+[[!toc]]
+
+##The simply-typed lambda calculus##
+
+The untyped lambda calculus is pure. Pure in many ways: all variables
+and lambdas, with no constants or other special symbols; also, all
+functions without any types. As we'll see eventually, pure also in
+the sense of having no side effects, no mutation, just pure
+computation.
+
+But we live in an impure world. It is much more common for practical
+programming languages to be typed, either implicitly or explicitly.
+Likewise, systems used to investigate philosophical or linguistic
+issues are almost always typed. Types will help us reason about our
+computations. They will also facilitate a connection between logic
+and computation.
+
+From a linguistic perspective, types are generalizations of (parts of)
+programs. To make this comment more concrete: types are to (e.g.,
+lambda) terms as syntactic categories are to expressions of natural
+language. If so, if it makes sense to gather a class of expressions
+together into a set of Nouns, or Verbs, it may also make sense to
+gather classes of terms into a set labelled with some computational type.
+
+Soon we will consider polymorphic type systems. First, however, we
+will consider the simply-typed lambda calculus.
+
+[Pedantic on. Why "simply typed"? Well, the type system is
+particularly simple. As mentioned in class by Koji Mineshima, Church
+tells us that "The simple theory of types was suggested as a
+modification of Russell's ramified theory of types by Leon Chwistek in
+1921 and 1922 and by F. P. Ramsey in 1926." This footnote appears in
+Church's 1940 paper [A formulation of the simple theory of
+types](church-simple-types.pdf). In this paper, as Will Starr
+mentioned in class, Church does indeed write types by simple
+apposition, without the ugly angle brackets and commas used by
+Montague. Furthermore, he omits parentheses under the convention that
+types associated to the *left*---the opposite of the modern
+convention. This is ok, however, because he also reverses the order,
+so that `te` is a function from objects of type `e` to objects of type
+`t`. Cool paper! If you ever want to see Church numerals in their
+native setting--but I'm getting ahead of my story. Pedantic off.]
+
+There's good news and bad news: the good news is that the simply-type
+lambda calculus is strongly normalizing: every term has a normal form.
+We shall see that self-application is outlawed, so Ω can't even
+be written, let alone undergo reduction. The bad news is that
+fixed-point combinators are also forbidden, so recursion is neither
+simple nor direct.
+
+#Types#
+
+We will have at least one ground type. For the sake of linguistic
+familiarity, we'll use `e`, the type of individuals, and `t`, the type
+of truth values.
+
+In addition, there will be a recursively-defined class of complex
+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 -->
+ 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
+
+and so on.
+
+#Typed lambda terms#
+
+Given a set of types `T`, we define the set of typed lambda terms <code>Λ_T</code>,
+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
+ 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 <code>λ a M</code> 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,
+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) x) o
+
+Excercise: write down terms that have the following types:
+
+ o --> o --> o
+ (o --> o) --> o --> o
+ (o --> o --> o) --> o
+
+#Associativity of types versus terms#
+
+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
+type of the complete term.
+
+It is a serious faux pas to associate to the left for types. You may
+as well use your salad fork to stir your tea.
+
+#The simply-typed lambda calculus is strongly normalizing#
+
+If `M` is a term with type τ in Λ_T, then `M` has a
+normal form. The proof is not particularly complex, but we will not
+present it here; see Berendregt or Hankin.
+
+Since Ω does not have a normal form, it follows that Ω
+cannot have a type in Λ_T. We can easily see why:
+
+ Ω = (\x.xx)(\x.xx)
+
+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 -->
+τ. By repeating this reasoning, `\x.xx` must also have type
+(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.
+
+In general, there is no way for a function to have a type that can
+take itself for an argument. It follows that there is no way to
+define the identity function in such a way that it can take itself as
+an argument. Instead, there must be many different identity
+functions, one for each type.
+
+#Typing numerals#
+
+Version 1 type numerals are not a good choice for the simply-typed
+lambda calculus. The reason is that each different numberal has a
+different type! For instance, if zero has type σ, then since
+one is represented by the function `\x.x false 0`, it must have type
+`b --> σ --> σ`, where `b` is the type of a boolean. But
+this is a different type than zero! Because each number has a
+different type, it becomes unbearable to write arithmetic operations
+that can combine zero with one, since we would need as many different
+addition operations as we had pairs of numbers that we wanted to add.
+
+Fortunately, the Church numerals are well behaved with respect to
+types. They can all be given the type (σ --> σ) -->
+σ --> σ.
+
+
+
+
+
+<!--
+
+Mau integrate some mention of this at some point.
+
+http://okmij.org/ftp/Computation/lambda-calc.html#predecessor
+
+
+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 [101] and [108] 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.
+
+
+-->