1 Basics of Lambda Calculus
2 =========================
4 We often talk about "*the* Lambda Calculus", as if there were just
5 one; but in fact, there are many, many variations. The one we will
6 start with, and that we will explore in some detail, is the "pure"
7 Lambda Calculus. And actually, there are many variations even in the
8 pure Lambda Calculus. But all of the variations share a strong family
9 resemblance, so what we learn now will apply to all of them.
11 The lambda calculus we'll be focusing on for the first part of the
12 course has no types. Some prefer to say it does have types, it's just
13 that there's only one type, so that every expression is a member of
14 that one type. If you say that, you have to say that functions from
15 this type to this type also belong to this type. Which is weird... In
16 fact, though, such types are studied, under the name "recursive
17 types." More about these later in the seminar.
22 <strong>Variables</strong>: <code>x</code>, <code>y</code>, <code>z</code>...
25 Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:
28 <strong>Abstract</strong>: <code>(λa M)</code>
31 We'll tend to write <code>(λa M)</code> as just `(\a M)`, so we don't have to write out the markup code for the <code>λ</code>. You can yourself write <code>(λa M)</code> or `(\a M)` or `(lambda a M)`.
34 <strong>Application</strong>: <code>(M N)</code>
38 Examples of expressions:
47 ((\x (x x)) (\x (x x)))
49 The lambda calculus has an associated proof theory. For now, we can regard the
50 proof theory as having just one rule, called the rule of **beta-reduction** or
51 "beta-contraction". Suppose you have some expression of the form:
55 that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.
57 The rule of beta-reduction permits a transition from that expression to the following:
61 What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
63 What is a free occurrence?
65 > An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
67 > If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`.
69 > An occurrence of a variable is **free** if it's not bound.
74 > T is defined to be `(x (\x (\y (x (y z)))))`
76 The first occurrence of `x` in T is free. The `\x` we won't regard as containing an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free.
80 * [[!wikipedia Free variables and bound variables]]
82 Here's an example of beta-reduction:
90 We'll write that like this:
92 ((\x (y x)) z) ~~> (y z)
94 Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write:
98 We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol <code><big><big>→</big></big></code> for one-step contraction, and the symbol <code><big><big>↠</big></big></code> for zero-or-more step reduction. Hindley and Seldin use <code><big><big><big>⊳</big></big></big><sub>1</sub></code> and <code><big><big><big>⊳</big></big></big></code>.
100 When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **beta-convertible**. We'll write that like this:
104 This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.)
106 In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol <code>≡</code> for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing the metatheory. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in:
108 > T is defined to be `(M N)`.
110 We'll regard the following two expressions:
116 as syntactically equivalent, since they only involve a typographic change of a bound variable. Read Hankin section 2.3 for discussion of different attitudes one can take about this.
118 Note that neither of those expressions are identical to:
122 because here it's a free variable that's been changed. Nor are they identical to:
126 because here the second occurrence of `y` is no longer free.
128 There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we've linked to about the lambda calculus. We expect you have a good intuitive understanding of what to do already, though, even if you're not able to articulate it rigorously.
130 * [More discussion in week 2 notes](/week2/#index1h1)
136 The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)
139 **Parentheses** Outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:
147 but you should include the parentheses in:
156 **Dot notation** Dot means "put a left paren here, and put the right
157 paren as far the right as possible without creating unbalanced
162 can be abbreviated as:
178 This on the other hand:
187 **Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
191 Similarly, `(\x (\y (\z M)))` can be abbreviated as:
196 Lambda terms represent functions
197 --------------------------------
199 The untyped lambda calculus is Turing complete: all (recursively computable) functions can be represented by lambda terms. For some lambda terms, it is easy to see what function they represent:
201 > `(\x x)` represents the identity function: given any argument `M`, this function
202 simply returns `M`: `((\x x) M) ~~> M`.
204 > `(\x (x x))` duplicates its argument:
205 `((\x (x x)) M) ~~> (M M)`
207 > `(\x (\y x))` throws away its second argument:
208 `(((\x (\y x)) M) N) ~~> M`
212 It is easy to see that distinct lambda expressions can represent the same
213 function, considered as a mapping from input to outputs. Obviously:
221 both represent the same function, the identity function. However, we said above that we would be regarding these expressions as synactically equivalent, so they aren't yet really examples of *distinct* lambda expressions representing a single function. However, all three of these are distinct lambda expressions:
229 yet when applied to any argument M, all of these will always return M. So they have the same extension. It's also true, though you may not yet be in a position to see, that no other function can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.
231 The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as proof-theoretically equivalent even though they're not syntactically identical. However, the proof theory we've given so far doesn't permit you to reduce the second expression to the third. So these lambda expressions are non-equivalent.
233 There's an extension of the proof-theory we've presented so far which does permit this further move. And in that extended proof theory, all computable functions with the same extension do turn out to be equivalent (convertible). However, at that point, we still won't be working with the traditional mathematical notion of a function as a set of ordered pairs. One reason is that the latter but not the former permits many uncomputable functions. A second reason is that the latter but not the former prohibits functions from applying to themselves. We discussed this some at the end of Monday's meeting (and further discussion is best pursued in person).
240 Our definition of these is reviewed in [[Assignment1]].
243 It's possible to do the assignment without using a Scheme interpreter, however
244 you should take this opportunity to [get Scheme installed on your
245 computer](/how_to_get_the_programming_languages_running_on_your_computer), and
246 [get started learning Scheme](/learning_scheme). It will help you test out
247 proposed answers to the assignment.
250 There's also a (slow, bare-bones, but perfectly adequate) version of Scheme available for online use at <http://tryscheme.sourceforge.net/>.