determine which expressions can combine with which other expressions.
If a word is a member of the category of prepositions, it had better
not try to combine (merge) with an expression in the category of, say,
-an auxilliary verb, since *under has* is not a well-formed constituent
+an auxilliary verb, since \**under has* is not a well-formed constituent
in English. Likewise, types in formal languages will determine which
expressions can be sensibly combined.
not correspond to any salient syntactic distinctions (as in any
analysis that involves silent type-shifters, such as Herman Hendriks'
theory of quantifier scope, in which expressions change their semantic
-type without any effect on the syntactic expressions they can combine
+type without any effect on the expressions they can combine
with syntactically). We will consider again the relationship between
syntactic types and semantic types later in the course.
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.]
+native setting--but we're getting ahead of our story. Pedantic off.]
There's good news and bad news: the good news is that the simply-typed
lambda calculus is strongly normalizing: every term has a normal form.
(o -> o) -> o -> o
(o -> o -> o) -> o
+#A first glipse of the connection between types and logic
+
+In the simply-typed lambda calculus, we write types like <code>σ
+-> τ</code>. This looks like logical implication. We'll take
+that resemblance seriously when we discuss the Curry-Howard
+correspondence. In the meantime, note that types respect modus
+ponens:
+
+<pre>
+Expression Type Implication
+-----------------------------------
+fn α -> β α ⊃ β
+arg α α
+------ ------ --------
+(fn arg) β β
+</pre>
+
+The implication in the right-hand column is modus ponens, of course.
+
+
#Associativity of types versus terms#
As we have seen many times, in the lambda calculus, function
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. Some of those types can be functions,
-and some of those functions can be (type-restricted) identity
-functions; but a simply-types identity function can never apply to itself.
+In fact, we can't even type the parts of Ω, that is, `ω
+\equiv \x.xx`. 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. Some of
+those types can be functions, and some of those functions can be
+(type-restricted) identity functions; but a simply-types identity
+function can never apply to itself.
#Typing numerals#
## Predecessor and lists are not representable in simply typed lambda-calculus ##
-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
Pierce 2002:547:
let zero = \s z. z in
- let snd = \a b. b in
- let pair = \a b. \v. v a b in
+ let fst = \x y. x in
+ let snd = \x y. y in
+ let pair = \x y . \f . f x y in
let succ = \n s z. s (n s z) in
- let shift = \p. p (\a b. pair (succ a) a)
+ let shift = \p. pair (succ (p fst)) (p fst) in
let pred = \n. n shift (pair zero zero) snd in
-Note that `shift` applies its argument p ("p" for "pair") to a
-function that ignores its second argument---why does it do that? In
-order to understand what this code is doing, it is helpful to go
-through a sample computation, the predecessor of 3:
+Note that `shift` takes a pair `p` as argument, but makes use of only
+the first element of the pair. Why does it do that? In order to
+understand what this code is doing, it is helpful to go through a
+sample computation, the predecessor of 3:
- pred (\s z.s(s(s z)))
- (\s z.s(s(s z))) (\n.n shift (\f.f 0 0) snd)
+ pred 3
+ 3 shift (pair zero zero) snd
+ (\s z.s(s(s z))) shift (pair zero zero) snd
shift (shift (shift (\f.f 0 0))) snd
- shift (shift ((\f.f 0 0) (\a b.pair(succ a) a))) snd
+ shift (shift (pair (succ ((\f.f 0 0) fst)) ((\f.f 0 0) fst))) snd
shift (shift (\f.f 1 0)) snd
shift (\f. f 2 1) snd
(\f. f 3 2) snd
+ snd 3 2
2
At each stage, `shift` sees an ordered pair that contains two numbers
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.
+See Oleg Kiselyov's discussion and works cited there for details:
+[[predecessor and lists can't be represented in the simply-typed
+lambda
+calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]].
Because lists are (in effect) a generalization of the Church numbers,
computing the tail of a list is likewise beyond the reach of the
if a and b are types, <a,b> is a type
So `<e,<e,t>>` and `<s,<<s,e>,t>>` are types. As we have mentioned,
-this paper is the source for the convention in linguistics that a type
-of the form `<a, b>` corresponds to a functional type that we will
-write here as `a -> b`. So the type `<a,b>` is the type of a function
-that maps objects of type `a` onto objects of type `b`.
+Montague's paper is the source for the convention in linguistics that
+a type of the form `<a, b>` corresponds to a functional type that we
+will write here as `a -> b`. So the type `<a, b>` is the type of a
+function that maps objects of type `a` onto objects of type `b`.
Montague gave rules for the types of various logical formulas. Of
particular interest here, he gave the following typing rules for
-functional application and for lambda abstracts:
+functional application and for lambda abstracts, which match the rules
+for the simply-typed lambda calculus exactly:
* If *α* is an expression of type *<a, b>*, and *β* is an
expression of type b, then *α(β)* has type *b*.