3 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.
5 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.
10 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.
12 Recall from earlier discussions that the following two forms are equivalent:
14 [A] let x be EXPRESSION in
17 (lambda (x) -> BODY) (EXPRESSION)
19 This should seem entirely familiar:
23 (x + y, x + 20) ==> (13, 23)
25 In fragment [B], we bound the variables `x` and `y` to `int`s. We can also bind variables to function values, as here:
27 [C] let f be (lambda (x, y) -> x + y + 1) in
28 (f (10, 2), f (20, 2))
31 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:
34 let f be (lambda (x) -> x + y) in
38 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.
40 In our next fragment, we re-use a variable that had been bound to another value in a wider context:
44 (x + 10, x + 20) ==> (13, 23)
46 As you can see, the narrowest assignment is what's effective. This is just like in predicate logic: consider <code>∃x (Fx and ∃x Gx)</code>. The computer-science terminology to describe this is that the narrower assignment of `x` to the value 3 **shadows** the wider assignment to 4.
48 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.
50 Sometimes the shadowing is merely temporary, as here:
53 let f be (lambda (x) ->
55 ; here the most local assignment to y applies
58 ; here the assignment of 3 to y has expired
59 (f (10), y, f (20)) ==> (13, 2, 23)
61 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:
64 let f be (lambda (x) ->
68 ; here the change in what value y was assigned *sticks*
69 ; because we *updated* the value of the original variable y
70 ; instead of introducing a new y with a narrower scope
71 (f (10), y, f (19)) ==> (13, 3, 23)
73 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:
76 int y = 2; // this is like "let y be 2 in ..."
78 y = 3; // this is like "change y to 3 then ..."
79 return x + y; // this is like "x + y"
82 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:
84 let ycell = ref 2 (* this creates a new reference cell *)
86 in let () = ycell := 3 (* this changes the contents of that cell to 3; the return value of doing so is () *)
87 (* other return values could also be reasonable: such as the old value of ycell, the new value, an arbitrary int, and so on *)
88 in x + !ycell;; (* the !ycell operation "dereferences" the cell---it retrieves the value it contains *)
90 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:
92 (let ([ycell (box 2)])
97 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.
100 ##Controlling order##
102 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.
104 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`:
106 let triple = (expression_a, expression_b, expression_c)
108 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`:
111 in let b = expression_b
114 Scheme does the same. (*If* you use Scheme's `let*`, but not if you use its `let`. I agree this is annoying.)
116 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:
118 let () = expression_a
119 in let () = expression_b
122 And OCaml has a syntactic shorthand for this form, namely to use semi-colons:
124 expression_a; expression_b; expression_c
126 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:
128 (expression_a; expression_b; expression_c)
130 begin expression_a; expression_b; expression_c end
132 Scheme has a construction similar to the latter:
134 (begin (expression_a) (expression_b) (expression_c))
136 Though often in Scheme, the `(begin ...)` is implicit and doesn't need to be explicitly inserted, as here:
138 (lambda (x) (expression_a) (expression_b) (expression_c))
140 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:
146 let f = fun () -> ...
148 In Scheme these are written as functions that take 0 arguments:
156 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:
159 in let f () = ycell := !ycell + 1
164 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.
166 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:
168 let factory (starting_value : int) =
169 let free_var = ref starting_value
172 in let setter (new_value : int) =
173 free_var := new_value
175 in let (getter, setter) = factory 1
176 in let first = getter ()
178 in let second = getter ()
180 in let third = getter ()
181 in (first, second, third)
183 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:
185 def factory (starting_value):
186 free_var = starting_value
189 def setter (new_value):
190 # the next line indicates that we're using the
191 # free_var from the surrounding function, not
192 # introducing a new local variable with the same name
195 return getter, setter
196 getter, setter = factory (1)
202 (first, second, third)
204 Here, too, just as in the OCaml fragment, all the calls to getter and setter are working with a single mutable variable `free_var`.
206 If however you called `factory` twice, you'd have different `getter`/`setter` pairs, each of which had their own, independent `free_var`. In OCaml:
208 let factory (starting_val : int) =
210 in let (getter, setter) = factory 1
211 in let (getter', setter') = factory 1
215 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.
217 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.
219 ##Referential opacity##
221 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:
226 should evaluate the same as:
235 Notice, however, that when mutable variables are present, the same substitution patterns can't always be relied on:
238 in ycell := 2; !ycell
241 (ref 1) := 2; !(ref 1)
245 ##How to implement explicit-style mutable variables##
247 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.
249 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.
251 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.
253 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.
255 Before we brought mutation on the scene, our language's semantics will have looked something like this:
257 > \[[expression]]<sub>g</sub> = value
259 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:
261 > \[[expression]]<sub>g s</sub> = (value, s')
263 With that kind of framework, we can interpret `newref`, `deref`, and `setref` as follows.
265 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:
270 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:
272 type store_index = Index of int;;
274 let rec eval expression g s =
275 match expression with
278 let (starting_val, s') = eval expr g s
279 (* note that s' may be different from s, if expr itself contained any mutation operations *)
280 (* now we want to retrieve the next free index in s' *)
281 in let new_index = List.length s'
282 (* now we want to insert starting_val there; the following is an easy but inefficient way to do it *)
283 in let s'' = List.append s' [starting_val]
284 (* now we return a pair of a wrapped new_index, and the new store *)
285 in (Index new_index, s'')
288 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.
290 let rec eval expression g s =
291 match expression with
294 let (Index n, s') = eval expr g s
295 (* note that s' may be different from s, if expr itself contained any mutation operations *)
296 in (List.nth s' n, s')
299 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`.
301 let rec eval expression g s =
302 match expression with
305 let (Index n, s') = eval expr1 g s
306 (* note that s' may be different from s, if expr itself contained any mutation operations *)
307 in let (new_value, s'') = eval expr2 g s'
308 (* now we create a list which is just like s'' except it has new_value in index n *)
309 in let rec replace_nth lst m =
311 | [] -> failwith "list too short"
312 | x::xs when m = 0 -> new_value :: xs
313 | x::xs -> x :: replace_nth xs (m - 1)
314 in let s''' = replace_nth s'' n
319 ##How to implement implicit-style mutable variables##
323 ##How to implicit mutation with a State monad##
327 ##Aliasing or Passing by reference##
331 [H] ; *** aliasing ***
335 (y, x, w) ==> (2, 2, 2)
337 [I] ; mutation plus aliasing
342 (y, x, w) ==> (3, 2, 3)
344 [J] let f be (lambda (y) -> BODY) in ; a
345 ... f (EXPRESSION) ...
347 (lambda (y) -> BODY) EXPRESSION
349 let y be EXPRESSION in ; b
352 [K] ; *** passing "by reference" ***
353 let f be (lambda (alias w) -> ; ?
361 [L] let f be (lambda (alias w) ->
370 [M] ; hyper-evaluativity
373 let f be (lambda (alias x, alias y) ->
374 ; contrast here: "let z be x + y + 1"
375 change y to y + 1 then
377 change y to y - 1 then
383 Notice: h, p have same value (1), but f (h, p) and f (h, h) differ
386 ##Five grades of mutation involvement##
390 0. Purely functional languages
391 1. Passing by reference
392 need primitive hyper-evaluative predicates for it to make a difference
395 - numerically distinct but indiscernible values
396 - two equality predicates
397 - examples: closures with currently-indiscernible but numerically distinct
398 environments, mutable lists
399 4. "references" as first-class values
400 - x not the same as !x, explicit deref operation
401 - can not only be assigned and passed as arguments, also returned (and manipulated?)
402 - can be compared for qualitative equality
403 5. structured references
404 (a) if `a` and `b` are mutable variables that uncoordinatedly refer to numerically the same value
405 then mutating `b` won't affect `a` or its value
406 (b) if however their value has a mutable field `f`, then mutating `b.f` does
407 affect their shared value; will see a difference in what `a.f` now evaluates to
412 * 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:
414 let rec factorial n =
415 if n = 0 then 1 else n * factorial (n - 1)
420 let rec helper n sofar =
421 if n = 0 then sofar else helper (n - 1) (n * sofar)
424 (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.)
426 When using mutable variables, on the other hand, this may be written as:
431 in while !current > 0 do
432 total := !total * !current; current := !current - 1
436 * Mutable variables also give us a way to achieve recursion, in a language that doesn't already have it. For example:
438 let fact_cell = ref None
440 if n = 0 then 1 else match !fact_cell with
441 | Some fact -> n * fact (n - 1)
442 | None -> failwith "can't happen"
443 in let () = fact_cell := Some factorial
446 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.
450 Fine and Pryor on "coordinated contents" (see, e.g., [Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt))