#Typing numerals#
-The Church numerals are well behaved with respect to types. They can
-all be given the type (σ --> σ) --> σ --> σ.
+The Church numerals are well behaved with respect to types.
+To see this, consider the first three Church numerals (starting with zero):
+
+ \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:
-## Predecessor and lists are not representable in 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:
+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.
+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.
+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.
-See also [[Oleg Kiselyov on the predecessor function in the lambda
-calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]].