-- FIXME --
- [H] ; *** aliasing ***
- let y be 2 in
- let x be y in
- let w alias y in
- (y, x, w) ==> (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) ==> (3, 2, 3)
-
- [J] let f be (lambda (y) -> BODY) in ; a
- ... f (EXPRESSION) ...
-
- (lambda (y) -> BODY) EXPRESSION
-
- let y be EXPRESSION in ; b
- ... BODY ...
-
- [K] ; *** passing "by reference" ***
- let f be (lambda (alias w) -> ; ?
- BODY
- ) in
- ... f (y) ...
-
- let w alias y in ; d
- ... BODY ...
-
- [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)
-
- [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
-
-
-Fine and Pryor on "coordinated contents" (see, e.g., [Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt))
+ [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)
+
+ [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)
+
+Notice: in [M], `h` and `p` have same value (1), but `f (h, p)` and `f (h, h)` differ.
+
+See Pryor's "[Hyper-Evaluativity](http://www.jimpryor.net/research/papers/Hyper-Evaluativity.txt)".
##Four grades of mutation involvement##
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 identical, but numerically distinct. However, arguably they're not *fully* qualitatively identical. 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.
+ 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 non-identity is `/=`) and Lua (physical (non)identity is `==` and `~=`).
xcell == wcell
xcell == zcell
- If we express qualitative identity using `=`, as OCaml does, then all of the salient comparisons would be true:
+ If we express qualitative indiscernibility using `=`, as OCaml does, then all of the salient comparisons would be true:
ycell = wcell
ycell = zcell
Of course, in most languages you wouldn't be able to evaluate a comparison like `getter = getter'`, because in general the question whether two computations always return the same values for the same argument 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* numerically equivalent to `getter'` (or `adder'`).
+ 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...
+* 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