[[!toc]]
-#These notes return to the topic of fixed point combiantors for one more return to the topic of fixed point combinators#
-
-Q: How do you know that every term in the untyped lambda calculus has
-a fixed point?
+#Q: How do you know that every term in the untyped lambda calculus has a fixed point?#
A: That's easy: let `T` be an arbitrary term in the lambda calculus. If
`T` has a fixed point, then there exists some `X` such that `X <~~>
Please slow down and make sure that you understand what justified each
of the equalities in the last line.
-Q: How do you know that for any term `T`, `YT` is a fixed point of `T`?
+#Q: How do you know that for any term `T`, `YT` is a fixed point of `T`?#
A: Note that in the proof given in the previous answer, we chose `T`
and then set `X = WW = (\x.T(xx))(\x.T(xx))`. If we abstract over
what argument `T` we feed Y, it returns some `X` that is a fixed point
of `T`, by the reasoning in the previous answer.
-Q: So if every term has a fixed point, even `Y` has fixed point.
+#Q: So if every term has a fixed point, even `Y` has fixed point.#
A: Right:
= Y(Y((\x.Y(xx))(\x.Y(xx))))
= Y(Y(Y(...(Y(YY))...)))
-Q: Ouch! Stop hurting my brain.
+#Q: Ouch! Stop hurting my brain.#
A: Let's come at it from the direction of arithmetic. Recall that we
claimed that even `succ`---the function that added one to any
You should see the close similarity with YY here.
-Q. So `Y` applied to `succ` returns a number that is not finite!
+#Q. So `Y` applied to `succ` returns a number that is not finite!#
A. Yes! Let's see why it makes sense to think of `Y succ` as a Church
numeral:
there will always be an opportunity for more beta reduction. (Lather,
rinse, repeat!)
-Q. That reminds me, what about [[evaluation order]]?
+#Q. That reminds me, what about [[evaluation order]]?#
A. For a recursive function that has a well-behaved base case, such as
the factorial function, evaluation order is crucial. In the following
start with the `isZero` predicate, and only produce a fresh copy of
`prefac` if we are forced to.
-Q. You claimed that the Ackerman function couldn't be implemented
-using our primitive recursion techniques (such as the techniques that
-allow us to define addition and multiplication). But you haven't
-shown that it is possible to define the Ackerman function using full
-recursion.
+#Q. You claimed that the Ackerman function couldn't be implemented using our primitive recursion techniques (such as the techniques that allow us to define addition and multiplication). But you haven't shown that it is possible to define the Ackerman function using full recursion.#
A. OK:
A 3 x is to A 2 x as exponentiation is to multiplication---
so A 4 x is to A 3 x as hyper-exponentiation is to exponentiation...
+#Q. What other questions should I be asking?#
+
+* What is it about the variant fixed-point combinators that makes
+ them compatible with a call-by-value evaluation strategy?
+
+* How do you know that the Ackerman function can't be computed
+ using primitive recursion techniques?
+
+* What *exactly* is primitive recursion?
+
+* I hear that `Y` delivers the *least* fixed point. Least
+ according to what ordering? How do you know it's least?
+ Is leastness important?
+
+
+##The simply-typed lambda calculus##
+
+The uptyped 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 <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 (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, on a par
+with using 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, and `false`
+has type `τ --> τ --> τ` for some τ, and one is
+represented by the function `\x.x false 0`, then one must have type
+`(τ --> τ --> &tau) --> &sigma --> σ`. But this is a
+different type than zero! Because numbers have different types, 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`.
+