X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek2_lambda_advanced.mdwn;h=d84d5ee7a14f6a4a12cd3da0c7eba9a08c569e5c;hp=1372e436c1306cbcd71b74b7bf29e21589779265;hb=020566910974049e4cd9cbc9279a781768e1817c;hpb=2a7dcdce0415bc9d65c204f42e218d864cc00550 diff --git a/topics/week2_lambda_advanced.mdwn b/topics/week2_lambda_advanced.mdwn index 1372e436..d84d5ee7 100644 --- a/topics/week2_lambda_advanced.mdwn +++ b/topics/week2_lambda_advanced.mdwn @@ -1,4 +1,4 @@ -## Fine points concerning the lambda calculus ## +## Advanced notes on the Lambda Calculus ## Hankin uses the symbol → for one-step contraction, @@ -7,10 +7,7 @@ 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: +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 @@ -31,11 +28,11 @@ 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)`. +> `T` is defined to be `(M N)`. or: -> Let T be `(M N)`. +> Let `T` be `(M N)`. We'll regard the following two expressions: @@ -92,7 +89,7 @@ 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 +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). @@ -110,10 +107,10 @@ 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 formula. As we'll say, the lambda formula written with (a) and -the lambda formula written with (b) are literally syntactically identical. +*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 formula not with classes of +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: @@ -147,7 +144,7 @@ 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, +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: @@ -158,16 +155,16 @@ This: N ~~> z y -means that N beta-reduces to `z y`. This: +means that `N` beta-reduces to `z y`. This: M <~~> N -means that M and N are beta-convertible, that is, that there's something they both reduce to in zero or more steps. +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. +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](https://en.wikipedia.org/wiki/Sequent_calculus) 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.