3 ##The simply-typed lambda calculus##
5 The untyped lambda calculus is pure computation. It is much more
6 common, however, for practical programming languages to be typed.
7 Likewise, systems used to investigate philosophical or linguistic
8 issues are almost always typed. Types will help us reason about our
9 computations. They will also facilitate a connection between logic
12 Soon we will consider polymorphic type systems. First, however, we
13 will consider the simply-typed lambda calculus. There's good news and
14 bad news: the good news is that the simply-type lambda calculus is
15 strongly normalizing: every term has a normal form. We shall see that
16 self-application is outlawed, so Ω can't even be written, let
17 alone undergo reduction. The bad news is that fixed-point combinators
18 are also forbidden, so recursion is neither simple nor direct.
22 We will have at least one ground type, `o`. From a linguistic point
23 of view, think of the ground types as the bar-level 0 categories, that
24 is, the lexical types, such as Noun, Verb, Preposition (glossing over
25 the internal complexity of those categories in modern theories).
27 In addition, there will be a recursively-defined class of complex
28 types `T`, the smallest set such that
30 * ground types, including `o`, are in `T`
32 * for any types σ and τ in `T`, the type σ -->
35 For instance, here are some types in `T`:
47 Given a set of types `T`, we define the set of typed lambda terms <code>Λ_T</code>,
48 which is the smallest set such that
50 * each type `t` has an infinite set of distinct variables, {x^t}_1,
53 * If a term `M` has type σ --> τ, and a term `N` has type
54 σ, then the application `(M N)` has type τ.
56 * If a variable `a` has type σ, and term `M` has type τ,
57 then the abstract <code>λ a M</code> has type σ --> τ.
59 The definitions of types and of typed terms should be highly familiar
60 to semanticists, except that instead of writing σ --> τ,
61 linguists (following Montague, who followed Church) write <σ,
62 τ>. We will use the arrow notation, since it is more iconic.
64 Some examples (assume that `x` has type `o`):
70 Excercise: write down terms that have the following types:
76 #Associativity of types versus terms#
78 As we have seen many times, in the lambda calculus, function
79 application is left associative, so that `f x y z == (((f x) y) z)`.
80 Types, *THEREFORE*, are right associative: if `f`, `x`, `y`, and `z`
81 have types `a`, `b`, `c`, and `d`, respectively, then `f` has type `a
82 --> b --> c --> d == (a --> (b --> (c --> d)))`.
84 It is a serious faux pas to associate to the left for types. You may
85 as well use your salad fork to stir your tea.
87 #The simply-typed lambda calculus is strongly normalizing#
89 If `M` is a term with type τ in Λ_T, then `M` has a
90 normal form. The proof is not particularly complex, but we will not
91 present it here; see Berendregt or Hankin.
93 Since Ω does not have a normal form, it follows that Ω
94 cannot have a type in Λ_T. We can easily see why:
96 Ω = (\x.xx)(\x.xx)
98 Assume Ω has type τ, and `\x.xx` has type σ. Then
99 because `\x.xx` takes an argument of type σ and returns
100 something of type τ, `\x.xx` must also have type σ -->
101 τ. By repeating this reasoning, `\x.xx` must also have type
102 (σ --> τ) --> τ; and so on. Since variables have
103 finite types, there is no way to choose a type for the variable `x`
104 that can satisfy all of the requirements imposed on it.
106 In general, there is no way for a function to have a type that can
107 take itself for an argument. It follows that there is no way to
108 define the identity function in such a way that it can take itself as
109 an argument. Instead, there must be many different identity
110 functions, one for each type.
114 Version 1 type numerals are not a good choice for the simply-typed
115 lambda calculus. The reason is that each different numberal has a
116 different type! For instance, if zero has type σ, then `false`
117 has type τ --> τ --> τ, for some τ. Since one is
118 represented by the function `\x.x false 0`, one must have type (τ
119 --> τ --> τ) --> σ --> σ. But this is a different
120 type than zero! Because each number has a different type, it becomes
121 impossible to write arithmetic operations that can combine zero with
122 one. We would need as many different addition operations as we had
123 pairs of numbers that we wanted to add.
125 Fortunately, the Church numberals are well behaved with respect to
126 types. They can all be given the type (σ --> σ) -->
135 Mau integrate some mention of this at some point.
137 http://okmij.org/ftp/Computation/lambda-calc.html#predecessor
140 Predecessor and lists are not representable in simply typed lambda-calculus
142 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:
144 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...]
145 Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81.
147 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.
149 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
151 data List a = Nil | Cons a (List a)
153 gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type).
155 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.