+ (* 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. \[[newref starting_val]] 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''
+
+
+##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.