want to make use of our `'a option` technique, but for this assignment, just
pick a strategy, no matter how clunky.
-Be sure to test your proposals with simple lists. (You'll have to make_list
+Be sure to test your proposals with simple lists. (You'll have to `make_list`
the lists yourself; don't expect OCaml to magically translate between its
native lists and the ones you buil.d)
Baby monads
-----------
-Read the lecture notes for week 6, then write a
-function `lift'` that generalized the correspondence between + and
-`add'`: that is, `lift'` takes any two-place operation on integers
-and returns a version that takes arguments of type `int option`
-instead, returning a result of `int option`. In other words,
-`lift'` will have type
+Read the material on dividing by zero/towards monads from the end of lecture
+notes for week 6, then write a function `lift'` that generalized the
+correspondence between + and `add'`: that is, `lift'` takes any two-place
+operation on integers and returns a version that takes arguments of type `int
+option` instead, returning a result of `int option`. In other words, `lift'`
+will have type:
(int -> int -> int) -> (int option) -> (int option) -> (int option)
-so that `lift' (+) (Some 3) (Some 4)` will evalute to `Some 7`.
+so that `lift' (+) (Some 3) (Some 4)` will evalute to `Some 7`.
Don't worry about why you need to put `+` inside of parentheses.
You should make use of `bind'` in your definition of `lift'`:
- let bind' (x: int option) (f: int -> (int option)) =
- match x with None -> None | Some n -> f n;;
+ let bind' (u: int option) (f: int -> (int option)) =
+ match u with None -> None | Some x -> f x;;
(27 Sept) Lecture notes for [[Week3]]; [[Assignment3]];
an evaluator with the definitions used for homework 3
-preloaded is available at [[assignment 3 evaluator]].
+preloaded is available at [[assignment 3 evaluator]].
> Topics: [[Evaluation Order]]; Recursion with Fixed Point Combinators
(4 Oct) Lecture notes for [[Week4]]; [[Assignment4]].
-> Topics: More on Fixed Points; Sets; Aborting List Traversals; [[Implementing Trees]]
+> Topics: More on Fixed Points; Sets; Aborting List Traversals; [[Implementing Trees]]
(18 Oct, 25 Oct) Lecture notes for [[Week5]] and [[Week6]]; [[Assignment5]].
-> Topics: Types, Polymorphism, Dividing by Zero
+> Topics: Types, Polymorphism, Unit and Bottom, Dividing by Zero/[[Towards Monads]]
(1 Nov) Lecture notes for Week7; Assignment6.
--- /dev/null
+Dividing by zero
+----------------
+
+Integer division operation presupposes that its second argument
+(the divisor) is not zero, upon pain of presupposition failure.
+Here's what my OCaml interpreter says:
+
+ # 12/0;;
+ Exception: Division_by_zero.
+
+So we want to explicitly allow for the possibility that
+division will return something other than a number.
+We'll use OCaml's option type, which works like this:
+
+ # type 'a option = None | Some of 'a;;
+ # None;;
+ - : 'a option = None
+ # Some 3;;
+ - : int option = Some 3
+
+So if a division is normal, we return some number, but if the divisor is
+zero, we return None. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
+
+<pre>
+let div' (x:int) (y:int) =
+ match y with 0 -> None |
+ _ -> Some (x / y);;
+
+(*
+val div' : int -> int -> int option = fun
+# div' 12 3;;
+- : int option = Some 4
+# div' 12 0;;
+- : int option = None
+# div' (div' 12 3) 2;;
+Characters 4-14:
+ div' (div' 12 3) 2;;
+ ^^^^^^^^^^
+Error: This expression has type int option
+ but an expression was expected of type int
+*)
+</pre>
+
+This starts off well: dividing 12 by 3, no problem; dividing 12 by 0,
+just the behavior we were hoping for. But we want to be able to use
+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>
+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));;
+
+(*
+val div' : int option -> int option -> int option = <fun>
+# div' (Some 12) (Some 4);;
+- : int option = Some 3
+# div' (Some 12) (Some 0);;
+- : int option = None
+# div' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+</pre>
+
+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 OCaml's
+built-in tuple type:
+
+<pre>
+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>
+
+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>
+let add' (x:int option) (y:int option) =
+ 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);;
+- : int option = Some 16
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+</pre>
+
+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.
+
+But we can automate the adjustment. The standard way in OCaml,
+Haskell, etc., is to define a `bind` operator (the name `bind` is not
+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>
+let bind' (x: int option) (f: int -> (int option)) =
+ match x with None -> None |
+ Some n -> f n;;
+
+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)));;
+
+(*
+# div' (div' (Some 12) (Some 2)) (Some 4);;
+- : int option = Some 1
+# div' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : 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
+survive in dangerous presupposition-filled world. Note that the new
+definition of `add'` does not need to test whether its arguments are
+None objects or real numbers---those details are hidden inside of the
+`bind'` function.
+
+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
+of presupposition projection (every predicate is a hole).
+
diverge. As we consider richer languages, thunks will become more useful.
+Towards Monads
+--------------
+
+This has now been moved to [its own page](/towards_monads).
-Dividing by zero: Towards Monads
---------------------------------
-
-So the integer division operation presupposes that its second argument
-(the divisor) is not zero, upon pain of presupposition failure.
-Here's what my OCaml interpreter says:
-
- # 12/0;;
- Exception: Division_by_zero.
-
-So we want to explicitly allow for the possibility that
-division will return something other than a number.
-We'll use OCaml's option type, which works like this:
-
- # type 'a option = None | Some of 'a;;
- # None;;
- - : 'a option = None
- # Some 3;;
- - : int option = Some 3
-
-So if a division is normal, we return some number, but if the divisor is
-zero, we return None. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
-
-<pre>
-let div' (x:int) (y:int) =
- match y with 0 -> None |
- _ -> Some (x / y);;
-
-(*
-val div' : int -> int -> int option = fun
-# div' 12 3;;
-- : int option = Some 4
-# div' 12 0;;
-- : int option = None
-# div' (div' 12 3) 2;;
-Characters 4-14:
- div' (div' 12 3) 2;;
- ^^^^^^^^^^
-Error: This expression has type int option
- but an expression was expected of type int
-*)
-</pre>
-
-This starts off well: dividing 12 by 3, no problem; dividing 12 by 0,
-just the behavior we were hoping for. But we want to be able to use
-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>
-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));;
-
-(*
-val div' : int option -> int option -> int option = <fun>
-# div' (Some 12) (Some 4);;
-- : int option = Some 3
-# div' (Some 12) (Some 0);;
-- : int option = None
-# div' (div' (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-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 OCaml's
-built-in tuple type:
-
-<pre>
-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>
-
-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>
-let add' (x:int option) (y:int option) =
- 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);;
-- : int option = Some 16
-# add' (div' (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-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.
-
-But we can automate the adjustment. The standard way in OCaml,
-Haskell, etc., is to define a `bind` operator (the name `bind` is not
-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>
-let bind' (x: int option) (f: int -> (int option)) =
- match x with None -> None |
- Some n -> f n;;
-
-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)));;
-
-(*
-# div' (div' (Some 12) (Some 2)) (Some 4);;
-- : int option = Some 1
-# div' (div' (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-# add' (div' (Some 12) (Some 0)) (Some 4);;
-- : 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
-survive in dangerous presupposition-filled world. Note that the new
-definition of `add'` does not need to test whether its arguments are
-None objects or real numbers---those details are hidden inside of the
-`bind'` function.
-
-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
-of presupposition projection (every predicate is a hole).
[[!toc]]
+
Monads
------
Start by (re)reading the discussion of monads in the lecture notes for
-week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
+week 6 [[Towards Monads]].
In those notes, we saw a way to separate thinking about error
conditions (such as trying to divide by zero) from thinking about
normal arithmetic computations. We did this by making use of the