X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week9.mdwn;h=546b2d2c8f10534c2afa178af9109881ad438bbf;hp=f5beebb4aa78e74e462a90bd9f5ce58869e37bdb;hb=b5672ffae330118100a8de0a656ee9584d0f7ee6;hpb=54f07a87abd25524b0e1599cc5208c0c88b7a8c1;ds=sidebyside
diff --git a/week9.mdwn b/week9.mdwn
index f5beebb4..546b2d2c 100644
--- a/week9.mdwn
+++ b/week9.mdwn
@@ -100,7 +100,13 @@ Scheme is similar. There are various sorts of reference cells available in Schem
(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.)
+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.
@@ -291,7 +297,11 @@ Now we're going to relativize our interpretations not only to the assignment fun
> \[[expression]]_{g s} = (value, s')
-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:
+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
@@ -304,10 +314,19 @@ For expressions we already know how to interpret, `s'` will usually just be `s`.
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:
+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 ...
@@ -332,7 +351,7 @@ Let's consider how to interpet our new syntactic forms `newref`, `deref`, and `s
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
@@ -343,14 +362,14 @@ Let's consider how to interpet our new syntactic forms `newref`, `deref`, and `s
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`.
+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 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 =
@@ -376,7 +395,7 @@ This brings up an interesting conceptual distinction. Formerly, we'd naturally t
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 [week7](/reader_monad_for_variable_binding).
+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
@@ -410,6 +429,8 @@ Here's how to implement these. We'll suppose that our assignment function is lis
(* 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, Stockhof 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##
@@ -487,86 +508,205 @@ To get the whole process started, the complex computation so defined will need t
-- FIXME --
- [H] ; *** aliasing ***
- let y be 2 in
- let x be y in
- let w alias y in
- (y, x, w) ==> (2, 2, 2)
+ [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)
- [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) ==> (3, 2, 3)
+ [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)
- [J] let f be (lambda (y) -> BODY) in ; a
- ... f (EXPRESSION) ...
+Notice: in [M], `h` and `p` have same value (1), but `f (h, p)` and `f (h, h)` differ.
- (lambda (y) -> BODY) EXPRESSION
+See Pryor's "[Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt)".
- let y be EXPRESSION in ; b
- ... BODY ...
- [K] ; *** passing "by reference" ***
- let f be (lambda (alias w) -> ; ?
- BODY
- ) in
- ... f (y) ...
+##Four grades of mutation involvement##
- let w alias y in ; d
- ... BODY ...
+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.
- [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) ==> (4, 2)
+* At the first stage, we have a purely functional language, like we've been working with up until this week.
- [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))
- ==> (3, 4)
- Notice: h, p have same value (1), but f (h, p) and f (h, h) differ
+* 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.
-Fine and Pryor on "coordinated contents" (see, e.g., [Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt))
+* 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.
-##Five grades of mutation involvement##
+ 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:
--- FIXME --
+ 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 `~=`).
- 0. Purely functional languages
- 1. Passing by reference
- need primitive hyper-evaluative predicates for it to make a difference
- 2. mutable variables
- 3. mutable values
- - numerically distinct but indiscernible values
- - two equality predicates
- - examples: closures with currently-indiscernible but numerically distinct
- environments, mutable lists
- 4. "references" as first-class values
- - x not the same as !x, explicit deref operation
- - can not only be assigned and passed as arguments, also returned (and manipulated?)
- - can be compared for qualitative equality
- 5. structured references
+ 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.
+
+ 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##
@@ -607,6 +747,12 @@ Fine and Pryor on "coordinated contents" (see, e.g., [Hyper-Evaluativity](http:/
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.
+* 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 [[Advanced Topics/Calculator Improvements]]. 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##