At the beginning of learning PLT, everyone would likely be confused by some program that didn't have a variable! Will, I mean they didn't use String
, Text
or something like that to define a variable. A direct mapping definition of lambda calculus(we would use LC as the short name in the following context, and if you unfamiliar with LC , you can read this article first) usually looks like this:
#lang typed/racket
(struct term [] #:transparent)
(struct term:var term [(name : String)] #:transparent)
(struct term:lambda term [(x : String) (m : term)] #:transparent)
(struct term:application term [(t1 : term) (t2 : term)] #:transparent)
But what we may find some definitions look like this:
#lang typed/racket
(struct bterm [] #:transparent)
(struct bterm:var bterm [(v : Integer)] #:transparent)
(struct bterm:lambda bterm [(m : bterm)] #:transparent)
(struct bterm:application bterm [(t1 : bterm) (t2 : bterm)] #:transparent)
There has two significant different:
 variable is an
Integer
.  lambda does not contain
x
(which means parameter in highlevel languages' concept).
This is De Bruijn index(we would use DBI as short name in the following context), we can write it out, for example, id function λx.x
can be rewritten with λ0
, Y combinator λf.(λx.f (x x)) (λx.f (x x))
can be rewritten with λ(λ0 (1 1))(λ0 (1 1))
, but why? To understand DBI , we need to know what was the problem in LC.
α\alphaαconversion
Usually, two id functions considered the same function. However, if we encode LC as the first definition written in racket , we would get into trouble: We may say λx.x
is not the same function as λy.y
, when they are the same. To solve this problem, we developed a conversion called α\alphaαconversion (or α\alphaαrenaming ), which renaming λx.x
and λy.y
to λa.a
(a
is an abstraction variable, we can use any, any character to replace it) let's say. Looks good, any problem else? Emm...yes, as we know, the real world never make thing easier, but that also means a challenge is coming, and we all love the challenge! When a λy.λx.x
be renamed to λa.λa.a
is fine, because of every programmer work with variableshadowing for a longlong time. However, there has a possible dangerous conversion is the renamed variable existed! For example, λx.λa.x
should not simply be rewritten with λa.λa.a
, because later when we rename a
, we would get λa.λb.b
, oops. λx.λa.x
definitely is not λa.λb.b
. To correct α\alphaαconversion is really hard, that's the main reason we introducing the De Bruijn index.
De Bruijn Index
We already seem some examples, but why it resolves the problem we mentioned in the previous section? We need to know those rules used by the conversion process:
 remember the level of
λ
, every time we found aλ
when converting recursively, it should increase(or decrease, it depends on index order) this level value.  when found a
λ
, replace it'sx
by variable using the De Bruijn index form, the value of the index is the current level.
Let's manually do this conversion:
λx.λy.λz.z
> λλy.λz.z // x = cur_level = 0, cur_level+1
> λλλz.z // y = cur_level = 1, cur_level+1
> λλλ2 // z = cur_level = 2
Notice that since new form of abstraction(a.k.a lambda) only needs M
part(a.k.a. body). Another important thing is some De Bruijn index use reverse order than we show at here, so would be λλλ0
, not λλλ2
.
Implementation
Now, it's time for the program:
(: convert (>* [term] [(ImmutableHashTable String Integer)] bterm))
(define (convert t [renameto (makeimmutablehash '())])
(match t
;; get index from environment
([term:var name] (bterm:var (hashref renameto name)))
([term:lambda p b]
(bterm:lambda
(convert b
;; bind parameter name to an index
(hashset renameto p (hashcount renameto)))))
([term:application t1 t2]
(bterm:application
(convert t1 renameto)
(convert t2 renameto)))))
>*
is a type constructor in Racket for optional parameters, should be read like (>* normalparametertypes optionalparametertypes returntype)
. I use optional parameters to help users don't need to remember they must provide an empty hash table. A tricky thing is I didn't record level, at least, not directly. Here I use an immutable hash table to remember level, since how many variables should be renamed exactly is level value. Then variable only need to replace its name with the index.
Congratulation, now you know everything about DBI!? No, not yet, there still one thing you need to know.
β\betaβreduction
β\betaβreduction? You might think how can such basic things make things go wrong. However, a naive implementation of β\betaβreduction can break structural equivalence of DBI form, which can make an annoying bug in those systems based on LC. The problem is lacklifting. For example, a normal implementation of β\betaβreduction would simply make λ(λλ2 0)
become λλ2
. However, the same form directly converted from λx.λz.z
would become λλ1
, and λλ2
will be considered as different value as λλ1
since 1
is not 2
. We can introduce another renaming for these, but if we can fix it in β\betaβreduction, why need another phase?
Implementation
(: betareduction (>* [bterm] [Integer (ImmutableHashTable Integer bterm)] bterm))
(define (betareduction t [debruijnlevel 0] [subst (makeimmutablehash '())])
(match t
([bterm:var i]
(hashref subst i (λ () t)))
([bterm:lambda body]
(bterm:lambda (betareduction body (+ 1 debruijnlevel) subst)))
([bterm:application t1 t2]
(match t1
([bterm:lambda body]
(let ([reducedterm (betareduction body (+ 1 debruijnlevel)
(hashset subst debruijnlevel t2))])
;;; dbi lifting by replace reducedterm (+ 1 dbi) with (var dbi)
(betareduction reducedterm debruijnlevel
(hashset subst (+ 1 debruijnlevel) (bterm:var debruijnlevel)))))
(_ (raise "cannot do application on nonlambda term"))))))
We have to record level independently since it has no relation with the substitution map this time. For variables, all need to do is apply substitution map to get value, if not, use origin form as a result. For the lambda, increase level is the only thing. For application, it's complicated. We need to be more careful with it. It contains three major parts:
 check
t1
is an abstraction(a.k.a lambda), line 9 and 16  do β\betaβreduction by add stuff into substitution map, line 11 to 12

DBI lifting (for example,
λλ2
should becomeλλ1
), line 14 to 15
Conclusion
DBI is a quite useful technology when implementing complicated AST conversion. It's not just easier to avoid rename conflicting, but also a less memory required form for implementations. I hope you enjoy the article and have a nice day if this even really helps you in a real task, would be awesome!
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