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-# System F and recursive types
+[[!toc levels=2]]
-In the simply-typed lambda calculus, we write types like σ
--> τ. This looks like logical implication. We'll take
-that resemblance seriously when we discuss the Curry-Howard
-correspondence. In the meantime, note that types respect modus
-ponens:
+# System F: the polymorphic lambda calculus
-
-Expression Type Implication
-----------------------------------
-fn α -> β α ⊃ β
-arg α α
------- ------ --------
-fn arg β β
-
+The simply-typed lambda calculus is beautifully simple, but it can't
+even express the predecessor function, let alone full recursion. And
+we'll see shortly that there is good reason to be unsatisfied with the
+simply-typed lambda calculus as a way of expressing natural language
+meaning. So we will need to get more sophisticated about types. The
+next step in that journey will be to consider System F.
-The implication in the right-hand column is modus ponens, of course.
+System F was discovered by Girard (the same guy who invented Linear
+Logic), but it was independently proposed around the same time by
+Reynolds, who called his version the *polymorphic lambda calculus*.
+(Reynolds was also an early player in the development of
+continuations.)
-System F is usually attributed to Girard, but was independently
-proposed around the same time by Reynolds. It enhances the
-simply-typed lambda calculus with quantification over types. In
-System F, you can say things like
+System F enhances the simply-typed lambda calculus with abstraction
+over types. Normal lambda abstraction abstracts (binds) an expression
+(a term); type abstraction abstracts (binds) a type.
-Λ α (\x.x):(α -> α)
+In order to state System F, we'll need to adopt the
+notational convention (which will last throughout the rest of the
+course) that "x:α" represents an expression `x`
+whose type is α.
-This says that the identity function maps arguments of type α to
-results of type α, for any choice of α. So the Λ is
-a universal quantifier over types.
+Then System F can be specified as follows:
+ System F:
+ ---------
+ types Ï ::= c | α | Ï1 -> Ï2 | âα.Ï
+ expressions e ::= x | λx:Ï.e | e1 e2 | Îα.e | e [Ï]
+
+In the definition of the types, "`c`" is a type constant. Type
+constants play the role in System F that base types play in the
+simply-typed lambda calculus. So in a lingusitics context, type
+constants might include `e` and `t`. "α" is a type variable. In
+various discussions, type variables are distinguished by using letters
+from the greek alphabet (α, β, etc.), as we do here, or by
+using capital roman letters (X, Y, etc.), or by adding a tick mark
+(`'a`, `'b`, etc.), as in OCaml. "`Ï1 -> Ï2`" is the type of a
+function from expressions of type `Ï1` to expressions of type `Ï2`.
+And "`âα.Ï`" is called a universal type, since it universally
+quantifies over the type variable `α`. You can expect that in
+`âα.Ï`, the type `Ï` will usually have at least one free occurrence of
+`α` somewhere inside of it.
+
+In the definition of the expressions, we have variables "`x`" as usual.
+Abstracts "`λx:Ï.e`" are similar to abstracts in the simply-typed lambda
+calculus, except that they have their shrug variable annotated with a
+type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
+
+In addition to variables, abstracts, and applications, we have two
+additional ways of forming expressions: "`Îα.e`" is called a *type
+abstraction*, and "`e [Ï]`" is called a *type application*. The idea
+is that Λ is a capital λ: just
+like the lower-case λ, Λ binds
+variables in its body, except that unlike λ,
+Λ binds type variables instead of expression
+variables. So in the expression
+
+Λ Î± (λ x:α. x)
+
+the Λ binds the type variable `α` that occurs in
+the λ abstract.
+
+This expression is a polymorphic version of the identity function. It
+defines one general identity function that can be adapted for use with
+expressions of any type. In order to get it ready to apply this
+identity function to, say, a variable of type boolean, just do this:
+
+(Λ Î± (λ x:α. x)) [t]
+
+This type application (where `t` is a type constant for Boolean truth
+values) specifies the value of the type variable `α`. Not
+surprisingly, the type of the expression that results from this type
+application is a function from Booleans to Booleans:
+
+((Λα (λ x:α . x)) [t]): (b->b)
+
+Likewise, if we had instantiated the type variable as an entity (base
+type `e`), the resulting identity function would have been a function
+of type `e -> e`:
+
+((Λα (λ x:α. x)) [e]): (e->e)
+
+Clearly, for any choice of a type `α`, the identity function can be
+instantiated as a function from expresions of type `α` to expressions
+of type `α`. In general, then, the type of the uninstantiated
+(polymorphic) identity function is
+
+(Λα (λx:α . x)): (∀α. α->α)
+
+Pred in System F
+----------------
+
+We saw that the predecessor function couldn't be expressed in the
+simply-typed lambda calculus. It *can* be expressed in System F,
+however. Here is one way:
+
+ let N = âα.(α->α)->α->α in
+ let Pair = (N->N->N)->N in
+
+ let zero = Îα. λs:α->α. λz:α. z in
+ let fst = λx:N. λy:N. x in
+ let snd = λx:N. λy:N. y in
+ let pair = λx:N. λy:N. λz:N->N->N. z x y in
+ let succ = λn:N. Îα. λs:α->α. λz:α. s (n [α] s z) in
+ let shift = λp:Pair. pair (succ (p fst)) (p fst) in
+ let pred = λn:N. n [Pair] shift (pair zero zero) snd in
+
+ pre (suc (suc (suc zero)));
+
+[If you want to run this code in
+[[Benjamin Pierce's type-checker and evaluator for
+System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]], the
+relevant evaluator is called "fullpoly", and you'll need to
+truncate the names of "suc(c)" and "pre(d)", since those are
+reserved words in Pierce's system.]
+
+Exercise: convince yourself that `zero` has type `N`.
+
+The key to the extra expressive power provided by System F is evident
+in the typing imposed by the definition of `pred`. The variable `n`
+is typed as a Church number, i.e., as `N ≡ âα.(α->α)->α->α`.
+The type application `n [Pair]` instantiates `n` in a way that allows
+it to manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`.
+In other words, the instantiation turns a Church number into a certain
+pair-manipulating function, which is the heart of the strategy for
+this version of computing the predecessor function.
+
+Could we try to accommodate the needs of the predecessor function by
+building a system for doing Church arithmetic in which the type for
+numbers always manipulated ordered pairs? The problem is that the
+ordered pairs we need here are pairs of numbers. If we tried to
+replace the type for Church numbers with a concrete (simple) type, we
+would have to replace each `N` with the type for Pairs, `(N -> N -> N)
+-> N`. But then we'd have to replace each of these `N`'s with the
+type for Church numbers, which we're imagining is `(Pair -> Pair) ->
+Pair -> Pair`. And then we'd have to replace each of these `Pairs`'s
+with... ad infinitum. If we had to choose a concrete type built
+entirely from explicit base types, we'd be unable to proceed.
+
+[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
+Press, chapter 23.]
+
+Typing ω
+--------------
+
+In fact, unlike in the simply-typed lambda calculus,
+it is even possible to give a type for ω in System F.
+
+ω = λx:(âα.α->α). x [âα.α->α] x
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+ω id ≡ (λx:(âα.α->α). x [âα.α->α] x) (Îα.λx:α.x)
+
+Since the type of the identity function is `âα.α->α`, it's the
+right type to serve as the argument to ω. The definition of
+ω instantiates the identity function by binding the type
+variable `α` to the universal type `âα.α->α`. Instantiating the
+identity function in this way results in an identity function whose
+type is (in some sense, only accidentally) the same as the original
+fully polymorphic identity function.
+
+So in System F, unlike in the simply-typed lambda calculus, it *is*
+possible for a function to apply to itself!
+
+Does this mean that we can implement recursion in System F? Not at
+all. In fact, despite its differences with the simply-typed lambda
+calculus, one important property that System F shares with the
+simply-typed lambda calculus is that they are both strongly
+normalizing: *every* expression in either system reduces to a normal
+form in a finite number of steps.
+
+Not only does a fixed-point combinator remain out of reach, we can't
+even construct an infinite loop. This means that although we found a
+type for ω, there is no general type for Ω ≡ ω
+ω. In fact, it turns out that no Turing complete system can be
+strongly normalizing, from which it follows that System F is not
+Turing complete.
+
+
+## Polymorphism in natural language
+
+Is the simply-typed lambda calclus enough for analyzing natural
+language, or do we need polymorphic types? Or something even more expressive?
+
+The classic case study motivating polymorphism in natural language
+comes from coordination. (The locus classicus is Partee and Rooth
+1983.)
+
+ Ann left and Bill left.
+ Ann left and slept.
+ Ann and Bill left.
+ Ann read and reviewed the book.
+
+In English (likewise, many other languages), *and* can coordinate
+clauses, verb phrases, determiner phrases, transitive verbs, and many
+other phrase types. In a garden-variety simply-typed grammar, each
+kind of conjunct has a different semantic type, and so we would need
+an independent rule for each one. Yet there is a strong intuition
+that the contribution of *and* remains constant across all of these
+uses. Can we capture this using polymorphic types?
+
+ Ann, Bill e
+ left, slept e -> t
+ read, reviewed e -> e -> t
+
+With these basic types, we want to say something like this:
+
+ and:t->t->t = λl:t. λr:t. l r false
+ and = Îα.Îβ.λl:α->β.λr:α->β.λx:α. and [β] (l x) (r x)
+
+The idea is that the basic *and* (the one defined in the first line)
+conjoins expressions of type `t`. But when *and* conjoins functional
+types (the definition in the second line), it builds a function that
+distributes its argument across the two conjuncts and conjoins the two
+results. The intention is that `Ann left and slept` will evaluate to
+`(\x.and(left x)(slept x)) ann`. Following the terminology of Partee
+and Rooth, this strategy of defining the coordination of expressions
+with complex types in terms of the coordination of expressions with
+less complex types is known as Generalized Coordination.
+
+But the definitions just given are not well-formed expressions in
+System F. There are three problems. The first is that we have two
+definitions of the same word. The intention is for one of the
+definitions to be operative when the type of its arguments is type
+`t`, but we have no way of conditioning evaluation on the *type* of an
+argument. The second is that for the polymorphic definition, the term
+*and* occurs inside of the definition. We know how to handle some
+cases of using a function name inside of its own definition in the
+untyped lambda calculus, but System F does not have
+recursion. [Exercise: convince yourself that the fixed-point
+combinator `Y` can't be typed in System F.]
+
+The third problem is more subtle. The defintion as given takes two
+types as parameters: the type of the first argument expected by each
+conjunct, and the type of the result of applying each conjunct to an
+argument of that type. We would like to instantiate the recursive use
+of *and* in the definition by using the result type, so that
+"and [β]" evaluates to the kind of *and* that
+coordinates expressions of type β. But fully instantiating the
+definition as given requires type application to a *pair* of types,
+not to just to a single type. We want to somehow guarantee that β
+will always itself be a complex type. This goes beyond the expressive
+power of System F.
+
+So conjunction and disjunction provide a compelling motivation for
+polymorphism in natural language, but we don't yet have the ability to
+build the polymorphism into a formal system.
+
+And in fact, discussions of generalized coordination in the
+linguistics literature are almost always left as a meta-level
+generalizations over a basic simply-typed grammar. For instance, in
+Hendriks' 1992:74 dissertation, generalized coordination is
+implemented as a method for generating a suitable set of translation
+rules, which are in turn expressed in a simply-typed grammar.
+
+There is some work using System F to express generalizations about
+natural language: Ponvert, Elias. 2005. Polymorphism in English Logical
+Grammar. In *Lambda Calculus Type Theory and Natural Language*: 47--60.
+
+Not incidentally, we're not aware of any programming language that
+makes generalized coordination available, despite is naturalness and
+ubiquity in natural language. That is, coordination in programming
+languages is always at the sentential level. You might be able to
+evaluate `(delete file1) and (delete file2)`, but never `delete (file1
+and file2)`.
+
+We'll return to thinking about generalized coordination as we get
+deeper into types. There will be an analysis in term of continuations
+that will be particularly satisfying.
+
+
+#Types in OCaml
+
+
+OCaml has type inference: the system can often infer what the type of
+an expression must be, based on the type of other known expressions.
+
+For instance, if we type
+
+ # let f x = x + 3;;
+
+The system replies with
+
+ val f : int -> int =
+
+Since `+` is only defined on integers, it has type
+
+ # (+);;
+ - : int -> int -> int =
+
+The parentheses are there to turn off the trick that allows the two
+arguments of `+` to surround it in infix (for linguists, SOV) argument
+order. That is,
+
+ # 3 + 4 = (+) 3 4;;
+ - : bool = true
+
+In general, tuples with one element are identical to their one
+element:
+
+ # (3) = 3;;
+ - : bool = true
+
+though OCaml, like many systems, refuses to try to prove whether two
+functional objects may be identical:
+
+ # (f) = f;;
+ Exception: Invalid_argument "equal: functional value".
+
+Oh well.
+
+[Note: There is a limited way you can compare functions, using the
+`==` operator instead of the `=` operator. Later when we discuss mutation,
+we'll discuss the difference between these two equality operations.
+Scheme has a similar pair, which they name `eq?` and `equal?`. In Python,
+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
+`(f) = f`. However, don't expect it to figure out in general when two functions
+are equivalent. (That question is not Turing computable.)
+
+ # (f) == (fun x -> x + 3);;
+ - : bool = false
+
+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.]
+
+
+
+Booleans in OCaml, and simple pattern matching
+----------------------------------------------
+
+Where we would write `true 1 2` in our pure lambda calculus and expect
+it to evaluate to `1`, in OCaml boolean types are not functions
+(equivalently, they're functions that take zero arguments). Instead, selection is
+accomplished as follows:
+
+ # if true then 1 else 2;;
+ - : int = 1
+
+The types of the `then` clause and of the `else` clause must be the
+same.
+
+The `if` construction can be re-expressed by means of the following
+pattern-matching expression:
+
+ match with true -> | false ->
+
+That is,
+
+ # match true with true -> 1 | false -> 2;;
+ - : int = 1
+
+Compare with
+
+ # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
+ - : int = 9
+
+Unit and thunks
+---------------
+
+All functions in OCaml take exactly one argument. Even this one:
+
+ # let f x y = x + y;;
+ # f 2 3;;
+ - : int = 5
+
+Here's how to tell that `f` has been curry'd:
+
+ # f 2;;
+ - : int -> int =
+
+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 OCaml called `unit`. There is exactly one
+object in this type, written `()`. So
+
+ # ();;
+ - : unit = ()
+
+Just as you can define functions that take constants for arguments
+
+ # let f 2 = 3;;
+ # f 2;;
+ - : int = 3;;
+
+you can also define functions that take the unit as its argument, thus
+
+ # let f () = 3;;
+ val f : unit -> int =
+
+Then the only argument you can possibly apply `f` to that is of the
+correct type is the unit:
+
+ # f ();;
+ - : int = 3
+
+Now why would that be useful?
+
+Let's have some fun: think of `rec` as our `Y` combinator. Then
+
+ # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
+ val f : int -> int =
+ # f 5;;
+ - : int = 120
+
+We can't define a function that is exactly analogous to our ω.
+We could try `let rec omega x = x x;;` what happens?
+
+[Note: if you want to learn more OCaml, you might come back here someday and try:
+
+ # let id x = x;;
+ val id : 'a -> 'a =
+ # let unwrap (`Wrap a) = a;;
+ val unwrap : [< `Wrap of 'a ] -> 'a =
+ # let omega ((`Wrap x) as y) = x y;;
+ val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b =
+ # unwrap (omega (`Wrap id)) == id;;
+ - : bool = true
+ # unwrap (omega (`Wrap omega));;
+
+
+But we won't try to explain this now.]
+
+
+Even if we can't (easily) express omega in OCaml, we can do this:
+
+ # let rec blackhole x = blackhole x;;
+
+By the way, what's the type of this function?
+
+If you then apply this `blackhole` function to an argument,
+
+ # blackhole 3;;
+
+the interpreter goes into an infinite loop, and you have to type control-c
+to break the loop.
+
+Oh, one more thing: lambda expressions look like this:
+
+ # (fun x -> x);;
+ - : 'a -> 'a =
+ # (fun x -> x) true;;
+ - : bool = true
+
+(But `(fun x -> x x)` still won't work.)
+
+You may also see this:
+
+ # (function x -> x);;
+ - : 'a -> 'a =
+
+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.
+
+We can try our usual tricks:
+
+ # (fun x -> true) blackhole;;
+ - : bool = true
+
+OCaml declined to try to fully reduce the argument before applying the
+lambda function. Question: Why is that? Didn't we say that OCaml is a call-by-value/eager language?
+
+Remember that `blackhole` is a function too, so we can
+reverse the order of the arguments:
+
+ # blackhole (fun x -> true);;
+
+Infinite loop.
+
+Now consider the following variations in behavior:
+
+ # let test = blackhole blackhole;;
+
+
+ # let test () = blackhole blackhole;;
+ val test : unit -> 'a =
+
+ # test;;
+ - : unit -> 'a =
+
+ # test ();;
+
+
+We can use functions that take arguments of type `unit` to control
+execution. In Scheme parlance, functions on the `unit` type are called
+*thunks* (which I've always assumed was a blend of "think" and "chunk").
+
+Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like:
+
+ let f = fun () -> blackhole ()
+ in true
+
+terminate?
+
+Bottom type, divergence
+-----------------------
+
+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:
+
+ type 'a option = None | Some of 'a;;
+ type 'a option = None | Some of 'a | bottom;;
+
+Here are some exercises that may help better understand this. Figure out what is the type of each of the following:
+
+ fun x y -> y;;
+
+ fun x (y:int) -> y;;
+
+ fun x y : int -> y;;
+
+ let rec blackhole x = blackhole x in blackhole;;
+
+ let rec blackhole x = blackhole x in blackhole 1;;
+
+ let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;;
+
+ let rec blackhole x = blackhole x in (blackhole 1) + 2;;
+
+ let rec blackhole x = blackhole x in (blackhole 1) || false;;
+
+ let rec blackhole x = blackhole x in 2 :: (blackhole 1);;
+
+By the way, what's the type of this:
+
+ let rec blackhole (x:'a) : 'a = blackhole x in blackhole
+
+
+Back to thunks: the reason you'd want to control evaluation with
+thunks is to manipulate when "effects" happen. In a strongly
+normalizing system, like the simply-typed lambda calculus or System F,
+there are no "effects." In Scheme and OCaml, on the other hand, we can
+write programs that have effects. One sort of effect is printing.
+Another sort of effect is mutation, which we'll be looking at soon.
+Continuations are yet another sort of effect. None of these are yet on
+the table though. The only sort of effect we've got so far is
+*divergence* or non-termination. So the only thing thunks are useful
+for yet is controlling whether an expression that would diverge if we
+tried to fully evaluate it does diverge. As we consider richer
+languages, thunks will become more useful.