X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week9.mdwn;h=568fc812dc66802eb89122197277c6f1a16e2995;hp=20c464dc387a58adec68645c04a145d2c68f1a86;hb=677d5bbd54d54101e2855598db2aaae0d9ed6bad;hpb=c9c7561e163ba3d0869a61b42d11118ded32ff23 diff --git a/week9.mdwn b/week9.mdwn index 20c464dc..568fc812 100644 --- a/week9.mdwn +++ b/week9.mdwn @@ -40,14 +40,14 @@ Other choices about how to interpret free variables are also possible (you can r In our next fragment, we re-use a variable that had been bound to another value in a wider context: - [E] let x be 4 in - let x be 3 in - (x + 10, x + 20) + [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 `∃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: @@ -98,7 +100,9 @@ Scheme is similar. There are various sorts of reference cells available in Schem (set-box! ycell 3) (+ x (unbox ycell))) -When dealing with explicit-style mutation, there's a difference between the types and values of `ycell` and `!ycell` (or `(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. +(C has explicit-style mutable variables, too, which it calls *pointers*. But simple variables in C are already mutable, in the implicit style.) + +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## @@ -222,7 +226,27 @@ Notice in these fragments that once we return from inside the call to `factory`, ##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. What they do mean is a kind of substitution principle, illustrated here: +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) @@ -243,14 +267,17 @@ 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. @@ -264,21 +291,46 @@ Now we're going to relativize our interpretations not only to the assignment fun > \[[expression]]g s = (value, s') -With that kind of framework, we can interpret `newref`, `deref`, and `setref` as follows. +For expressions we already know how to interpret, `s'` will usually just be `s`. One 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 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 ... - | Newref expr -> + | 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' *) @@ -289,12 +341,12 @@ With that kind of framework, we can interpret `newref`, `deref`, and `setref` as 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. +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 -> + | 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') @@ -305,9 +357,9 @@ With that kind of framework, we can interpret `newref`, `deref`, and `setref` as let rec eval expression g s = match expression with ... - | Setref expr1 expr2 + | Setref (expr1, expr2) -> let (Index n, s') = eval expr1 g s - (* note that s' may be different from s, if expr itself contained any mutation operations *) + (* 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 = @@ -320,56 +372,61 @@ With that kind of framework, we can interpret `newref`, `deref`, and `setref` as ... + + + ##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 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). +Here's how to implement these. We'll suppose that our assignment function is list of pairs, 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 - 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 ((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_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 -> + | 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'' -##How to implicit mutation with a State monad## +##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 evironment, 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. +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: @@ -400,38 +457,37 @@ 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_box >>= fun _ -> get_state >>= (fun cur_store -> ...) -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_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: +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 = - fun s -> (dummy, value);; + let set_state (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 >>= fun _ -> set_state 100 >>= ... + some_existing_state_monad_box >>= 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_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 `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_box >>= fun _ -> get_state >>= (fun cur_store -> set_state (cur_store + 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. +We can of course define more complex functions that perform the `get_state >>= (fun cur_store -> set_state (cur_store + 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 **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 (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 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_value >>= operation >>= operation + let computation = some_state_monadic_box >>= operation >>= operation in computation initial_store;; @@ -495,6 +551,9 @@ To get the whole process started, the complex computation so defined will need t Notice: h, p have same value (1), but f (h, p) and f (h, h) differ +Fine and Pryor on "coordinated contents" (see, e.g., [Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt)) + + ##Five grades of mutation involvement## -- FIXME -- @@ -558,7 +617,29 @@ To get the whole process started, the complex computation so defined will need t 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. +##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) +