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
+types](church-simple-types.pdf). In this paper, Church writes 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
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)
+<code>Ω = (\x.xx)(\x.xx)</code>
Assume Ω has type τ, and `\x.xx` has type σ. Then
because `\x.xx` takes an argument of type σ and returns
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.
+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#
-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.
+The Church numerals are well behaved with respect to types.
+To see this, consider the first three Church numerals (starting with zero):
-Fortunately, the Church numerals are well behaved with respect to
-types. They can all be given the type (σ --> σ) -->
-σ --> σ.
+ \s z . z
+ \s z . s z
+ \s z . s (s z)
+Given the internal structure of the term we are using to represent
+zero, its type must have the form ρ --> σ --> σ for
+some ρ and σ. This type is consistent with term for one,
+but the structure of the definition of one is more restrictive:
+because the first argument (`s`) must apply to the second argument
+(`z`), the type of the first argument must describe a function from
+expressions of type σ to some result type. So we can refine
+ρ by replacing it with the more specific type σ --> τ.
+At this point, the overall type is (σ --> τ) --> σ -->
+σ. Note that this refined type remains compatible with the
+definition of zero. Finally, by examinining the definition of two, we
+see that expressions of type τ must be suitable to serve as
+arguments to functions of type σ --> τ, since the result of
+applying `s` to `z` serves as the argument of `s`. The most general
+way for that to be true is if τ ≡ σ. So at this
+point, we have the overall type of (σ --> σ) --> σ
+--> σ.
+<!-- Make sure there is talk about unification and computation of the
+principle type-->
+## 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]].
+The reason is that ...
-<!--
+Need to digest the following, which is quoted from Oleg's page:
-Mau integrate some mention of this at some point.
-http://okmij.org/ftp/Computation/lambda-calc.html#predecessor
+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.
-Predecessor and lists are not representable in simply typed lambda-calculus
+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.
- 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
+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).
+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.
+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.
--->