4 Towards Monads: Safe division
5 -----------------------------
7 Integer division presupposes that its second argument
8 (the divisor) is not zero, upon pain of presupposition failure.
9 Here's what my OCaml interpreter says:
12 Exception: Division_by_zero.
14 So we want to explicitly allow for the possibility that
15 division will return something other than a number.
16 We'll use OCaml's `option` type, which works like this:
18 # type 'a option = None | Some of 'a;;
22 - : int option = Some 3
24 So if a division is normal, we return some number, but if the divisor is
25 zero, we return `None`. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
28 let div' (x:int) (y:int) =
34 val div' : int -> int -> int option = fun
36 - : int option = Some 6
39 # div' (div' 12 2) 3;;
43 Error: This expression has type int option
44 but an expression was expected of type int
48 This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
49 just the behavior we were hoping for. But we want to be able to use
50 the output of the safe-division function as input for further division
51 operations. So we have to jack up the types of the inputs:
54 let div' (u:int option) (v:int option) =
58 | Some y -> (match u with
60 | Some x -> Some (x / y));;
63 val div' : int option -> int option -> int option = <fun>
64 # div' (Some 12) (Some 2);;
65 - : int option = Some 6
66 # div' (Some 12) (Some 0);;
68 # div' (div' (Some 12) (Some 0)) (Some 3);;
73 Beautiful, just what we need: now we can try to divide by anything we
74 want, without fear that we're going to trigger any system errors.
76 I prefer to line up the `match` alternatives by using OCaml's
80 let div' (u:int option) (v:int option) =
85 | (Some x, Some y) -> Some (x / y);;
88 So far so good. But what if we want to combine division with
89 other arithmetic operations? We need to make those other operations
90 aware of the possibility that one of their arguments has triggered a
91 presupposition failure:
94 let add' (u:int option) (v:int option) =
98 | (Some x, Some y) -> Some (x + y);;
101 val add' : int option -> int option -> int option = <fun>
102 # add' (Some 12) (Some 4);;
103 - : int option = Some 16
104 # add' (div' (Some 12) (Some 0)) (Some 4);;
105 - : int option = None
109 This works, but is somewhat disappointing: the `add'` operation
110 doesn't trigger any presupposition of its own, so it is a shame that
111 it needs to be adjusted because someone else might make trouble.
113 But we can automate the adjustment. The standard way in OCaml,
114 Haskell, etc., is to define a `bind` operator (the name `bind` is not
115 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.
118 let bind' (u: int option) (f: int -> (int option)) =
123 let add' (u: int option) (v: int option) =
124 bind' u (fun x -> bind' v (fun y -> Some (x + y)));;
126 let div' (u: int option) (v: int option) =
127 bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));;
130 # div' (div' (Some 12) (Some 2)) (Some 3);;
131 - : int option = Some 2
132 # div' (div' (Some 12) (Some 0)) (Some 3);;
133 - : int option = None
134 # add' (div' (Some 12) (Some 0)) (Some 3);;
135 - : int option = None
139 Compare the new definitions of `add'` and `div'` closely: the definition
140 for `add'` shows what it looks like to equip an ordinary operation to
141 survive in dangerous presupposition-filled world. Note that the new
142 definition of `add'` does not need to test whether its arguments are
143 None objects or real numbers---those details are hidden inside of the
146 The definition of `div'` shows exactly what extra needs to be said in
147 order to trigger the no-division-by-zero presupposition.
149 For linguists: this is a complete theory of a particularly simply form
150 of presupposition projection (every predicate is a hole).
158 Start by (re)reading the discussion of monads in the lecture notes for
159 week 6 [[Towards Monads]].
160 In those notes, we saw a way to separate thinking about error
161 conditions (such as trying to divide by zero) from thinking about
162 normal arithmetic computations. We did this by making use of the
163 `option` type: in each place where we had something of type `int`, we
164 put instead something of type `int option`, which is a sum type
165 consisting either of one choice with an `int` payload, or else a `None`
166 choice which we interpret as signaling that something has gone wrong.
168 The goal was to make normal computing as convenient as possible: when
169 we're adding or multiplying, we don't have to worry about generating
170 any new errors, so we do want to think about the difference between
171 `int`s and `int option`s. We tried to accomplish this by defining a
172 `bind` operator, which enabled us to peel away the `option` husk to get
173 at the delicious integer inside. There was also a homework problem
174 which made this even more convenient by mapping any binary operation
175 on plain integers into a lifted operation that understands how to deal
176 with `int option`s in a sensible way.
178 [Linguitics note: Dividing by zero is supposed to feel like a kind of
179 presupposition failure. If we wanted to adapt this approach to
180 building a simple account of presupposition projection, we would have
181 to do several things. First, we would have to make use of the
182 polymorphism of the `option` type. In the arithmetic example, we only
183 made use of `int option`s, but when we're composing natural language
184 expression meanings, we'll need to use types like `N option`, `Det option`,
185 `VP option`, and so on. But that works automatically, because we can use
186 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
187 theory of accommodation, and a theory of the situations in which
188 material within the sentence can satisfy presuppositions for other
189 material that otherwise would trigger a presupposition violation; but,
190 not surprisingly, these refinements will require some more
191 sophisticated techniques than the super-simple option monad.]
193 So what exactly is a monad? We can consider a monad to be a system
194 that provides at least the following three elements:
196 * A complex type that's built around some more basic type. Usually
197 the complex type will be polymorphic, and so can apply to different basic types.
198 In our division example, the polymorphism of the `'a option` type
199 provides a way of building an option out of any other type of object.
200 People often use a container metaphor: if `u` has type `int option`,
201 then `u` is a box that (may) contain an integer.
203 type 'a option = None | Some of 'a;;
205 * A way to turn an ordinary value into a monadic value. In OCaml, we
206 did this for any integer `x` by mapping it to
207 the option `Some x`. In the general case, this operation is
208 known as `unit` or `return.` Both of those names are terrible. This
209 operation is only very loosely connected to the `unit` type we were
210 discussing earlier (whose value is written `()`). It's also only
211 very loosely connected to the "return" keyword in many other
212 programming languages like C. But these are the names that the literature
215 The unit/return operation is a way of lifting an ordinary object into
216 the monadic box you've defined, in the simplest way possible. You can think
217 of the singleton function as an example: it takes an ordinary object
218 and returns a set containing that object. In the example we've been
221 let unit x = Some x;;
222 val unit : 'a -> 'a option = <fun>
224 So `unit` is a way to put something inside of a monadic box. It's crucial
225 to the usefulness of monads that there will be monadic boxes that
226 aren't the result of that operation. In the option/maybe monad, for
227 instance, there's also the empty box `None`. In another (whimsical)
228 example, you might have, in addition to boxes merely containing integers,
229 special boxes that contain integers and also sing a song when they're opened.
231 The unit/return operation will always be the simplest, conceptually
232 most straightforward way to lift an ordinary value into a monadic value
233 of the monadic type in question.
235 * Thirdly, an operation that's often called `bind`. This is another
236 unfortunate name: this operation is only very loosely connected to
237 what linguists usually mean by "binding." In our option/maybe monad, the
240 let bind u f = match u with None -> None | Some x -> f x;;
241 val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
243 Note the type: `bind` takes two arguments: first, a monadic box
244 (in this case, an `'a option`); and second, a function from
245 ordinary objects to monadic boxes. `bind` then returns a monadic
246 value: in this case, a `'b option` (you can start with, e.g., `int option`s
247 and end with `bool option`s).
249 Intuitively, the interpretation of what `bind` does is this:
250 the first argument is a monadic value `u`, which
251 evaluates to a box that (maybe) contains some ordinary value, call it `x`.
252 Then the second argument uses `x` to compute a new monadic
253 value. Conceptually, then, we have
255 let bind u f = (let x = unbox u in f x);;
257 The guts of the definition of the `bind` operation amount to
258 specifying how to unbox the monadic value `u`. In the `bind`
259 operator for the option monad, we unboxed the monadic value by
260 matching it with the pattern `Some x`---whenever `u`
261 happened to be a box containing an integer `x`, this allowed us to
262 get our hands on that `x` and feed it to `f`.
264 If the monadic box didn't contain any ordinary value,
265 we instead pass through the empty box unaltered.
267 In a more complicated case, like our whimsical "singing box" example
268 from before, if the monadic value happened to be a singing box
269 containing an integer `x`, then the `bind` operation would probably
270 be defined so as to make sure that the result of `f x` was also
271 a singing box. If `f` also wanted to insert a song, you'd have to decide
272 whether both songs would be carried through, or only one of them.
274 There is no single `bind` function that dictates how this must go.
275 For each new monadic type, this has to be worked out in an
278 So the "option/maybe monad" consists of the polymorphic `option` type, the
279 `unit`/return function, and the `bind` function.
282 A note on notation: Haskell uses the infix operator `>>=` to stand
283 for `bind`. Chris really hates that symbol. Following Wadler, he prefers to
284 use an infix five-pointed star ⋆, or on a keyboard, `*`. Jim on the other hand
285 thinks `>>=` is what the literature uses and students won't be able to
286 avoid it. Moreover, although ⋆ is OK (though not a convention that's been picked up), overloading the multiplication symbol invites its own confusion
287 and Jim feels very uneasy about that. If not `>>=` then we should use
288 some other unfamiliar infix symbol (but `>>=` already is such...)
290 In any case, the course leaders will work this out somehow. In the meantime,
291 as you read around, wherever you see `u >>= f`, that means `bind u f`. Also,
292 if you ever see this notation:
298 That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`.
306 is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u (fun x
307 -> bind v (fun y -> f x y))`. Those who did last week's homework may recognize
308 this last expression.
310 (Note that the above "do" notation comes from Haskell. We're mentioning it here
311 because you're likely to see it when reading about monads. It won't work in
312 OCaml. In fact, the `<-` symbol already means something different in OCaml,
313 having to do with mutable record fields. We'll be discussing mutation someday
316 As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the list monad. Here the monadic type is:
320 The `unit`/return operation is:
323 val unit : 'a -> 'a list = <fun>
325 That is, the simplest way to lift an `'a` into an `'a list` is just to make a
326 singleton list of that `'a`. Finally, the `bind` operation is:
328 # let bind u f = List.concat (List.map f u);;
329 val bind : 'a list -> ('a -> 'b list) -> 'b list = <fun>
331 What's going on here? Well, consider `List.map f u` first. This goes through all
332 the members of the list `u`. There may be just a single member, if `u = unit x`
333 for some `x`. Or on the other hand, there may be no members, or many members. In
334 any case, we go through them in turn and feed them to `f`. Anything that gets fed
335 to `f` will be an `'a`. `f` takes those values, and for each one, returns a `'b list`.
336 For example, it might return a list of all that value's divisors. Then we'll
337 have a bunch of `'b list`s. The surrounding `List.concat ( )` converts that bunch
338 of `'b list`s into a single `'b list`:
340 # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
341 - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
343 So now we've seen two monads: the option/maybe monad, and the list monad. For any
344 monadic system, there has to be a specification of the complex monad type,
345 which will be parameterized on some simpler type `'a`, and the `unit`/return
346 operation, and the `bind` operation. These will be different for different
349 Many monadic systems will also define special-purpose operations that only make
350 sense for that system.
352 Although the `unit` and `bind` operation are defined differently for different
353 monadic systems, there are some general rules they always have to follow.
359 Just like good robots, monads must obey three laws designed to prevent
360 them from hurting the people that use them or themselves.
362 * **Left identity: unit is a left identity for the bind operation.**
363 That is, for all `f:'a -> 'a m`, where `'a m` is a monadic
364 type, we have `(unit x) * f == f x`. For instance, `unit` is itself
365 a function of type `'a -> 'a m`, so we can use it for `f`:
367 # let unit x = Some x;;
368 val unit : 'a -> 'a option = <fun>
369 # let ( * ) u f = match u with None -> None | Some x -> f x;;
370 val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
372 The parentheses is the magic for telling OCaml that the
373 function to be defined (in this case, the name of the function
374 is `*`, pronounced "bind") is an infix operator, so we write
375 `u * f` or `( * ) u f` instead of `* u f`. Now:
378 - : int option = Some 2
380 - : int option = Some 2
382 # let divide x y = if 0 = y then None else Some (x/y);;
383 val divide : int -> int -> int option = <fun>
385 - : int option = Some 3
386 # unit 2 * divide 6;;
387 - : int option = Some 3
390 - : int option = None
391 # unit 0 * divide 6;;
392 - : int option = None
395 * **Associativity: bind obeys a kind of associativity**. Like this:
397 (u * f) * g == u * (fun x -> f x * g)
399 If you don't understand why the lambda form is necessary (the "fun
400 x" part), you need to look again at the type of `bind`.
402 Some examples of associativity in the option monad:
404 # Some 3 * unit * unit;;
405 - : int option = Some 3
406 # Some 3 * (fun x -> unit x * unit);;
407 - : int option = Some 3
409 # Some 3 * divide 6 * divide 2;;
410 - : int option = Some 1
411 # Some 3 * (fun x -> divide 6 x * divide 2);;
412 - : int option = Some 1
414 # Some 3 * divide 2 * divide 6;;
415 - : int option = None
416 # Some 3 * (fun x -> divide 2 x * divide 6);;
417 - : int option = None
419 Of course, associativity must hold for *arbitrary* functions of
420 type `'a -> 'a m`, where `m` is the monad type. It's easy to
421 convince yourself that the `bind` operation for the option monad
422 obeys associativity by dividing the inputs into cases: if `u`
423 matches `None`, both computations will result in `None`; if
424 `u` matches `Some x`, and `f x` evalutes to `None`, then both
425 computations will again result in `None`; and if the value of
426 `f x` matches `Some y`, then both computations will evaluate
429 * **Right identity: unit is a right identity for bind.** That is,
430 `u * unit == u` for all monad objects `u`. For instance,
433 - : int option = Some 3
438 More details about monads
439 -------------------------
441 If you studied algebra, you'll remember that a *monoid* is an
442 associative operation with a left and right identity. For instance,
443 the natural numbers along with multiplication form a monoid with 1
444 serving as the left and right identity. That is, temporarily using
445 `*` to mean arithmetic multiplication, `1 * u == u == u * 1` for all
446 `u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`. As
447 presented here, a monad is not exactly a monoid, because (unlike the
448 arguments of a monoid operation) the two arguments of the bind are of
449 different types. But it's possible to make the connection between
450 monads and monoids much closer. This is discussed in [Monads in Category
451 Theory](/advanced_notes/monads_in_category_theory).
452 See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
454 Here are some papers that introduced monads into functional programming:
456 * [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991.
458 * [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
459 in M. Broy, editor, *Marktoberdorf Summer School on Program Design
460 Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems
461 sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer,
462 editors, *Advanced Functional Programming*, Springer Verlag,
463 LNCS 925, 1995. Some errata fixed August 2001. This paper has a great first
464 line: **Shall I be pure, or impure?**
465 <!-- The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.-->
467 * [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
468 invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
469 <!-- This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
470 Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
471 The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.-->
473 * [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from <https://www.cs.indiana.edu/cgi-pub/c311/doku.php?id=home> but the link above is to a local copy.
475 There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
477 In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category
478 Theory](/advanced_notes/monads_in_category_theory) notes do so, for example.
480 Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
482 You may sometimes see:
494 You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.
496 The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type. Then we "lift" that function into an operation on the monad. For example:
498 # let even x = (x mod 2 = 0);;
499 val g : int -> bool = <fun>
501 `even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
503 # let lift g = fun u -> bind u (fun x -> Some (g x));;
504 val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
506 `lift even` will now be a function from `int option`s to `bool option`s. We can
507 also define a lift operation for binary functions:
509 # let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));;
510 val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = <fun>
512 `lift2 (+)` will now be a function from `int option`s and `int option`s to `int option`s. This should look familiar to those who did the homework.
514 The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the list monad, `lift f` is exactly `List.map f`!
516 Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
518 In general, any lift/map operation can be relied on to satisfy these laws:
521 * lift (compose f g) = compose (lift f) (lift g)
523 where `id` is `fun x -> x` and `compose f g` is `fun x -> f (g x)`. If you think about the special case of the map operation on lists, this should make sense. `List.map id lst` should give you back `lst` again. And you'd expect these
524 two computations to give the same result:
526 List.map (fun x -> f (g x)) lst
527 List.map f (List.map g lst)
529 Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this:
531 ap [f] [x; y] = [f x; f y]
532 ap (Some f) (Some x) = Some (f x)
534 and so on. Here are the laws that any `ap` operation can be relied on to satisfy:
537 ap (ap (ap (unit compose) u) v) w = ap u (ap v w)
538 ap (unit f) (unit x) = unit (f x)
539 ap u (unit x) = ap (unit (fun f -> f x)) u
541 Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the list monad, there was a step where
544 [[1]; [1;2]; [1;3]; [1;2;4]]
548 [1; 1; 2; 1; 3; 1; 2; 4]
550 That is the `join` operation.
552 All of these operations can be defined in terms of `bind` and `unit`; or alternatively, some of them can be taken as primitive and `bind` can be defined in terms of them. Here are various interdefinitions:
554 lift f u = u >>= compose unit f
555 lift f u = ap (unit f) u
556 lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y)))
557 lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v
558 ap u v = u >>= (fun f -> lift f v)
559 ap u v = lift2 id u v
561 u >>= f = join (lift f u)
562 u >> v = u >>= (fun _ -> v)
563 u >> v = lift2 (fun _ -> id) u v
570 We're going to be using monads for a number of different things in the
571 weeks to come. The first main application will be the State monad,
572 which will enable us to model mutation: variables whose values appear
573 to change as the computation progresses. Later, we will study the
576 In the meantime, we'll look at several linguistic applications for monads, based
577 on what's called the *reader monad*.
581 ##[[Intensionality monad]]##