Advanced notes on the Lambda Calculus

Hankin uses the symbol → for one-step contraction, and the symbol ↠ for zero-or-more step reduction. Hindley and Seldin use ⊳1 and ⊳.

As we said in the main notes, 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 write that like this:

M <~~> N


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, we keep confusing that with the relation to be described next, so let's use the <~~> 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.)

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 ≡ 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:

T is defined to be (M N).

or:

Let T be (M N).

We'll regard the following two expressions:

(\x (x y))

(\z (z y))


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.

Note that neither of the above expressions are identical to:

(\x (x w))


because here it's a free variable that's been changed. Nor are they identical to:

(\y (y y))


because here the second occurrence of y is no longer free.

There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we'll link 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.

Substitution and Alpha-Conversion

Intuitively, (a) and (b) express the application of the same function to the argument y:

1. (\x. \z. z x) y
2. (\x. \y. y x) y

One can't just rename variables freely. (a) and (b) are different than what's expressed by:

1. (\z. (\z. z z) y

Substituting y into the body of (a) (\x. \z. z x) is unproblematic:

(\x. \z. z x) y ~~> \z. z y


However, with (b) we have to be more careful. If we just substituted blindly, then we might take the result to be \y. y y. But this is the self-application function, not the function which accepts an arbitrary argument and applies that argument to the free variable y. In fact, the self-application function is what (c) reduces to. So if we took (b) to reduce to \y. y y, we'd wrongly be counting (b) to be equivalent to (c), instead of (a).

To reduce (b), then, we need to be careful to that no free variables in what we're substituting in get "captured" by binding λs that they shouldn't be captured by.

In practical terms, you'd just replace (b) with (a) and do the unproblematic substitution into (a).

How should we think about the explanation and justification for that practical procedure?

One way to think about things here is to identify expressions of the lambda calculus with particular alphabetic sequences. Then (a) and (b) would be distinct expressions, and we'd have to have an explicit rule permitting us to do the kind of variable-renaming that takes us from (a) to (b) (or vice versa). This kind of renaming is called "alpha-conversion." Look in the standard treatments of the lambda calculus for detailed discussion of this.

Another way to think of it is to identify expressions not with particular alphabetic sequences, but rather with classes of alphabetic sequences, which stand to each other in the way that (a) and (b) do. That's the way we'll talk. We say that (a) and (b) are just typographically different notations for a single lambda term. As we'll say, the lambda term written with (a) and the lambda term written with (b) are literally syntactically identical.

A third way to think is to identify the lambda term not with classes of alphabetic sequences, but rather with abstract structures that we might draw like this:


(λ. λ. _ _) y
^  ^  | |
|  |__| |
|_______|


Here there are no bound variables, but bound positions remain. We can regard formula like (a) and (b) as just helpfully readable ways to designate these abstract structures.

A version of this last approach is known as de Bruijn notation for the lambda calculus.

It doesn't seem to matter which of these approaches one takes; the logical properties of the systems are exactly the same. It just affects the particulars of how one states the rules for substitution, and so on. And whether one talks about expressions being literally "syntactically identical," or whether one instead counts them as "equivalent modulu alpha-conversion."

(Linguistic trivia: some linguistic discussions do suppose that alphabetic variance has important linguistic consequences; see Ivan Sag's dissertation.)

Next week, we'll discuss other systems that lack variables. Those systems will not just lack variables in the sense that de Bruijn notation does; they will furthermore lack any notion of a bound position.

Review: syntactic equality, reduction, convertibility

Define N to be (\x. x y) z. Then N and (\x. x y) z are syntactically equal, and we're counting them as syntactically equal to (\z. z y) z as well. We'll express all these claims in our metalanguage as:

N ≡ (\x. x y) z ≡ (\z. z y) z


This:

N ~~> z y


means that N beta-reduces to z y. This:

M <~~> N


means that M and N are beta-convertible, that is, that there's some common term they both reduce to in zero or more steps.

The symbols ~~> and <~~> aren't part of what we're calling "the Lambda Calculus". In our mouths, they're just part of our metatheory for talking about it. In the uses of the Lambda Calculus as a formal proof theory, one or the other of these symbols (or some notational variant of them) is added to the object language. But only in outermost contexts. It's like the "sequent" symbol (written => or ⊢) in Gentzen-style proof systems for logic. You can't embed the ~~> or <~~> symbol inside lambda terms.

See Hankin Sections 2.2 and 2.4 for the proof theory using <~~> (which he writes as =). He discusses the proof theory using ~~> in his Chapter 3. This material is covered in slightly different ways (different organization and some differences in terminology and notation) in Chapters 1, 6, and 7 of the Hindley & Seldin.