## 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. 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 Λ_T, 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 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, 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:

 &Omega; = (\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 --> &sigma; --> &sigma;, 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.