X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_simply_typed_lambda.mdwn;h=14e21729d3c573984875fcdd4dca223008622f1a;hp=2be19fff807a09c96e151d2c126aa75b51d09c72;hb=c70ea8653fd391ba094c8a64419f3532af2076ce;hpb=9895cbdfbcb62fcc80c7e8e26c7bf27f9c8c8116 diff --git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn index 2be19fff..14e21729 100644 --- a/topics/_week5_simply_typed_lambda.mdwn +++ b/topics/_week5_simply_typed_lambda.mdwn @@ -81,16 +81,16 @@ types `T`, the smallest set such that * ground types, including `e` and `t`, are in `T` -* for any types σ and τ in `T`, the type σ --> +* for any types σ and τ in `T`, the type σ -> τ is in `T`. For instance, here are some types in `T`: e - e --> t - e --> e --> t - (e --> t) --> t - (e --> t) --> e --> t + e -> t + e -> e -> t + (e -> t) -> t + (e -> t) -> e -> t and so on. @@ -102,28 +102,28 @@ which is the smallest set such that * each type `t` has an infinite set of distinct variables, {x^t}_1, {x^t}_2, {x^t}_3, ... -* If a term `M` has type σ --> τ, and a term `N` has type +* If a term `M` has type σ -> τ, and a term `N` has type σ, then the application `(M N)` has type τ. * If a variable `a` has type σ, and term `M` has type τ, - then the abstract `λ a M` has type σ --> τ. + then the abstract `λ a M` has type σ -> τ. The definitions of types and of typed terms should be highly familiar -to semanticists, except that instead of writing σ --> τ, +to semanticists, except that instead of writing σ -> τ, linguists write <σ, τ>. We will use the arrow notation, since it is more iconic. Some examples (assume that `x` has type `o`): x o - \x.x o --> o + \x.x o -> o ((\x.x) x) o Excercise: write down terms that have the following types: - o --> o --> o - (o --> o) --> o --> o - (o --> o --> o) --> o + o -> o -> o + (o -> o) -> o -> o + (o -> o -> o) -> o #Associativity of types versus terms# @@ -131,7 +131,7 @@ As we have seen many times, in the lambda calculus, function application is left associative, so that `f x y z == (((f x) y) z)`. Types, *THEREFORE*, are right associative: if `x`, `y`, and `z` have types `a`, `b`, and `c`, respectively, then `f` has type -`a --> b --> c --> d == (a --> (b --> (c --> d)))`, where `d` is the +`a -> b -> c -> d == (a -> (b -> (c -> d)))`, where `d` is the type of the complete term. It is a serious faux pas to associate to the left for types. You may @@ -150,9 +150,9 @@ cannot have a type in Λ_T. We can easily see why: Assume Ω has type τ, and `\x.xx` has type σ. Then because `\x.xx` takes an argument of type σ and returns -something of type τ, `\x.xx` must also have type σ --> +something of type τ, `\x.xx` must also have type σ -> τ. By repeating this reasoning, `\x.xx` must also have type -(σ --> τ) --> τ; and so on. Since variables have +(σ -> τ) -> τ; and so on. Since variables have finite types, there is no way to choose a type for the variable `x` that can satisfy all of the requirements imposed on it. @@ -166,29 +166,101 @@ functions; but a simply-types identity function can never apply to itself. #Typing numerals# -The Church numerals are well behaved with respect to types. They can -all be given the type (σ --> σ) --> σ --> σ. - - - +The Church numerals are well behaved with respect to types. +To see this, consider the first three Church numerals (starting with zero): + + \s z . z + \s z . s z + \s z . s (s z) + +Given the internal structure of the term we are using to represent +zero, its type must have the form ρ -> σ -> σ for +some ρ and σ. This type is consistent with term for one, +but the structure of the definition of one is more restrictive: +because the first argument (`s`) must apply to the second argument +(`z`), the type of the first argument must describe a function from +expressions of type σ to some result type. So we can refine +ρ by replacing it with the more specific type σ -> τ. +At this point, the overall type is (σ -> τ) -> σ -> +σ. Note that this refined type remains compatible with the +definition of zero. Finally, by examinining the definition of two, we +see that expressions of type τ must be suitable to serve as +arguments to functions of type σ -> τ, since the result of +applying `s` to `z` serves as the argument of `s`. The most general +way for that to be true is if τ ≡ σ. So at this +point, we have the overall type of (σ -> σ) -> σ +-> σ. + + ## Predecessor and lists are not representable in simply typed lambda-calculus ## -The predecessor of a Church-encoded numeral, or, generally, the encoding of a list with the car and cdr operations are both impossible in the simply typed lambda-calculus. Henk Barendregt's ``The impact of the lambda-calculus in logic and computer science'' (The Bulletin of Symbolic Logic, v3, N2, June 1997) has the following phrase, on p. 186: - -Even for a function as simple as the predecessor lambda definability remained an open problem for a while. From our present knowledge it is tempting to explain this as follows. Although the lambda calculus was conceived as an untyped theory, typeable terms are more intuitive. Now the functions addition and multiplication are defineable by typeable terms, while  and  have characterized the lambda-defineable functions in the (simply) typed lambda calculus and the predecessor is not among them [the story of the removal of Kleene's four wisdom teeth is skipped...] -Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81. - -Since list is a generalization of numeral -- with cons being a successor, append being the addition, tail (aka cdr) being the predecessor -- it follows then the list cannot be encoded in the simply typed lambda-calculus. - -To encode both operations, we need either inductive (generally, recursive) types, or System F with its polymorphism. The first approach is the most common. Indeed, the familiar definition of a list - - data List a = Nil | Cons a (List a) - -gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type). - -Lists can also be represented in System F. As a matter of fact, we do not need the full System F (where the type reconstruction is not decidable). We merely need the extension of the Hindley-Milner system with higher-ranked types, which requires a modicum of type annotations and yet is able to infer the types of all other terms. This extension is supported in Haskell and OCaml. With such an extension, we can represent a list by its fold, as shown in the code below. It is less known that this representation is faithful: we can implement all list operations, including tail, drop, and even zip. - -See also [[Oleg Kiselyov on the predecessor function in the lambda +As Oleg Kiselyov points out, [[predecessor and lists can't be +represented in the simply-typed lambda calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]]. - +This is not because there is any difficulty typing what the functions +involved do "from the outside": for instance, the predecessor function +is a function from numbers to numbers, or τ -> τ, where τ +is our type for Church numbers (i.e., (σ -> σ) -> σ +-> σ). (Though this type will only be correct if we decide that +the predecessor of zero should be a number, perhaps zero.) + +Rather, the problem is that the definition of the function requires +subterms that can't be simply-typed. We'll illustrate with our +implementation of the predecessor, sightly modified in inessential +ways to suit present purposes: + + let zero = \s z. z in + let snd = \a b. b in + let pair = \a b. \v. v a b in + let succ = \n s z. s (n s z) in + let collect = \p. p (\a b. pair (succ a) a) + let pred = \n. n collect (pair zero zero) snd in + +Let's see how far we can get typing these terms. `zero` is the Church +encoding of zero. Using `N` as the type for Church numbers (i.e., +`N ≡ (σ -> σ) -> σ -> σ` for some +σ, `zero` has type `N`. `snd` takes two numbers, and returns +the second, so `snd` has type `N -> N -> N`. Then the type of `pair` +is `N -> N -> (type(snd)) -> N`, that is, `N -> N -> (N -> N -> N) -> +N`. Likewise, `succ` has type `N -> N`, and `collect` has type `pair +-> pair`, where `pair` is the type of an ordered pair of numbers, +namely, `pair ≡ (N -> N -> N) -> N`. So far so good. + +The problem is the way in which `pred` puts these parts together. In +particular, `pred` applies its argument, the number `n`, to the +`collect` function. Since `n` is a number, its type is ```(σ +-> σ) -> σ -> σ```. This means that the type of +`collect` has to match `σ -> σ`. But we +concluded above that the type of `collect` also had to be `pair -> +pair`. Putting these constraints together, it appears that +`σ` must be the type of a pair of numbers. But we +already decided that the type of a pair of numbers is `(N -> N -> N) +-> N`. Here's the difficulty: `N` is shorthand for a type involving +`σ`. If `σ` turns out to depend on +`N`, and `N` depends in turn on `σ`, then +`σ` is a proper subtype of itself, which is not +allowed in the simply-typed lambda calculus. + +The way we got here is that the `pred` function relies on the built-in +right-fold structure of the Church numbers to recursively walk down +the spine of its argument. In order to do that, the argument had to +apply to the `collect` operation. And since `collect` had to be the +sort of operation that manipulates numbers, the infinite regress is +established. + +Now, of course, this is only one of myriad possible implementations of +the predecessor function in the lambda calculus. Could one of them +possibly be simply-typeable? It turns out that this can't be done. +See the works cited by Oleg for details. + +Because lists are (in effect) a generalization of the Church numbers, +computing the tail of a list is likewise beyond the reach of the +simply-typed lambda calculus. + +This result is surprising. It illustrates how recursion is built into +the structure of the Church numbers (and lists). Most importantly for +the discussion of the simply-typed lambda calculus, it demonstrates +that even fairly basic recursive computations are beyond the reach of +a simply-typed system.