[[!toc]] ##The simply-typed lambda calculus## The untyped lambda calculus is pure computation. It is much more common, however, for practical programming languages to be typed. 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. Soon we will consider polymorphic type systems. First, however, we will consider the simply-typed lambda calculus. 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, `o`. From a linguistic point of view, think of the ground types as the bar-level 0 categories, that is, the lexical types, such as Noun, Verb, Preposition (glossing over the internal complexity of those categories in modern theories). In addition, there will be a recursively-defined class of complex types `T`, the smallest set such that * ground types, including `o`, 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`: o o --> o o --> o --> o (o --> o) --> o (o --> o) --> o --> o 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 (following Montague, who followed Church) 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 `f`, `x`, `y`, and `z` have types `a`, `b`, `c`, and `d`, respectively, then `f` has type `a --> b --> c --> d == (a --> (b --> (c --> d)))`. 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 `false` has type τ --> τ --> τ, for some τ. Since one is represented by the function `\x.x false 0`, one must have type (τ --> τ --> τ) --> σ --> σ. But this is a different type than zero! Because each number has a different type, it becomes impossible to write arithmetic operations that can combine zero with one. We would need as many different addition operations as we had pairs of numbers that we wanted to add. Fortunately, the Church numberals are well behaved with respect to types. They can all be given the type (σ --> σ) --> σ --> σ.