`∃x (Fx and ∃x Gx)`

. The computer-science terminology to describe this is that the narrower assignment of `x` to the value 3 **shadows** the wider assignment to 4.
+As you can see, the narrowest assignment is what's effective. This is just like in predicate logic: consider `∃y (Fy and ∃y ~Fy)`

. 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 x is assigned to." What we'll go on to see is a more exotic phenomenon that merits that description better.
+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:
@@ -61,7 +61,7 @@ Sometimes the shadowing is merely temporary, as here:
(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 assignemnt:
+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) ->
@@ -83,12 +83,14 @@ In languages that have native syntax for this, there are two styles in which it
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 explicitly referred to in the object language. In explicit-style mutation, they are. OCaml has explicit-style mutation. It looks like this:
+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 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:
@@ -243,14 +245,15 @@ Notice, however, that when mutable variables are present, the same substitution
(* evaluates to 2 *)
(ref 1) := 2; !(ref 1)
- (* evaluates to 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'd have to 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 both much at this point with the assignment function. Below we'll pay more attention to it.
+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.
@@ -271,9 +274,11 @@ With that kind of framework, we can interpret `newref`, `deref`, and `setref` as
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 just `int`s. Somehow we'd have to 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:
+ 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
@@ -324,11 +329,11 @@ With that kind of framework, we can interpret `newref`, `deref`, and `setref` as
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 those values by virtue of the joint work of the assignment function and the store.
+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 novelty. 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 can 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 might be some other type, such as a `bool` or a `string`.
+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`, `let x = expr1 in expr2`. We will have just one new syntactic form, `change x to expr1 then expr2`.
+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 in [week6](/reader_monad_for_variable_binding).
@@ -337,28 +342,30 @@ Here's how to implement these. We'll suppose that our assignment function is lis
...
| Var (c : char) ->
let index = List.assoc c g
- in List.nth s index
+ (* 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_value, s') = eval expr1 g s
+ let (starting_val, s') = eval expr1 g s
(* get next free index in s' *)
in let new_index = List.length s'
- (* insert starting_value there *)
- in let s'' = List.append s' [starting_value]
+ (* 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 n = List.assoc c g
- (* now we create a list which is just like s' except it has new_value in index n *)
+ 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' n
+ in let s'' = replace_nth s' index
(* evaluate expr2 using original assignment function and new store *)
in eval expr2 g s''
@@ -400,34 +407,33 @@ Notice the similarities (and differences) between the implementation of these tw
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 get_state : 'a -> store state =
- fun _ ->
+ let get_state : store state =
fun s -> (s, s);;
-This passes through the current store unaltered, and also returns a copy of the store as its value. Note the beginning `fun _ ->` part. That's so we can use this operation like this:
+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 >>= get_state >>= (fun cur_state -> ...)
+ some_existing_state_monad >>= fun _ -> get_state >>= (fun cur_state -> ...)
-The `get_state` operation ignores the value wrapped by `some_existing_state_monad`. It just passes through whatever store is generated by `some_existing_state_monad`. It also wraps that store as its own value, which can be retrieved by further operations in the `... >>= ...` chain, such as the `(fun cur_state -> ...)`.
+The `fun _ ->` part here discards the value wrapped by `some_existing_state_monad`. We're only going to pass through, unaltered, whatever *store* is generated by that monadic value. We also wrap that store as *our own value*, which can be retrieved by further operations in the `... >>= ...` chain, such as `(fun cur_state -> ...)`.
-The other operation for the state monad will be to update the existing store to a new one. This operation looks like this:
+The other operation for the State monad will be to update the existing store to a new one. This operation looks like this:
- let set_state (value : int) : 'a -> dummy state =
+ let set_state (value : int) : dummy state =
fun s -> (dummy, value);;
If we want to stick this in a `... >>= ...` chain, we'll need to prefix it with `fun _ ->` too, like this:
some_existing_state_monad >>= fun _ -> set_state 100 >>= ...
-In this kind of usage, we don't care what value is wrapped by `some_existing_state_monad`. 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 `set_state` or `update_state` operation might use as a new store not just some constant value, but rather something which is 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:
+In this usage, we don't care what value is wrapped by `some_existing_state_monad`. 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 `set_state` 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 >>= get_state >>= (fun cur_state -> set_state (cur_state + 1) >>= ...
+ some_existing_state_monad >>= fun _ -> get_state >>= (fun cur_state -> set_state (cur_state + 1) >>= ...
We can of course define more complex functions that perform the `get_state >>= (fun cur_state -> set_state (cur_state + 1)` as a single operation.
-In general, a State monadic value (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 an encoding of a operation on the store as a box wrapped around a value.
+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 value plus a possibly modified store. This can be thought of as an encoding of some operation on a store serving as a box wrapped around a value.
-State monadic operations (what appears anywhere in the middle or end of a `... >>= ... >>= ...` chain) are operations that generate new State monadic values, based on what value was wrapped by the preceding elements in the `... >>= ... >>= ...` chain. The computations on the 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.
+State monadic **operations** (type 'a -> 'b state, what appears anywhere in the middle or end of a `... >>= ... >>= ...` chain) are operations that generate new State monadic values, 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: