable's value, we leave the variable assigned to the same reference cell, but we modify that reference cell's contents. The same thing happens in the semantic machinery underlying implicit-style mutable variables, but there it's implicit --- the reference cells aren't themselves expressed by any term in the object language. In explicit-style mutation, they are. OCaml has explicit-style mutation. It looks like this:
+
+ let ycell = ref 3 (* this creates a new reference cell *)
+ ... in
+ let () = ycell := 2 in (* this changes the contents of that cell to 2 *)
+ (* the return value of doing so is () *)
+ (* other return values could also be reasonable: *)
+ (* such as the old value of ycell, the new value, an arbitrary int, and so on *)
+ x + !ycell (* the !ycell operation "dereferences" the cell---it retrieves the value it contains *)
+
+Scheme is similar. There are various sorts of reference cells available in Scheme. The one most like OCaml's `ref` is a `box`. Here's how we'd write the same fragment in Scheme:
+
+ (let ([ycell (box 3)])
+ ...
+ (set-box! ycell 2)
+ (+ x (unbox ycell)))
+
+C has explicit-style mutable variables, too, which it calls *pointers*. But simple variables in C are already mutable, in the implicit style. Scheme also has both styles of mutation. In addition to the explicit boxes, Scheme also lets you mutate unboxed variables:
+
+ (begin
+ (define y 3)
+ (set! y 2)
+ y)
+ ; evaluates to 2
+
+When dealing with explicit-style mutation, there's a difference between the types and values of `ycell` and `!ycell` (or in Scheme, `(unbox ycell)`). The former has the type `int ref`: the variable `ycell` is assigned a reference cell that contains an `int`. The latter has the type `int`, and has whatever value is now stored in the relevant reference cell. In an implicit-style framework though, we only have the resources to refer to the contents of the relevant reference cell. The variables `y` in fragment [F] or in the C snippet above have the type `int`, and only ever evaluate to `int` values.
+
+
+##Controlling order##
+
+When we're dealing with mutable variables (or any other kind of effect), order matters. For example, it would make a big difference whether I evaluated `let z = !ycell` before or after evaluating `ycell := !ycell + 1`. Before this point, order never mattered except sometimes it played a role in avoiding divergence.
+
+OCaml does *not* guarantee what order expressions will be evaluated in arbitrary contexts. For example, in the following fragment, you cannot rely on `expression_a` being evaluated before `expression_b` before `expression_c`:
+
+ let triple = (expression_a, expression_b, expression_c)
+
+OCaml does however guarantee that different let-expressions are evaluated in the order they lexically appear. So in the following fragment, `expression_a` *will* be evaluated before `expression_b` and that before `expression_c`:
+
+ let a = expression_a in
+ let b = expression_b in
+ expression_c
+
+Scheme does the same. (*If* you use Scheme's `let*`, but not if you use its `let`. I agree this is annoying.)
+
+If `expression_a` and `expression_b` evaluate to (), for instance if they're something like `ycell := !ycell + 1`, that can also be expressed in OCaml as:
+
+ let () = expression_a in
+ let () = expression_b in
+ expression_c
+
+And OCaml has a syntactic shorthand for this form, namely to use semi-colons:
+
+ expression_a; expression_b; expression_c
+
+This is not the same role that semi-colons play in list expressions, like `[1; 2; 3]`. To be parsed correctly, these semi-colon'ed complexes sometimes need to be enclosed in parentheses or a `begin ... end` construction:
+
+ (expression_a; expression_b; expression_c)
+
+ begin expression_a; expression_b; expression_c end
+
+Scheme has a construction similar to the latter:
+
+ (begin (expression_a) (expression_b) (expression_c))
+
+Though often in Scheme, the `(begin ...)` is implicit and doesn't need to be explicitly inserted, as here:
+
+ (lambda (x) (expression_a) (expression_b) (expression_c))
+
+Another way to control evaluation order, you'll recall from previous discussion, is to use **thunks**. These are functions that only take the uninformative `()` as an argument, such as this:
+
+ let f () = ... in
+ ...
+
+or this:
+
+ let f = fun () -> ... in
+ ...
+
+In Scheme these are written as functions that take 0 arguments:
+
+ (let* ([f (lambda () ...)]) ...)
+
+or:
+
+ (define (f) ...)
+ ...
+
+How could such functions be useful? Well, as always, the context in which you build a function need not be the same as the one in which you apply it to arguments. So for example:
+
+ let ycell = ref 1 in
+ let incr_y () = ycell := !ycell + 1 in
+ let y = !ycell in
+ incr_y () in
+ y
+
+We don't apply (or call or execute or however you want to say it) the function `incr_y` until after we've extracted `ycell`'s value and assigned it to `y`. So `y` will get assigned `1`. If on the other hand we called `incr_y ()` before evaluating `let y = !ycell`, then `y` would have gotten assigned a different value.
+
+In languages with mutable variables, the free variables in a function definition are often taken to refer back to the same *reference cells* they had in their lexical contexts, and not just their original value. So if we do this for instance:
+
+ let factory (starting_value : int) =
+ let free_var = ref starting_value in
+ (* `free_var` will be free in the bodies of the next two functions *)
+ let getter () = !free_var in
+ let setter (new_value : int) = free_var := new_value in
+ (* here's what `factory starting_value` returns *)
+ (getter, setter) in
+ let (getter, setter) = factory 1 in
+ let first = getter () in
+ let () = setter 2 in
+ let second = getter () in
+ let () = setter 3 in
+ let third = getter () in
+ (first, second, third)
+
+At the end, we'll get `(1, 2, 3)`. The reference cell that gets updated when we call `setter` is the same one that gets fetched from when we call `getter`. This should seem very intuitive here, since we're working with explicit-style mutation. When working with a language with implicit-style mutation, it can be more surprising. For instance, here's the same fragment in Python, which has implicit-style mutation:
+
+ def factory (starting_value):
+ free_var = starting_value
+ def getter ():
+ return free_var
+ def setter (new_value):
+ # the next line indicates that we're using the
+ # free_var from the surrounding function, not
+ # introducing a new local variable with the same name
+ nonlocal free_var
+ free_var = new_value
+ return getter, setter
+ getter, setter = factory (1)
+ first = getter ()
+ setter (2)
+ second = getter ()
+ setter (3)
+ third = getter ()
+ (first, second, third)
+
+Here, too, just as in the OCaml fragment, all the calls to getter and setter are working with a single mutable variable `free_var`.
+
+If you've got a copy of *The Seasoned Schemer*, which we recommended for the seminar, see the discussion at pp. 91-118 and 127-137.
+
+If however you call the `factory` function twice, even if you supply the same `starting_value`, you'll get independent `getter`/`setter` pairs, each of which have their own, separate `free_var`. In OCaml:
+
+ let factory (starting_val : int) =
+ ... (* as above *) in
+ let (getter, setter) = factory 1 in
+ let (getter', setter') = factory 1 in
+ let () = setter 2 in
+ getter' ()
+
+Here, the call to `setter` only mutated the reference cell associated with the `getter`/`setter` pair. The reference cell associated with `getter'` hasn't changed, and so `getter' ()` will still evaluate to `1`.
+
+Notice in these fragments that once we return from inside the call to `factory`, the `free_var` mutable variable is no longer accessible, except through the helper functions `getter` and `setter` that we've provided. This is another way in which a thunk like `getter` can be useful: it still has access to the `free_var` reference cell that was created when it was, because its free variables are interpreted relative to the context in which `getter` was built, even if that context is otherwise no longer accessible. What `getter ()` evaluates to, however, will very much depend on *when* we evaluate it --- in particular, it will depend on which calls to the corresponding `setter` were evaluated first.
+
+
+##Referential opacity##
+
+In addition to order-sensitivity, when you're dealing with mutable variables you also give up a property that computer scientists call "referential transparency." It's not obvious whether they mean exactly the same by that as philosophers and linguists do, or only something approximately the same.
+
+The core idea to referential transparency is that when the same value is supplied to a context, the whole should always evaluate the same way. Mutation makes it possible to violate this. Consider:
+
+ let ycell = ref 1 in
+ let plus_y x = x + !ycell in
+ let first = plus_y 1 in (* first is assigned the value 2 *)
+ ycell := 2; let second = plus_y 1 in (* second is assigned the value 3 *)
+ first = second (* not true! *)
+
+Notice that the two invocations of `plus_y 1` yield different results, even though the same value is being supplied as an argument to the same function.
+
+Similarly, functions like these:
+
+ let f cell = !cell
+
+ let g cell = cell := !cell + 1; !cell
+
+may return different results each time they're invoked, even if they're always supplied one and the same reference cell as argument.
+
+Computer scientists also associate referential transparency with a kind of substitution principle, illustrated here:
+
+ let x = 1 in
+ (x, x)
+
+should evaluate the same as:
+
+ let x = 1 in
+ (x, 1)
+
+or:
+
+ (1, 1)
+
+Notice, however, that when mutable variables are present, the same substitution patterns can't always be relied on:
+
+ let ycell = ref 1 in
+ ycell := 2; !ycell
+ (* evaluates to 2 *)
+
+ (ref 1) := 2; !(ref 1)
+ (* creates a ref 1 cell and changes its contents *)
+ (* then creates a *new* ref 1 cell and returns *its* contents *)
+ (* so evaluates to 1 *)
+
+
+
+##How to implement explicit-style mutable variables##
+
+We'll think about how to implement explicit-style mutation first. We suppose that we add some new syntactic forms to a language, let's call them `newref`, `getref`, and `putref`. And now we want to expand the semantics for the language so as to interpret these new forms.
+
+Well, part of our semantic machinery will be an assignment function or environment, call it `e`. Perhaps we should keep track of the *types* of the variables and values we're working with, but we won't pay much attention to that now. In fact, we won't even bother much at this point with the assignment function. Below we'll pay more attention to it.
+
+In addition to the assignment function, we'll also need a way to keep track of how many reference cells have been "allocated" (using `newref`), and what their current values are. We'll suppose all the reference cells are organized in a single data structure we might call a table or **store**. This might be a big heap of memory. For our purposes, we'll suppose that reference cells only ever contain `int`s, and we'll let the store be a list of `int`s.
+
+We won't suppose that the metalanguage we use to express the semantics of our mutation-language itself has any mutation facilities. Instead, we'll think about how to model mutation in a wholly declarative or functional or *static* metalanguage.
+
+In many languages, including OCaml, the first position in a list is indexed `0`, the second is indexed `1` and so on. If a list has length `2`, then there won't be any value at index `2`; that will be the "next free location" in the list.
+
+Before we brought mutation on the scene, our language's semantics will have looked something like this:
+
+> \[[expression]]<sub>e</sub> = result
+
+Now we're going to relativize our interpretations not only to the environment `e`, but also to the current store, which I'll label `s`. Additionally, we're going to want to allow that evaluating some functions might *change* the store, perhaps by allocating new reference cells or perhaps by modifying the contents of some existing cells. So the interpretation of an expression won't just return a result; it will also return a possibly updated store. We'll suppose that our interpretation function does this quite generally, even though for many expressions in the language, the store that's returned will be the same one that the interpretation function started with:
+
+> \[[expression]]<sub>e s</sub> = (result, s')
+
+For expressions we already know how to interpret, you can by default expect `s'` to just be `s`.
+An exception is complex expressions like `let var = expr1 in expr2`. Part of
+interpreting this will be to interpret the sub-expression `expr1`, and we have
+to allow that in doing that, the store may have already been updated. We want
+to use that possibly updated store when interpreting `expr2`. Like this:
+
+ let rec eval term e s = match term with
+ ...
+ | Let (var, expr1, expr2) ->
+ let (res1, s') = eval expr1 e s
+ (* s' may be different from s *)
+ (* now we evaluate expr2 in a new environment where var has been associated
+ with the result of evaluating expr1 in the current environment *)
+ eval expr2 ((var, res1) :: e) s'
+ ...
+
+Similarly:
+
+ ...
+ | Apply (Apply(PrimitiveAddition, expr1), expr2) ->
+ let (res1, s') = eval expr1 e s in
+ let (res2, s'') = eval expr2 e s' in
+ (res1 + res2, s'')
+ ...
+
+Let's consider how to interpet our new syntactic forms `newref`, `getref`, and `putref`:
+
+
+1. When `expr` evaluates to `starting_val`, then **newref expr** should allocate a new reference cell in the store and insert `starting_val` into that cell. It should return some "key" or "index" or "pointer" to the newly created reference cell, so that we can do things like:
+
+ let ycell = newref 1 in
+ ...
+
+ and be able to refer back to that cell later by using the result that we assigned to the variable `ycell`. In our simple implementation, we're letting the store just be an `int list`, and we can let the "keys" be indexes in that list, which are (also) just `int`s. Somehow we should keep track of which variables are assigned `int`s as `int`s and which are assigned `int`s as indexes into the store. So we'll create a special type to wrap the latter:
+
+ type store_index = Index of int
+
+ Our interpretation function will look something like this:
+
+ let rec eval term e s = match term with
+ ...
+ | Newref (expr) ->
+ let (starting_val, s') = eval expr e s in
+ (* note that s' may be different from s, if expr itself contained any mutation operations *)
+ (* now we want to retrieve the next free index in s' *)
+ let new_index = List.length s' in
+ (* now we want to insert starting_val there; the following is an easy but inefficient way to do it *)
+ let s'' = List.append s' [starting_val] in
+ (* now we return a pair of a wrapped new_index, and the new store *)
+ (Index new_index, s'')
+ ...
+
+2. When `expr` evaluates to a `store_index`, then **getref expr** should evaluate to whatever value is at that index in the current store. (If `expr` evaluates to a value of another type, `getref expr` is undefined.) In this operation, we don't change the store at all; we're just reading from it. So we'll return the same store back unchanged (assuming it wasn't changed during the evaluation of `expr`).
+
+ let rec eval term e s = match term with
+ ...
+ | Getref (expr) ->
+ let (Index n, s') = eval expr e s in
+ (* s' may be different from s, if expr itself contained any mutation operations *)
+ (List.nth s' n, s')
+ ...
+
+3. When `expr1` evaluates to a `store_index` and `expr2` evaluates to an `int`, then **putref expr1 expr2** should have the effect of changing the store so that the reference cell at that index now contains that `int`. We have to make a decision about what result the `putref ...` call should itself evaluate to; OCaml makes this `()` but other choices are also possible. Here I'll just suppose we've got some appropriate value in the variable `dummy`.
+
+ let rec eval term e s = match term with
+ ...
+ | Putref (expr1, expr2) ->
+ let (Index n, s') = eval expr1 e s in
+ (* note that s' may be different from s, if expr1 itself contained any mutation operations *)
+ let (new_value, s'') = eval expr2 e s' in
+ (* now we create a list which is just like s'' except it has new_value in index n *)
+ (* the following could be expressed in Juli8 as `modify m (fun _ -> new_value) xs` *)
+ let rec replace_nth xs m = match xs with
+ | [] -> failwith "list too short"
+ | x::xs when m = 0 -> new_value :: xs
+ | x::xs -> x :: replace_nth xs (m - 1) in
+ let s''' = replace_nth s'' n in
+ (dummy, s''')
+ ...
+
+
+
+
+##How to implement implicit-style mutable variables##
+
+With implicit-style mutation, we don't have new syntactic forms like `newref` and `getref`. Instead, we just treat ordinary variables as being mutable. You could if you wanted to have some variables be mutable and others not; perhaps the first sort are written in Greek and the second in Latin. But for present purposes, we will suppose all variables in our language are mutable.
+
+We will still need a store to keep track of reference cells and their current values, just as in the explicit-style implementation. This time, every variable will be associated with an index into the store. So this is what we'll have our assignment function keep track of. The assignment function will bind variables to indexes into the store, rather than to the variables' current values. The variables will only indirectly be associated with "their values" by virtue of the joint work of the assignment function and the store.
+
+This brings up an interesting conceptual distinction. Formerly, we'd naturally think that a variable `x` is associated with only one type, and that that's the type that the expression `x` would *evaluate to*, and also the type of value that the assignment function *bound* `x` to. However, in the current framework these two types come apart. The assignment function binds `x` to an index into the store, and what the expression `x` evaluates to will be the value at that location in the store, which will usually be some type other than an index into a store, such as a `bool` or a `string`.
+
+To handle implicit-style mutation, we'll need to re-implement the way we interpret expressions like `x` and `var x = expr1 in expr2`. We will also have just one new syntactic form, `set x to expr1 then expr2`. (The `then` here is playing the role of the sequencing semicolon in OCaml.)
+
+Here's how to implement these. We'll suppose that our assignment function is list of pairs, as above and as in [week7](/reader_monad_for_variable_binding). LINK TODO
+
+ let rec eval term e s = match term with
+ ...
+ | Var (var : identifier) ->
+ let index = List.assoc var e in
+ (* retrieve the value at that index in the current store *)
+ let res = List.nth s index in
+ (res, s)
+
+ (* instead of `let x = ...` we now have `var x = ...`, for which I'll use the `Letvar` tag *)
+ | Letvar ((var : identifier), expr1, expr2) ->
+ let (starting_val, s') = eval expr1 e s in
+ (* get next free index in s' *)
+ let new_index = List.length s' in
+ (* insert starting_val there *)
+ let s'' = List.append s' [starting_val] in
+ (* evaluate expr2 using a new assignment function and store *)
+ eval expr2 ((var, new_index) :: e) s''
+
+ | Set ((var : identifier), expr1, expr2) ->
+ let (new_value, s') = eval expr1 e s in
+ (* lookup which index is associated with Var var *)
+ let index = List.assoc var e in
+ (* now we create a list which is just like s' except it has new_value at index *)
+ let rec replace_nth xs m = match xs with
+ | [] -> failwith "list too short"
+ | x::xs when m = 0 -> new_value :: xs
+ | x::xs -> x :: replace_nth xs (m - 1) in
+ let s'' = replace_nth s' index in
+ (* evaluate expr2 using original assignment function and new store *)
+ eval expr2 e s''
+
+
+##How to implement mutation with a State monad##
+
+It's possible to do all of this monadically, instead of adding new syntactic forms and new interpretation rules to a language's semantics. The patterns we use to do this in fact closely mirror the machinery described above.
+
+We call this a State monad. It's a lot like the Reader monad, except that with the Reader monad, we could only read from the environment. We did have the possibility of interpreting sub-expressions inside a "shifted" environment, but as you'll see, that corresponds to the "shadowing" behavior described before, not to the mutation behavior that we're trying to implement now.
+
+With a State monad, we call our book-keeping apparatus a "store" instead of an environment, and this time we are able to both read from it and write to it. To keep things simple, we'll work here with the simplest possible kind of store, which only holds a single value. One could also have stores that were composed of a list of values, of a length that could expand or shrink, or even more complex structures.
+
+Here's the implementation of the State monad, together with an implementation of the Reader monad for comparison:
+
+ type env = (identifier * int) list
+ (* alternatively, an env could be implemented as type identifier -> int *)
+
+ type 'a reader = env -> 'a
+ let reader_mid (x : 'a) : 'a reader =
+ fun e -> x
+ let reader_mbind (xx : 'a reader) (k : 'a -> 'b reader) : 'b reader =
+ fun e -> let x = xx e in
+ let yy = k x in
+ yy e
+
+ type store = int
+ (* very simple store, holds only a single int *)
+ (* this corresponds to having only a single mutable variable *)
+
+ type 'a state = store -> ('a, store)
+ let state_mid (x : 'a) : 'a state =
+ fun s -> (x, s)
+ let state_mbind (xx : 'a state) (k : 'a -> 'b state) : 'b state =
+ fun s -> let (x, s') = xx s in
+ let yy = k x in
+ yy s'
+
+Notice the similarities (and differences) between the implementation of these two monads.
+
+With the Reader monad, we also had some special-purpose operations, beyond its general monadic operations. Two to focus on were `asks` and `shift`. We would call `asks` with a helper function like `lookup "x"` that looked up a given variable in an environment. And we would call `shift` with a helper function like `insert "x" new_value` that operated on an existing environment to return a new one.
+
+With the State monad, we'll also have some special-purpose operations. We'll consider two basic ones here. One will be to retrieve what is the current store. This is like the Reader monad's `asks (lookup "x")`, except in this simple implementation there's only a single location for a value to be looked up from. Here's how we'll do it:
+
+ let state_get : store state = fun s -> (s, s)
+
+This passes through the current store unaltered, and also returns a copy of the store as its payload. (What exactly corresponds to this is the simpler Reader operation `ask`.) We can use the `state_get` operation like this:
+
+ some_existing_state_monad_value >>= fun _ -> state_get >>= (fun cur_store -> ...)
+
+The `fun _ ->` part here discards the payload wrapped by `some_existing_state_monad_value`. We're only going to pass through, unaltered, whatever *store* is generated by that monadic box. We also wrap that store as *our own payload*, which can be retrieved by further operations in the `... >>= ...` chain, such as `(fun cur_store -> ...)`.
+
+As we've mentioned elsewhere, `xx >>= fun _ -> yy` can be abbreviated as `xx >> yy`.
+
+The other operation for the State monad will be to update the existing store to a new one. This operation looks like this:
+
+ let state_put (new_store : int) : dummy state =
+ fun s -> (dummy, new_store)
+
+If we want to stick this in a `... >>= ...` chain, we'll need to prefix it with `fun _ ->` too, like this:
+
+ some_existing_state_monad_value >>= fun _ -> state_put 100 >>= ...
+
+Or:
+
+ some_existing_state_monad_value >> state_put 100 >>= ...
+
+In this usage, we don't care what payload is wrapped by `some_existing_state_monad_value`. We don't even care what store it generates, since we're going to replace that store with our own new store. A more complex kind of `state_put` operation might insert not just some constant value as the new store, but rather the result of applying some function to the existing store. For example, we might want to increment the current store. Here's how we could do that:
+
+<pre>
+some_existing_state_monad_value >> <span class=ul>state_get >>= (fun cur_store -> state_put (succ cur_store))</span> >>= ...
+</pre>
+
+We can define more complex functions that perform the underlined part `state_get >>= (fun cur_store -> state_put (succ cur_store))` as a single operation. In the Juli8 and Haskell monad libraries, this is expressed by the State monad operation `modify succ`.
+
+In general, a State monadic **value** (type `'a state`, what appears at the start of a `... >>= ... >>= ...` chain) is an operation that accepts some starting store as input --- where the store might be simple as it is here, or much more complex --- and returns a payload plus a possibly modified store. This can be thought of as a static encoding of some computation on a store, which encoding is used as a box wrapped around a value of type `'a`. (And also it's a burrito.)
+
+State monadic **operations** or Kleisli arrows (type `'a -> 'b state`, what appears anywhere in the middle or end of a `... >>= ... >>= ...` chain) are operations that generate new State monad boxes, based on what payload was wrapped by the preceding elements in the `... >>= ... >>= ...` chain. The computations on a store that such operations encode (which their payloads may or may not be sensitive to) will be chained in the order given by their position in the `... >>= ... >>= ...` chain. That is, the computation encoded by the first element in the chain will accept a starting store `s0` as input, and will return (a payload and) a new store `s1` as output, the next computation will get `s1` as input and will return `s2` as part of its output, the next computation will get `s2` as input, ... and so on.
+
+To get the whole process started, the complex computation so defined will need to be given a starting store. So we'd need to do something like this:
+
+ let computation = some_state_monad_value >>= operation >>= operation in
+ computation initial_store
+
+
+* See also our [[State Monad Tutorial]]. LINK TODO
+
+
+
+##Some grades of mutation involvement##
+
+Programming languages tend to provide a bunch of mutation-related capabilities at once, if they provide any. For conceptual clarity, however, it's helped me to distill these into several small increments. This is a list of some different ways in which languages might involve mutation-like idioms. (It doesn't exhaust all the interesting such ways, but only the ones we've so far touched on.)
+
+* At the zeroth stage, we have a purely functional language, like we've been working with up until this week.
+
+
+* One increment would be to add implicit-style mutable variables, as we explained above.
+
+ The semantic machinery for implicit-style mutable variables will have something playing the role of a reference cell. However these won't be **first-class values** in the language. For something to be a first-class value, it has to be possible to assign that value to variables, to pass it as an argument to functions, and to return it as the result of a function call. Now for some of these criteria it's debatable that they are already here satisfied. For example, in some sense the introduction of a new implicitly mutable variable (`var x = 1 in ...`) will associate a reference cell with `x`. That won't be what `x` evaluates to, but it will be what the assignment function *binds* `x` to, behind the scenes.
+
+ However, in language with implicit-style mutation, what you're clearly not able to do is to return a reference cell as the result of a function call, or indeed of any expression. This is connected to --- perhaps it's the same point as --- the fact that `x` doesn't evalute to a reference cell, but rather to the value that the reference cell it's implicitly associated with contains, at that stage in the computation.
+
+* Another grade of mutation involvement is to have explicit-style mutation. Here we might say we have not just mutable variables but also first-class values whose contents can be altered. That is, we have not just mutable variables but **mutable values**.
+
+ This introduces some interesting new conceptual possibilities. For example, what should be the result of the following fragment?
+
+ let ycell = ref 1 in
+ let xcell = ref 1 in
+ ycell = xcell
+
+ Are the two reference cell values equal or aren't they? Well, at this stage in the computation, they're qualitatively indiscernible. They're both `int ref`s containing the same `int`. And that is in fact the relation that `=` expresses in OCaml. In Scheme the analogous relation is spelled `equal?` Computer scientists sometimes call this relation "structural equality."
+
+ On the other hand, these are numerically *two* reference cells. If we mutate one of them, the other one doesn't change. For example:
+
+ let ycell = ref 1 in
+ let xcell = ref 1 in
+ ycell := 2; !xcell
+ (* evaluates to 1, not to 2 *)
+
+ So we have here the basis for introducing a new kind of equality predicate into our language, which tests not for qualitative indiscernibility but for numerical equality. In OCaml this relation is expressed by the double equals `==`. In Scheme it's spelled `eq?` Computer scientists sometimes call this relation "physical equality". Using this equality predicate, our comparison of `ycell` and `xcell` will be `false`, even if they then happen to contain the same `int`.
+
+ Isn't this interesting? Intuitively, elsewhere in math, you might think that qualitative indicernibility always suffices for numerical identity. Well, perhaps this needs discussion. In some sense the imaginary numbers ι and -ι are qualitatively indiscernible, but numerically distinct. However, arguably they're not *fully* qualitatively indiscernible. They don't both bear all the same relations to ι for instance. But then, if we include numerical/physical identity as a relation, then `ycell` and `xcell` don't both bear all the same relations to `ycell`, either. Yet there is still a useful sense in which they can be understood to be qualitatively equal --- at least, at a given stage in a computation.
+
+ **Terminological aside**: in OCaml, `=` and `<>` express the qualitative (in)discernibility relations, also expressed in Scheme with `equal?`. In OCaml, `==` and `!=` express the numerical (non)identity relations, also expressed in Scheme with `eq?`. `=` also has other syntactic roles in OCaml, such as in the form `let x = value in ...`. In other languages, like C and Python, `=` is commonly used just for assignment (of either of the sorts we've now seen: `var x = value in ...` or `set x to value in ...`). The symbols `==` and `!=` are commonly used to express qualitative (in)discernibility in these languages. Python expresses numerical (non)identity with `is` and `is not`. What an unattractive mess. Don't get me started on Haskell (qualitative discernibility is `/=`) and Lua (physical (non)identity is `==` and `~=`).
+
+ In the following fragment:
+
+ let ycell = ref 1 in
+ let xcell = ref 1 in
+ let zcell = ycell in
+ ...
+
+ If we express numerical identity using `==`, as OCaml does, then this (and its converse) would be true:
+
+ ycell == zcell
+
+ but these would be false:
+
+ xcell == ycell
+ xcell == zcell
+
+ If we express qualitative indiscernibility using `=`, as OCaml does, then all of the salient comparisons would be true:
+
+ ycell = zcell
+ xcell = ycell
+ xcell = zcell
+
+ Because of the particular way the numerical identity predicates are implemented in all of these languages, it doesn't quite match our conceptual expectations. For instance, For instance, if `ycell` is a reference cell, then `ref !ycell` will always be a numerically distinct reference cell containing the same value. We get this pattern of comparisons in OCaml:
+
+ ycell == ycell (* of course true *)
+ ycell != ref !ycell (* true, these aren't numerically identical *)
+
+ ycell = ycell (* of course true *)
+ ycell = ref !ycell (* true, they are qualitatively indiscernible *)
+
+ But now what about?
+
+ (0, 1, ycell) ? (0, 1, ycell)
+ (0, 1, ycell) ? (0, 1, ref !ycell)
+
+ You might expect the first pair to be numerically identical too --- after all, they involve the same structure (an immutable triple) each of whose components is numerically identical. But OCaml's "physical identity" predicate `==` does not detect that identity. It counts both of these comparisons as false. OCaml's `=` predicate does count the first pair as equal, but only because it's insensitive to numerical identity; it also counts the second pair as equal. This odd pattern shows up in many other languages, too. In Python, `y = []; (0, 1, y) is (0, 1, y)` evaluates to false. In Racket, `(define y (box 1)) (eq? (cons 0 y) (cons 0 y))` also evaluates to false (and in Racket, unlike traditional Schemes, `cons` is creating immutable pairs). All these languages chose an implementation for their numerical identity predicates that is especially efficient and does the right thing in the common cases, but doesn't quite match our mathematical expectations.
+
+ Another interesting example of "mutable values" that illustrate the coming apart of qualitative indiscernibility and numerical identity are the `getter`/`setter` pairs we discussed earlier. Recall:
+
+ let factory (starting_val : int) =
+ let free_var = ref starting_value in
+ let getter () = !free_var in
+ let setter (new_value : int) = free_var := new_value in
+ (getter, setter) in
+ let (getter, setter) = factory 1 in
+ let (getter', setter') = factory 1 in
+ ...
+
+ After this, `getter` and `getter'` would (at least, temporarily) be qualitatively indiscernible. They'd return the same value whenever called with the same argument (`()`). So too would `adder` and `adder'` in the following example:
+
+ let factory (starting_val : int) =
+ let free_var = ref starting_value in
+ let adder x = x + !free_var in
+ let setter (new_value : int) = free_var := new_value in
+ (adder, setter) in
+ let (adder, setter) = factory 1 in
+ let (adder', setter') = factory 1 in
+ ...
+
+ Of course, in most languages you wouldn't be able to evaluate a comparison like `getter = getter'`, because in general the question whether two functions always return the same values for the same arguments is not decidable. So typically languages don't even try to answer that question. However, it would still be true that `getter` and `getter'` (and `adder` and `adder'`) were extensionally equivalent; you just wouldn't be able to establish so.
+
+ However, they're not numerically identical, because by calling `setter 2` (but not calling `setter' 2`) we can mutate the function value `getter` (and `adder`) so that it's *no longer* qualitatively indiscernible from `getter'` (or `adder'`).
+
+There are several more layers and complexity to the way different languages engage with mutation. But this exhausts what we're in a position to consider now.
+
+
+##Miscellany##
+
+* When using mutable variables, programmers will often write using *imperatival loops* that repeatedly mutate a variable, rather than the recursive techniques we've been using so far. For example, we'd define the factorial function like this:
+
+ let rec factorial n =
+ if n = 0 then 1 else n * factorial (n - 1)
+
+ or like this:
+
+ let factorial n =
+ let rec aux n sofar =
+ if n = 0 then sofar else aux (n - 1) (n * sofar) in
+ aux n 1
+
+ (The second version is more efficient than the first; so you may sometimes see this programming style. But for our purposes, these can be regarded as equivalent.)
+
+ When using mutable variables, on the other hand, this may be written as:
+
+ let factorial n =
+ let current = ref n in
+ let total = ref 1 in
+ while !current > 0 do
+ total := !total * !current; current := !current - 1
+ done; !total
+
+ This is often referred to as an *iterative* as opposed to a *recursive* algorithm.
+
+
+* Mutable variables also give us a way to achieve recursion, in a language that doesn't already have it. For example:
+
+ let fact_cell = ref None in
+ let factorial n =
+ if n = 0 then 1 else match !fact_cell with
+ | Some fact -> n * fact (n - 1)
+ | None -> failwith "can't happen" in
+ let () = fact_cell := Some factorial in
+ ...
+
+ We use the `None`/`Some factorial` option type here just as a way to ensure that the contents of `fact_cell` are of the same type both at the start and the end of the block.
+
+ If you've got a copy of *The Seasoned Schemer*, which we recommended for the seminar, see the discussion at pp. 118-125.
+
+<!--
+* Now would be a good time to go back and review some material from [[week1]], and seeing how much we've learned. There's discussion back then of declarative or functional languages versus languages using imperatival features, like mutation. Mutation is distinguished from shadowing. There's discussion of sequencing, and of what we mean by saying "order matters."
+
+ In point 7 of the Rosetta Stone discussion, the contrast between call-by-name and call-by-value evaluation order appears (though we don't yet call it that). We'll be discussing that more in coming weeks. In the [[damn]] example, continuations and other kinds of side effects (namely, printing) make an appearance. These too will be center-stage in coming weeks.
+
+* Now would also be a good time to read [Calculator Improvements](/week10). This reviews the different systems discussed above, as well as other capabilities we can add to the calculators introduced in [week7](/reader_monad_for_variable_binding). We will be building off of that in coming weeks.
+-->
+
+
+##Offsite Reading##
+
+* [[!wikipedia Declarative programming]]
+* [[!wikipedia Functional programming]]
+* [[!wikipedia Purely functional]]
+* [[!wikipedia Side effect (computer science) desc="Side effects"]]
+* [[!wikipedia Referential transparency (computer science)]]
+* [[!wikipedia Imperative programming]]
+* [[!wikipedia Reference (computer science) desc="References"]]
+* [[!wikipedia Pointer (computing) desc="Pointers"]]
+* [Pointers in OCaml](http://caml.inria.fr/resources/doc/guides/pointers.html)
+
+<!--
+# General issues about variables and scope in programming languages #
+
+* [[!wikipedia Variable (programming) desc="Variables"]]
+* [[!wikipedia Free variables and bound variables]]
+* [[!wikipedia Variable shadowing]]
+* [[!wikipedia Name binding]]
+* [[!wikipedia Name resolution]]
+* [[!wikipedia Parameter (computer science) desc="Function parameters"]]
+* [[!wikipedia Scope (programming) desc="Variable scope"]]
+* [[!wikipedia Closure (computer science) desc="Closures"]]
+
+-->
+
+
+
--- /dev/null
+This walks through most of [A State Monad Tutorial](http://strabismicgobbledygook.wordpress.com/2010/03/06/a-state-monad-tutorial/), which is addressed to a Haskell-using audience. But we convert it to OCaml. See our page on [[comparing OCaml and Haskell|/rosetta2]].
+
+Some of what we do here will make use of our monad library for OCaml. See:
+
+* [[Installing and Using the Juli8 Libraries|/juli8]]
+* [[Using the OCaml Monad library|/topics/week9_using_the_monad_library]]
+
+As we discussed in [[week9]] TODO LINK, a State monad is implemented with the type:
+
+ store -> ('a * store)
+
+It's common practice to encapsulate this in some way, so that the interpreter knows the difference between arbitrary functions from a `blah` to a pair of something and a `blah` and the values that you've specially designated as being State monadic values.
+
+The most lightweight way encapsulate it would be just to add a data constructor to the type. In the same way that the `'a option` type has the `None` and `Some` data constructors, we give our `'a state` type a `State` data constructor:
+
+ (* we assume that the store type has already been declared *)
+ type 'a state = State of (store -> ('a * store))
+
+Then a function expecting an `'a state` will look for a value with the structure `State ...` rather than just one with the structure `...`.
+
+To take a `State (s -> (a,s))` and get at the `s -> (a,s)` it wraps, you use the same techniques you use to take an `Some int` and get at the `int` it wraps:
+
+ let xx = State (fun s -> (1, s)) in
+ let State unwrapped_xx = xx in
+ ...
+
+or:
+
+ let xx = State (fun s -> (1, s)) in
+ match xx with
+ | State unwrapped_xx -> ...
+
+There are two heavierweight ways to encapsulate the type of the State monad. One is used by our [[monad library]] TODO LINK --- the type is hidden from the outside user, and only gets exposed by the `run` function. But the result of `run xx` is not itself recognized as a monadic value any longer. You can't replace `xx` in:
+
+ SomeMonad.(xx >>= ...)
+
+with `run xx`. So you should only apply `run` when you've finished building up a monadic value. (That's why it's called `run`.)
+
+Of course you can do this:
+
+ let xx = SomeMonad.(...) in
+ let intermediate_result = SomeMonad.run xx 0 in
+ let yy = SomeMonad.(xx >>= ...) in
+ let final_result = SomeMonad.run yy 0 in
+ ...
+
+The other heavyweight way to encapsulate the type of a monad is to use records. See [here](/translating_between_OCaml_Scheme_and_Haskell) and [here](/coroutines_and_aborts/) for some introduction to these. TODO LINKS We don't use this design in our OCaml monad library, but the Haskell monad libraries do, and it would be good for you to get acquainted with it so that you can see how to ignore it when you come across it in Haskell-based literature. (Or you might want to learn Haskell, who knows?)
+
+We'll briefly illustrate this technique in OCaml code, for uniformity. See the [translation page](/translating_between_OCaml_Scheme_and_Haskell) TODO LINKS about how this looks in Haskell.
+
+To use the record technique, instead of saying;
+
+ type 'a state = State of (store -> ('a * store))
+
+you'd say:
+
+ type 'a state = { state : store -> ('a * store) }
+
+and instead of saying:
+
+ let xx = State (fun s -> (1, s)) in
+ let State unwrapped_xx = xx in
+ ...
+
+you'd say:
+
+ let xx = { state = fun s -> (1, s) } in
+ let unwrapped_xx = xx.state in
+ ...
+
+That's basically it. As with the other two techniques, the type of `xx` is not the same as `store -> ('a * store)`, but the relevant code will know how to convert between them.
+
+The main benefit of these techniques is that it gives you better type-checking: it makes sure that you're only using your monadic values in the hygenic ways you're supposed to. Perhaps you don't care about that. Well, then, if you want to write all your own monadic code, you can proceed as you like. If you ever want to use other people's code, though, or read papers or web posts about monads, you will encounter one or more of these techniques, and so you need to get comfortable enough with them not to let them confuse you.
+
+OK, back to our walk-through of "A State Monad Tutorial". What shall we use for a store? Instead of a plain `int`, let's suppose our store is a structure of two values: a running total, and a count of how many times the store has been modified. We'll implement this with a record. Hence:
+
+ type store' = { total : int; modifications: int }
+
+State monads employing this store will then have *three* salient values at any point in the computation: the `total` and `modifications` field in the store, and also the `'a` payload that is currently wrapped in the monadic box.
+
+Here's a monadic value that encodes the operation of incrementing the store's `total` and wrapping the value that was the former `total`:
+
+ let increment_store : store' -> (int * store') = fun s ->
+ let value = s.total in
+ let s' = { total = succ s.total; modifications = succ s.modifications } in
+ (value, s')
+
+If we wanted to work with one of the encapsulation techniques described above, we'd have to proceed a bit differently. Here is how to do it with the first, lightweight technique:
+
+ let increment_store' : 'a state = State (fun s ->
+ let value = s.total in
+ let s' = { total = succ s.total; modifications = succ s.modifications } in
+ (value, s'))
+
+
+Here is how you'd have to do it using our OCaml/Juli8 monad library:
+
+ (* have Juli8 loaded, as explained elsewhere *)
+ # module S = Monad.State(struct type store = store' end);;
+ # let increment_store'' : 'a S.t =
+ S.(get >>= fun cur ->
+ let value = cur.total in
+ let s' = { total = succ cur.total; modifications = succ cur.modifications } in
+ put s' >> mid value);;
+
+Let's try it out:
+
+ # let s0 = { total = 42; modifications = 3 };;
+ # increment_store s0;;
+ - : int * store' = (42, {total = 43; modifications = 4})
+
+Or if you used the OCaml/Juli8 monad library:
+
+ # S.run increment_store'' s0;;
+ - : int * S.store = (42, {total = 43; modifications = 4})
+
+Great!
+
+Can you write a monadic value that instead of incrementing each of the `total` and `modifications` fields in the store, doubles the `total` field and increments the `modifications` field?
+
+What about a value that increments each of `total` and `modifications` twice? Well, you could custom-write that, as with the previous question. But we already have the tools to express it easily, using our existing `increment_store` value:
+
+ increment_store >>= fun value -> increment_store >> mid value
+
+That ensures that the value we get at the end is the value returned by the first application of `increment_store`, that is, the contents of the `total` field in the store before we started modifying the store at all.
+
+You should start to see here how chaining monadic values together gives us a kind of programming language. Of course, it's a cumbersome programming language. It'd be much easier to write, directly in OCaml:
+
+ let value = s0.total
+ in (value, { total = s0.total + 2; modifications = s0.modifications + 2})
+
+or, using pattern-matching on the record (you don't have to specify every field in the record):
+
+ let { total = value; _ } = s0
+ in (value, { total = s0.total + 2; modifications = s0.modifications + 2})
+
+But **the point of learning how to do this monadically** is that (1) monads show us how to embed more sophisticated programming techniques, such as imperative state and continuations, into frameworks that don't natively possess them (such as the set-theoretic metalanguage of Groenendijk, Stokhof and Veltman's paper); (2) becoming familiar with monads will enable you to see patterns you'd otherwise miss, and implement some seemingly complex computations using the same simple patterns (same-fringe is an example); and finally, of course (3) monads are delicious.
+
+Keep in mind that the final result of a `mbind` chain doesn't have to be the same type as the starting value:
+
+ increment_store >>= fun value -> increment_store >> mid (string_of_int value)
+
+Or:
+
+ mid 1 >> mid "blah"
+
+The store keeps the same type throughout the computation, but the type of the wrapped value can change.
+
+What are the special-purpose operations that the `Monad.State` module defines for us?
+
+* `get` is a monadic value that passes through the existing store unchanged, and also wraps that same store as its boxed payload. You use it like this:
+
+ ... >>= fun _ -> get >>= fun cur -> ... here we can use cur ...
+
+ As we've said, that's equivalent to:
+
+ ... >> get >>= fun cur -> ...
+
+ You can also get the current store at the start of the computation:
+
+ get >> = fun cur -> ...
+
+* `gets selector` is like `get`, but it additionally applies the `selector` function to the store before depositing it in the box. If your store is structured, you can use this to only extract a piece of the structure:
+
+ ... >> gets (fun cur -> cur.total) >>= fun total -> ...
+
+ For more complex structured stores, consider using the `Monad.Ref` variant of the State monad in the OCaml library.
+
+* `put new_store` replaces the existing store with `new_store`. Use it like this:
+
+ ... >> put new_store >> fun () -> ...
+
+ As that code snippet suggests, the boxed payload after the application of `modify new_store` is just `()`. If you want to preserve the existing payload but replace the store, do this:
+
+ ... >>= fun value -> put new_store >> mid value >>= ...
+
+* Finally, `modify modifier` applies `modifier` to whatever the existing store is, and substitutes that as the new store. As with `put`, the boxed payload afterwards is `()`.
+
+ <!--
+ Haskell calls this operation `modify`. We've called it `puts` because it seems to fit naturally with the convention of `get` vs `gets`. (See also `ask` vs `asks` in `Monad.Reader`, which are also the names used in Haskell.)
+ -->
+
+Here's an example from "A State Monad Tutorial":
+
+ increment_store >> get >>= fun cur ->
+ State (fun s -> ((), { total = s.total / 2; modifications = succ s.modifications })) >>
+ increment_store >> mid cur.total
+
+Or, as you'd have to write it using our OCaml monad library:
+
+ increment_store'' >> get >>= fun cur ->
+ put { total = cur.total / 2; modifications = succ cur.modifications } >>
+ increment_store'' >> mid cur.total
+
+
+The last topic covered in "A State Monad Tutorial" is the use of do-notation to work with monads in Haskell. We discuss that on our [translation page](/translating_between_OCaml_Scheme_and_Haskell).
+
+