3 Polymorphic Types and System F
4 ------------------------------
6 [Notes still to be added. Hope you paid attention during seminar.]
10 1. Product or record types, e.g. pairs and triples
11 2. Sum or variant types; tagged or "disjoint" unions
12 3. Maybe/option types; representing "out-of-band" values
15 6. Inductive types (numbers, lists)
16 7. "Pattern-matching" or type unpacking<p>
17 8. The simply-typed lambda calculus<p>
18 9. Parametric polymorphism, System F, "type inference"<p>
19 10. [Phil/ling application] inner/outer domain semantics for positive free logic
20 <http://philosophy.ucdavis.edu/antonelli/papers/pegasus-JPL.pdf>
21 11. [Phil/ling application] King vs Schiffer in King 2007, pp 103ff. [which paper?](http://rci.rutgers.edu/~jeffreck/pub.php)
22 12. [Phil/ling application] King and Pryor on that clauses, predicates vs singular property-designators
23 13. Possible excursion: [Frege's "On Concept and Object"](http://www.persiangig.com/pages/download/?dl=http://sahmir.persiangig.com/document/Frege%27s%20Articles/On%20Concept%20And%20object%20%28Jstore%29.pdf)<p>
31 OCaml has type inference: the system can often infer what the type of
32 an expression must be, based on the type of other known expressions.
34 For instance, if we type
38 The system replies with
40 val f : int -> int = <fun>
42 Since `+` is only defined on integers, it has type
45 - : int -> int -> int = <fun>
47 The parentheses are there to turn off the trick that allows the two
48 arguments of `+` to surround it in infix (for linguists, SOV) argument
54 In general, tuples with one element are identical to their one
60 though OCaml, like many systems, refuses to try to prove whether two
61 functional objects may be identical:
64 Exception: Invalid_argument "equal: functional value".
68 [Note: There is a limited way you can compare functions, using the
69 `==` operator instead of the `=` operator. Later when we discuss mutation,
70 we'll discuss the difference between these two equality operations.
71 Scheme has a similar pair, which they name `eq?` and `equal?`. In Python,
72 these are `is` and `==` respectively. It's unfortunate that OCaml uses `==` for the opposite operation that Python and many other languages use it for. In any case, OCaml will accept `(f) == f` even though it doesn't accept
73 `(f) = f`. However, don't expect it to figure out in general when two functions
74 are equivalent. (That question is not Turing computable.)
76 # (f) == (fun x -> x + 3);;
79 Here OCaml says (correctly) that the two functions don't stand in the `==` relation, which basically means they're not represented in the same chunk of memory. However as the programmer can see, the functions are extensionally equivalent. The meaning of `==` is rather weird.]
83 Booleans in OCaml, and simple pattern matching
84 ----------------------------------------------
86 Where we would write `true 1 2` in our pure lambda calculus and expect
87 it to evaluate to `1`, in OCaml boolean types are not functions
88 (equivalently, they're functions that take zero arguments). Instead, selection is
89 accomplished as follows:
91 # if true then 1 else 2;;
94 The types of the `then` clause and of the `else` clause must be the
97 The `if` construction can be re-expressed by means of the following
98 pattern-matching expression:
100 match <bool expression> with true -> <expression1> | false -> <expression2>
104 # match true with true -> 1 | false -> 2;;
109 # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
115 All functions in OCaml take exactly one argument. Even this one:
117 # let f x y = x + y;;
121 Here's how to tell that `f` has been curry'd:
124 - : int -> int = <fun>
126 After we've given our `f` one argument, it returns a function that is
127 still waiting for another argument.
129 There is a special type in OCaml called `unit`. There is exactly one
130 object in this type, written `()`. So
135 Just as you can define functions that take constants for arguments
141 you can also define functions that take the unit as its argument, thus
144 val f : unit -> int = <fun>
146 Then the only argument you can possibly apply `f` to that is of the
147 correct type is the unit:
152 Now why would that be useful?
154 Let's have some fun: think of `rec` as our `Y` combinator. Then
156 # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
157 val f : int -> int = <fun>
161 We can't define a function that is exactly analogous to our ω.
162 We could try `let rec omega x = x x;;` what happens?
164 [Note: if you want to learn more OCaml, you might come back here someday and try:
167 val id : 'a -> 'a = <fun>
168 # let unwrap (`Wrap a) = a;;
169 val unwrap : [< `Wrap of 'a ] -> 'a = <fun>
170 # let omega ((`Wrap x) as y) = x y;;
171 val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b = <fun>
172 # unwrap (omega (`Wrap id)) == id;;
174 # unwrap (omega (`Wrap omega));;
175 <Infinite loop, need to control-c to interrupt>
177 But we won't try to explain this now.]
180 Even if we can't (easily) express omega in OCaml, we can do this:
182 # let rec blackhole x = blackhole x;;
184 By the way, what's the type of this function?
186 If you then apply this `blackhole` function to an argument,
190 the interpreter goes into an infinite loop, and you have to type control-c
193 Oh, one more thing: lambda expressions look like this:
197 # (fun x -> x) true;;
200 (But `(fun x -> x x)` still won't work.)
202 You may also see this:
204 # (function x -> x);;
207 This works the same as `fun` in simple cases like this, and slightly differently in more complex cases. If you learn more OCaml, you'll read about the difference between them.
209 We can try our usual tricks:
211 # (fun x -> true) blackhole;;
214 OCaml declined to try to fully reduce the argument before applying the
215 lambda function. Question: Why is that? Didn't we say that OCaml is a call-by-value/eager language?
217 Remember that `blackhole` is a function too, so we can
218 reverse the order of the arguments:
220 # blackhole (fun x -> true);;
224 Now consider the following variations in behavior:
226 # let test = blackhole blackhole;;
227 <Infinite loop, need to control-c to interrupt>
229 # let test () = blackhole blackhole;;
230 val test : unit -> 'a = <fun>
233 - : unit -> 'a = <fun>
236 <Infinite loop, need to control-c to interrupt>
238 We can use functions that take arguments of type `unit` to control
239 execution. In Scheme parlance, functions on the `unit` type are called
240 *thunks* (which I've always assumed was a blend of "think" and "chunk").
242 Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like:
244 let f = fun () -> blackhole ()
249 Bottom type, divergence
250 -----------------------
252 Expressions that don't terminate all belong to the **bottom type**. This is a subtype of every other type. That is, anything of bottom type belongs to every other type as well. More advanced type systems have more examples of subtyping: for example, they might make `int` a subtype of `real`. But the core type system of OCaml doesn't have any general subtyping relations. (Neither does System F.) Just this one: that expressions of the bottom type also belong to every other type. It's as if every type definition in OCaml, even the built in ones, had an implicit extra clause:
254 type 'a option = None | Some of 'a;;
255 type 'a option = None | Some of 'a | bottom;;
257 Here are some exercises that may help better understand this. Figure out what is the type of each of the following:
265 let rec blackhole x = blackhole x in blackhole;;
267 let rec blackhole x = blackhole x in blackhole 1;;
269 let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;;
271 let rec blackhole x = blackhole x in (blackhole 1) + 2;;
273 let rec blackhole x = blackhole x in (blackhole 1) || false;;
275 let rec blackhole x = blackhole x in 2 :: (blackhole 1);;
277 By the way, what's the type of this:
279 let rec blackhole (x:'a) : 'a = blackhole x in blackhole
282 Back to thunks: the reason you'd want to control evaluation with thunks is to
283 manipulate when "effects" happen. In a strongly normalizing system, like the
284 simply-typed lambda calculus or System F, there are no "effects." In Scheme and
285 OCaml, on the other hand, we can write programs that have effects. One sort of
286 effect is printing (think of the [[damn]] example at the start of term).
287 Another sort of effect is mutation, which we'll be looking at soon.
288 Continuations are yet another sort of effect. None of these are yet on the
289 table though. The only sort of effect we've got so far is *divergence* or
290 non-termination. So the only thing thunks are useful for yet is controlling
291 whether an expression that would diverge if we tried to fully evaluate it does
292 diverge. As we consider richer languages, thunks will become more useful.
298 This has now been moved to [its own page](/towards_monads).