3 ##The simply-typed lambda calculus##
5 The untyped lambda calculus is pure. Pure in many ways: nothing but
6 variables and lambdas, with no constants or other special symbols;
7 also, all functions without any types. As we'll see eventually, pure
8 also in 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 To develop this analogy just a bit further, syntactic categories
26 determine which expressions can combine with which other expressions.
27 If a word is a member of the category of prepositions, it had better
28 not try to combine (merge) with an expression in the category of, say,
29 an auxilliary verb, since *under has* is not a well-formed constituent
30 in English. Likewise, types in formal languages will determine which
31 expressions can be sensibly combined.
33 Now, of course it is common linguistic practice to supply an analysis
34 of natural language both with syntactic categories and with semantic
35 types. And there is a large degree of overlap between these type
36 systems. However, there are mismatches in both directions: there are
37 syntactic distinctions that do not correspond to any salient semantic
38 difference (why can't adjectives behave syntactically like verb
39 phrases, since they both denote properties with (extensional) type
40 `<e,t>`?); and in some analyses there are semantic differences that do
41 not correspond to any salient syntactic distinctions (as in any
42 analysis that involves silent type-shifters, such as Herman Hendriks'
43 theory of quantifier scope, in which expressions change their semantic
44 type without any effect on the syntactic expressions they can combine
45 with syntactically). We will consider again the relationship between
46 syntactic types and semantic types later in the course.
48 Soon we will consider polymorphic type systems. First, however, we
49 will consider the simply-typed lambda calculus.
51 [Pedantic on. Why "*simply* typed"? Well, the type system is
52 particularly simple. As mentioned to us by Koji Mineshima, Church
53 tells us that "The simple theory of types was suggested as a
54 modification of Russell's ramified theory of types by Leon Chwistek in
55 1921 and 1922 and by F. P. Ramsey in 1926." This footnote appears in
56 Church's 1940 paper [A formulation of the simple theory of
57 types](church-simple-types.pdf). In this paper, as Will Starr
58 mentioned in class, Church does indeed write types by simple
59 apposition, without the ugly angle brackets and commas used by
60 Montague. Furthermore, he omits parentheses under the convention that
61 types associated to the *left*---the opposite of the modern
62 convention. This is ok, however, because he also reverses the order,
63 so that `te` is a function from objects of type `e` to objects of type
64 `t`. Cool paper! If you ever want to see Church numerals in their
65 native setting--but I'm getting ahead of my story. Pedantic off.]
67 There's good news and bad news: the good news is that the simply-typed
68 lambda calculus is strongly normalizing: every term has a normal form.
69 We shall see that self-application is outlawed, so Ω can't even
70 be written, let alone undergo reduction. The bad news is that
71 fixed-point combinators are also forbidden, so recursion is neither
76 We will have at least one ground type. For the sake of linguistic
77 familiarity, we'll use `e`, the type of individuals, and `t`, the type
80 In addition, there will be a recursively-defined class of complex
81 types `T`, the smallest set such that
83 * ground types, including `e` and `t`, are in `T`
85 * for any types σ and τ in `T`, the type σ -->
88 For instance, here are some types in `T`:
100 Given a set of types `T`, we define the set of typed lambda terms <code>Λ_T</code>,
101 which is the smallest set such that
103 * each type `t` has an infinite set of distinct variables, {x^t}_1,
104 {x^t}_2, {x^t}_3, ...
106 * If a term `M` has type σ --> τ, and a term `N` has type
107 σ, then the application `(M N)` has type τ.
109 * If a variable `a` has type σ, and term `M` has type τ,
110 then the abstract <code>λ a M</code> has type σ --> τ.
112 The definitions of types and of typed terms should be highly familiar
113 to semanticists, except that instead of writing σ --> τ,
114 linguists write <σ, τ>. We will use the arrow notation,
115 since it is more iconic.
117 Some examples (assume that `x` has type `o`):
123 Excercise: write down terms that have the following types:
126 (o --> o) --> o --> o
127 (o --> o --> o) --> o
129 #Associativity of types versus terms#
131 As we have seen many times, in the lambda calculus, function
132 application is left associative, so that `f x y z == (((f x) y) z)`.
133 Types, *THEREFORE*, are right associative: if `x`, `y`, and `z`
134 have types `a`, `b`, and `c`, respectively, then `f` has type
135 `a --> b --> c --> d == (a --> (b --> (c --> d)))`, where `d` is the
136 type of the complete term.
138 It is a serious faux pas to associate to the left for types. You may
139 as well use your salad fork to stir your tea.
141 #The simply-typed lambda calculus is strongly normalizing#
143 If `M` is a term with type τ in Λ_T, then `M` has a
144 normal form. The proof is not particularly complex, but we will not
145 present it here; see Berendregt or Hankin.
147 Since Ω does not have a normal form, it follows that Ω
148 cannot have a type in Λ_T. We can easily see why:
150 Ω = (\x.xx)(\x.xx)
152 Assume Ω has type τ, and `\x.xx` has type σ. Then
153 because `\x.xx` takes an argument of type σ and returns
154 something of type τ, `\x.xx` must also have type σ -->
155 τ. By repeating this reasoning, `\x.xx` must also have type
156 (σ --> τ) --> τ; and so on. Since variables have
157 finite types, there is no way to choose a type for the variable `x`
158 that can satisfy all of the requirements imposed on it.
160 In general, there is no way for a function to have a type that can
161 take itself for an argument. It follows that there is no way to
162 define the identity function in such a way that it can take itself as
163 an argument. Instead, there must be many different identity
164 functions, one for each type.
168 Version 1 type numerals are not a good choice for the simply-typed
169 lambda calculus. The reason is that each different numberal has a
170 different type! For instance, if zero has type σ, then since
171 one is represented by the function `\x.x false 0`, it must have type
172 `b --> σ --> σ`, where `b` is the type of a boolean. But
173 this is a different type than zero! Because each number has a
174 different type, it becomes unbearable to write arithmetic operations
175 that can combine zero with one, since we would need as many different
176 addition operations as we had pairs of numbers that we wanted to add.
178 Fortunately, the Church numerals are well behaved with respect to
179 types. They can all be given the type (σ --> σ) -->
188 Mau integrate some mention of this at some point.
190 http://okmij.org/ftp/Computation/lambda-calc.html#predecessor
193 Predecessor and lists are not representable in simply typed lambda-calculus
195 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:
197 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...]
198 Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81.
200 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.
202 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
204 data List a = Nil | Cons a (List a)
206 gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type).
208 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.