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post ass5
[lambda.git]
/
week6.mdwn
diff --git
a/week6.mdwn
b/week6.mdwn
index
b97a09f
..
7a5ae2c
100644
(file)
--- a/
week6.mdwn
+++ b/
week6.mdwn
@@
-1,16
+1,16
@@
[[!toc]]
[[!toc]]
-Types, OC
AML
+Types, OC
aml
------------
------------
-OC
AML
has type inference: the system can often infer what the type of
+OC
aml
has type inference: the system can often infer what the type of
an expression must be, based on the type of other known expressions.
an expression must be, based on the type of other known expressions.
-For instance, if we type
+For instance, if we type
# let f x = x + 3;;
# let f x = x + 3;;
-The system replies with
+The system replies with
val f : int -> int = <fun>
val f : int -> int = <fun>
@@
-32,7
+32,7
@@
element:
# (3) = 3;;
- : bool = true
# (3) = 3;;
- : bool = true
-though OC
AML
, like many systems, refuses to try to prove whether two
+though OC
aml
, like many systems, refuses to try to prove whether two
functional objects may be identical:
# (f) = f;;
functional objects may be identical:
# (f) = f;;
@@
-41,11
+41,11
@@
functional objects may be identical:
Oh well.
Oh well.
-Booleans in OC
AML
, and simple pattern matching
+Booleans in OC
aml
, and simple pattern matching
----------------------------------------------
Where we would write `true 1 2` in our pure lambda calculus and expect
----------------------------------------------
Where we would write `true 1 2` in our pure lambda calculus and expect
-it to evaluate to `1`, in OC
AML
boolean types are not functions
+it to evaluate to `1`, in OC
aml
boolean types are not functions
(equivalently, are functions that take zero arguments). Selection is
accomplished as follows:
(equivalently, are functions that take zero arguments). Selection is
accomplished as follows:
@@
-65,7
+65,7
@@
That is,
# match true with true -> 1 | false -> 2;;
- : int = 1
# match true with true -> 1 | false -> 2;;
- : int = 1
-Compare with
+Compare with
# match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
- : int = 9
# match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
- : int = 9
@@
-73,7
+73,7
@@
Compare with
Unit and thunks
---------------
Unit and thunks
---------------
-All functions in OC
AML
take exactly one argument. Even this one:
+All functions in OC
aml
take exactly one argument. Even this one:
# let f x y = x + y;;
# f 2 3;;
# let f x y = x + y;;
# f 2 3;;
@@
-87,7
+87,7
@@
Here's how to tell that `f` has been curry'd:
After we've given our `f` one argument, it returns a function that is
still waiting for another argument.
After we've given our `f` one argument, it returns a function that is
still waiting for another argument.
-There is a special type in OC
AML
called `unit`. There is exactly one
+There is a special type in OC
aml
called `unit`. There is exactly one
object in this type, written `()`. So
# ();;
object in this type, written `()`. So
# ();;
@@
-112,7
+112,7
@@
correct type is the unit:
Let's have some fn: think of `rec` as our `Y` combinator. Then
Let's have some fn: think of `rec` as our `Y` combinator. Then
- # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
+ # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
val f : int -> int = <fun>
# f 5;;
- : int = 120
val f : int -> int = <fun>
# f 5;;
- : int = 120
@@
-145,7
+145,7
@@
So we can try our usual tricks:
# (fun x -> true) omega;;
- : bool = true
# (fun x -> true) omega;;
- : bool = true
-OC
AML
declined to try to evaluate the argument before applying the
+OC
aml
declined to try to evaluate the argument before applying the
functor. But remember that `omega` is a function too, so we can
reverse the order of the arguments:
functor. But remember that `omega` is a function too, so we can
reverse the order of the arguments:
@@
-176,14
+176,14
@@
Towards Monads
So the integer division operation presupposes that its second argument
(the divisor) is not zero, upon pain of presupposition failure.
So the integer division operation presupposes that its second argument
(the divisor) is not zero, upon pain of presupposition failure.
-Here's what my OC
AML
interpreter says:
+Here's what my OC
aml
interpreter says:
# 12/0;;
Exception: Division_by_zero.
# 12/0;;
Exception: Division_by_zero.
-So we want to explicitly allow for the possibility that
+So we want to explicitly allow for the possibility that
division will return something other than a number.
division will return something other than a number.
-We'll use OC
AML
's option type, which works like this:
+We'll use OC
aml
's option type, which works like this:
# type 'a option = None | Some of 'a;;
# None;;
# type 'a option = None | Some of 'a;;
# None;;
@@
-192,22
+192,22
@@
We'll use OCAML's option type, which works like this:
- : int option = Some 3
So if a division is normal, we return some number, but if the divisor is
- : int option = Some 3
So if a division is normal, we return some number, but if the divisor is
-zero, we return None
:
+zero, we return None
. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
<pre>
<pre>
-let div
(x:int) (y:int) =
+let div
' (x:int) (y:int) =
match y with 0 -> None |
_ -> Some (x / y);;
(*
match y with 0 -> None |
_ -> Some (x / y);;
(*
-val div : int -> int -> int option = fun
-# div 12 3;;
+val div
'
: int -> int -> int option = fun
+# div
'
12 3;;
- : int option = Some 4
- : int option = Some 4
-# div 12 0;;
+# div
'
12 0;;
- : int option = None
- : int option = None
-# div
(div
12 3) 2;;
+# div
' (div'
12 3) 2;;
Characters 4-14:
Characters 4-14:
- div
(div
12 3) 2;;
+ div
' (div'
12 3) 2;;
^^^^^^^^^^
Error: This expression has type int option
but an expression was expected of type int
^^^^^^^^^^
Error: This expression has type int option
but an expression was expected of type int
@@
-220,19
+220,19
@@
the output of the safe-division function as input for further division
operations. So we have to jack up the types of the inputs:
<pre>
operations. So we have to jack up the types of the inputs:
<pre>
-let div
(x:int option) (y:int option) =
+let div
' (x:int option) (y:int option) =
match y with None -> None |
Some 0 -> None |
Some n -> (match x with None -> None |
Some m -> Some (m / n));;
(*
match y with None -> None |
Some 0 -> None |
Some n -> (match x with None -> None |
Some m -> Some (m / n));;
(*
-val div : int option -> int option -> int option = <fun>
-# div (Some 12) (Some 4);;
+val div
'
: int option -> int option -> int option = <fun>
+# div
'
(Some 12) (Some 4);;
- : int option = Some 3
- : int option = Some 3
-# div (Some 12) (Some 0);;
+# div
'
(Some 12) (Some 0);;
- : int option = None
- : int option = None
-# div
(div
(Some 12) (Some 0)) (Some 4);;
+# div
' (div'
(Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
</pre>
- : int option = None
*)
</pre>
@@
-240,74
+240,74
@@
val div : int option -> int option -> int option = <fun>
Beautiful, just what we need: now we can try to divide by anything we
want, without fear that we're going to trigger any system errors.
Beautiful, just what we need: now we can try to divide by anything we
want, without fear that we're going to trigger any system errors.
-I prefer to line up the `match` alternatives by using OC
AML's
+I prefer to line up the `match` alternatives by using OC
aml's
built-in tuple type:
<pre>
built-in tuple type:
<pre>
-let div
(x:int option) (y:int option) =
+let div
' (x:int option) (y:int option) =
match (x, y) with (None, _) -> None |
(_, None) -> None |
(_, Some 0) -> None |
(Some m, Some n) -> Some (m / n);;
</pre>
match (x, y) with (None, _) -> None |
(_, None) -> None |
(_, Some 0) -> None |
(Some m, Some n) -> Some (m / n);;
</pre>
-So far so good. But what if we want to combine division with
-other arithmetic operations? We need to make those other operations
+So far so good. But what if we want to combine division with
+other arithmetic operations? We need to make those other operations
aware of the possibility that one of their arguments will trigger a
presupposition failure:
<pre>
aware of the possibility that one of their arguments will trigger a
presupposition failure:
<pre>
-let add
(x:int option) (y:int option) =
+let add
' (x:int option) (y:int option) =
match (x, y) with (None, _) -> None |
(_, None) -> None |
(Some m, Some n) -> Some (m + n);;
(*
match (x, y) with (None, _) -> None |
(_, None) -> None |
(Some m, Some n) -> Some (m + n);;
(*
-val add : int option -> int option -> int option = <fun>
-# add (Some 12) (Some 4);;
+val add
'
: int option -> int option -> int option = <fun>
+# add
'
(Some 12) (Some 4);;
- : int option = Some 16
- : int option = Some 16
-# add
(div
(Some 12) (Some 0)) (Some 4);;
+# add
' (div'
(Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
</pre>
- : int option = None
*)
</pre>
-This works, but is somewhat disappointing: the `add` operation
+This works, but is somewhat disappointing: the `add
'
` operation
doesn't trigger any presupposition of its own, so it is a shame that
it needs to be adjusted because someone else might make trouble.
doesn't trigger any presupposition of its own, so it is a shame that
it needs to be adjusted because someone else might make trouble.
-But we can automate the adjustment. The standard way in OC
AML
,
+But we can automate the adjustment. The standard way in OC
aml
,
Haskell, etc., is to define a `bind` operator (the name `bind` is not
Haskell, etc., is to define a `bind` operator (the name `bind` is not
-well chosen to resonate with linguists, but what can you do)
:
+well chosen to resonate with linguists, but what can you do)
. To continue our mnemonic association, we'll put a `'` after the name "bind" as well.
<pre>
<pre>
-let bind
(x: int option) (f: int -> (int option)) =
- match x with None -> None |
+let bind
' (x: int option) (f: int -> (int option)) =
+ match x with None -> None |
Some n -> f n;;
Some n -> f n;;
-let add (x: int option) (y: int option) =
- bind
x (fun x -> bind
y (fun y -> Some (x + y)));;
+let add
'
(x: int option) (y: int option) =
+ bind
' x (fun x -> bind'
y (fun y -> Some (x + y)));;
-let div (x: int option) (y: int option) =
- bind
x (fun x -> bind
y (fun y -> if (0 = y) then None else Some (x / y)));;
+let div
'
(x: int option) (y: int option) =
+ bind
' x (fun x -> bind'
y (fun y -> if (0 = y) then None else Some (x / y)));;
(*
(*
-# div
(div
(Some 12) (Some 2)) (Some 4);;
+# div
' (div'
(Some 12) (Some 2)) (Some 4);;
- : int option = Some 1
- : int option = Some 1
-# div
(div
(Some 12) (Some 0)) (Some 4);;
+# div
' (div'
(Some 12) (Some 0)) (Some 4);;
- : int option = None
- : int option = None
-# add
(div
(Some 12) (Some 0)) (Some 4);;
+# add
' (div'
(Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
</pre>
- : int option = None
*)
</pre>
-Compare the new definitions of `add
` and `div
` closely: the definition
-for `add` shows what it looks like to equip an ordinary operation to
+Compare the new definitions of `add
'` and `div'
` closely: the definition
+for `add
'
` shows what it looks like to equip an ordinary operation to
survive in dangerous presupposition-filled world. Note that the new
survive in dangerous presupposition-filled world. Note that the new
-definition of `add` does not need to test whether its arguments are
+definition of `add
'
` does not need to test whether its arguments are
None objects or real numbers---those details are hidden inside of the
None objects or real numbers---those details are hidden inside of the
-`bind` function.
+`bind
'
` function.
-The definition of `div` shows exactly what extra needs to be said in
+The definition of `div
'
` shows exactly what extra needs to be said in
order to trigger the no-division-by-zero presupposition.
For linguists: this is a complete theory of a particularly simply form
order to trigger the no-division-by-zero presupposition.
For linguists: this is a complete theory of a particularly simply form