3 ##The simply-typed lambda calculus##
5 The untyped lambda calculus is pure. Pure in many ways: all variables
6 and lambdas, with no constants or other special symbols; also, all
7 functions without any types. As we'll see eventually, pure also in
8 the sense of having no side effects, no mutation, just pure
11 But we live in an impure world. It is much more common for practical
12 programming languages to be typed, either implicitly or explicitly.
13 Likewise, systems used to investigate philosophical or linguistic
14 issues are almost always typed. Types will help us reason about our
15 computations. They will also facilitate a connection between logic
18 From a linguistic perspective, types are generalizations of (parts of)
19 programs. To make this comment more concrete: types are to (e.g.,
20 lambda) terms as syntactic categories are to expressions of natural
21 language. If so, if it makes sense to gather a class of expressions
22 together into a set of Nouns, or Verbs, it may also make sense to
23 gather classes of terms into a set labelled with some computational type.
25 Soon we will consider polymorphic type systems. First, however, we
26 will consider the simply-typed lambda calculus.
28 [Pedantic on. Why "simply typed"? Well, the type system is
29 particularly simple. As mentioned in class by Koji Mineshima, Church
30 tells us that "The simple theory of types was suggested as a
31 modification of Russell's ramified theory of types by Leon Chwistek in
32 1921 and 1922 and by F. P. Ramsey in 1926." This footnote appears in
33 Church's 1940 paper [A formulation of the simple theory of
34 types](church-simple-types.pdf). In this paper, as Will Starr
35 mentioned in class, Church does indeed write types by simple
36 apposition, without the ugly angle brackets and commas used by
37 Montague. Furthermore, he omits parentheses under the convention that
38 types associated to the *left*---the opposite of the modern
39 convention. This is ok, however, because he also reverses the order,
40 so that `te` is a function from objects of type `e` to objects of type
41 `t`. Cool paper! If you ever want to see Church numerals in their
42 native setting--but I'm getting ahead of my story. Pedantic off.]
44 There's good news and bad news: the good news is that the simply-type
45 lambda calculus is strongly normalizing: every term has a normal form.
46 We shall see that self-application is outlawed, so Ω can't even
47 be written, let alone undergo reduction. The bad news is that
48 fixed-point combinators are also forbidden, so recursion is neither
53 We will have at least one ground type. For the sake of linguistic
54 familiarity, we'll use `e`, the type of individuals, and `t`, the type
57 In addition, there will be a recursively-defined class of complex
58 types `T`, the smallest set such that
60 * ground types, including `e` and `t`, are in `T`
62 * for any types σ and τ in `T`, the type σ -->
65 For instance, here are some types in `T`:
77 Given a set of types `T`, we define the set of typed lambda terms <code>Λ_T</code>,
78 which is the smallest set such that
80 * each type `t` has an infinite set of distinct variables, {x^t}_1,
83 * If a term `M` has type σ --> τ, and a term `N` has type
84 σ, then the application `(M N)` has type τ.
86 * If a variable `a` has type σ, and term `M` has type τ,
87 then the abstract <code>λ a M</code> has type σ --> τ.
89 The definitions of types and of typed terms should be highly familiar
90 to semanticists, except that instead of writing σ --> τ,
91 linguists write <σ, τ>. We will use the arrow notation,
92 since it is more iconic.
94 Some examples (assume that `x` has type `o`):
100 Excercise: write down terms that have the following types:
103 (o --> o) --> o --> o
104 (o --> o --> o) --> o
106 #Associativity of types versus terms#
108 As we have seen many times, in the lambda calculus, function
109 application is left associative, so that `f x y z == (((f x) y) z)`.
110 Types, *THEREFORE*, are right associative: if `x`, `y`, and `z`
111 have types `a`, `b`, and `c`, respectively, then `f` has type
112 `a --> b --> c --> d == (a --> (b --> (c --> d)))`, where `d` is the
113 type of the complete term.
115 It is a serious faux pas to associate to the left for types. You may
116 as well use your salad fork to stir your tea.
118 #The simply-typed lambda calculus is strongly normalizing#
120 If `M` is a term with type τ in Λ_T, then `M` has a
121 normal form. The proof is not particularly complex, but we will not
122 present it here; see Berendregt or Hankin.
124 Since Ω does not have a normal form, it follows that Ω
125 cannot have a type in Λ_T. We can easily see why:
127 Ω = (\x.xx)(\x.xx)
129 Assume Ω has type τ, and `\x.xx` has type σ. Then
130 because `\x.xx` takes an argument of type σ and returns
131 something of type τ, `\x.xx` must also have type σ -->
132 τ. By repeating this reasoning, `\x.xx` must also have type
133 (σ --> τ) --> τ; and so on. Since variables have
134 finite types, there is no way to choose a type for the variable `x`
135 that can satisfy all of the requirements imposed on it.
137 In general, there is no way for a function to have a type that can
138 take itself for an argument. It follows that there is no way to
139 define the identity function in such a way that it can take itself as
140 an argument. Instead, there must be many different identity
141 functions, one for each type.
145 Version 1 type numerals are not a good choice for the simply-typed
146 lambda calculus. The reason is that each different numberal has a
147 different type! For instance, if zero has type σ, then since
148 one is represented by the function `\x.x false 0`, it must have type
149 `b --> σ --> σ`, where `b` is the type of a boolean. But
150 this is a different type than zero! Because each number has a
151 different type, it becomes unbearable to write arithmetic operations
152 that can combine zero with one, since we would need as many different
153 addition operations as we had pairs of numbers that we wanted to add.
155 Fortunately, the Church numerals are well behaved with respect to
156 types. They can all be given the type (σ --> σ) -->
165 Mau integrate some mention of this at some point.
167 http://okmij.org/ftp/Computation/lambda-calc.html#predecessor
170 Predecessor and lists are not representable in simply typed lambda-calculus
172 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:
174 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...]
175 Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81.
177 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.
179 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
181 data List a = Nil | Cons a (List a)
183 gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type).
185 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.