Lab 12: Tail Recursion, Scheme
Due by 11:59pm on Tuesday, April 13.
Starter Files
Download lab12.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the Ok autograder.
Topics
Tail Recursion Minilecture Video
# Python
def fact(n):
if n == 0:
return 1
return n * fact(n - 1)
# Scheme
(define (fact n)
(if (= n 0)
1
(* n (fact (- n 1)))))
Notice that in this version of factorial
, the return expressions both
use recursive calls, and then use the values for more "work." In other
words, the current frame needs to sit around, waiting for the recursive
call to return with a value. Then the current frame can use that value
to calculate the final answer.
As an example, consider a call to fact(5)
(Shown with Scheme below).
We make a new frame for the call, and in carrying out the body of the
function, we hit the recursive case, where we want to multiply 5 by the
return value of the call to fact(4)
. Then we want to return this
product as the answer to fact(5)
. However, before calculating this
product, we must wait for the call to fact(4)
. The current frame
stays while it waits. This is true for every successive recursive
call, so by calling fact(5)
, at one point we will have the frame of
fact(5)
as well as the frames of fact(4)
, fact(3)
, fact(2)
, and
fact(1)
, all waiting for fact(0)
.
(fact 5)
(* 5 (fact 4))
(* 5 (* 4 (fact 3)))
(* 5 (* 4 (* 3 (fact 2))))
(* 5 (* 4 (* 3 (* 2 (fact 1)))))
(* 5 (* 4 (* 3 (* 2 (* 1 (fact 0))))))
(* 5 (* 4 (* 3 (* 2 (* 1 1)))))
(* 5 (* 4 (* 3 (* 2 1))))
(* 5 (* 4 (* 3 2)))
(* 5 (* 4 6))
(* 5 24)
120
Keeping all these frames around wastes a lot of space, so our goal is to come up with an implementation of factorial that uses a constant amount of space. We notice that in Python we can do this with a while loop:
def fact_while(n):
total = 1
while n > 0:
total = total * n
n = n - 1
return total
However, Scheme doesn't have for
and while
constructs. No problem!
Everything that can be written with while and for
loops and also be
written recursively. Instead of a variable, we introduce a new
parameter to keep track of the total.
def fact(n):
def fact_optimized(n, total):
if n == 0:
return total
return fact_optimized(n - 1, total * n)
return fact_optimized(n, 1)
(define (fact n)
(define (fact-optimized n total)
(if (= n 0)
total
(fact-optimized (- n 1) (* total n))))
(fact-optimized n 1))
Why is this better? Consider calling fact(n)
on based on this
definition:
(fact 5)
(fact-optimized 5 1)
(fact-optimized 4 5)
(fact-optimized 3 20)
(fact-optimized 2 60)
(fact-optimized 1 120)
(fact-optimized 0 120)
120
Because Scheme supports tail-call optimization (note that Python does
not), it knows when it no longer needs to keep around frames because
there is no further calculation to do. Looking at the last line in
fact_optimized
, we notice that it returns the same thing that the
recursive call does, no more work required. Scheme realizes that there
is no reason to keep around a frame that has no work left to do, so it
just has the return of the recursive call return directly to whatever
called the current frame.
Therefore the last line in fact_optimized
is a tail-call.
So far we've been able to define our own procedures in Scheme using the
define
special form. When we call these procedures, we have to follow
the rules for evaluating call expressions, which involve evaluating all the
operands.
We know that special form expressions do not follow the evaluation rules of call expressions. Instead, each special form has its own rules of evaluation, which may include not evaluating all the operands. Wouldn't it be cool if we could define our own special forms where we decide which operands are evaluated? Consider the following example where we attempt to write a function that evaluates a given expression twice:
scm> (define (twice f) (begin f f))
twice
scm> (twice (print 'woof))
woof
Since twice
is a regular procedure, a call to twice
will
follow the same rules of evaluation as regular call expressions; first we
evaluate the operator and then we evaluate the operands. That means that
woof
was printed when we evaluated the operand (print 'woof)
.
Inside the body of twice
, the name f
is bound to the value
undefined
, so the expression (begin f f)
does nothing at all!
The problem here is clear: we need to prevent the given expression from
evaluating until we're inside the body of the procedure. This is where the
define-macro
special form, which has identical syntax to the regular
define
form, comes in:
scm> (define-macro (twice f) (list 'begin f f))
twice
define-macro
allows us to define what's known as a macro
,
which is simply a way for us to combine unevaluated input expressions together
into another expression. When we call macros, the operands are not evaluated,
but rather are treated as Scheme data. This means that any operands that are
call expressions or special form expression are treated like lists.
If we call (twice (print 'woof))
, f
will actually be bound to
the list (print 'woof)
instead of the value undefined
.
Inside the body of define-macro
, we can insert these expressions into
a larger Scheme expression. In our case, we would want a begin
expression that looks like the following:
(begin (print 'woof) (print 'woof))
As Scheme data, this expression is really just a list containing three
elements: begin
and (print 'woof)
twice, which is exactly
what (list 'begin f f)
returns. Now, when we call twice
,
this list is evaluated as an expression and (print 'woof)
is evaluated
twice.
scm> (twice (print 'woof))
woof
woof
To recap, macros are called similarly to regular procedures, but the rules for evaluating them are different. We evaluated lambda procedures in the following way:
- Evaluate operator
- Evaluate operands
- Apply operator to operands, evaluating the body of the procedure
However, the rules for evaluating calls to macro procedures are:
- Evaluate operator
- Apply operator to unevaluated operands
- Evaluate the expression returned by the macro in the frame it was called in.
Questions
Q1: WWSD: Macros
One thing to keep in mind when doing this question, builtins get rendered as such:
scm> +
#[+]
scm> list
#[list]
If evaluating an expression causes an error, type
SchemeError
. If nothing is displayed, typeNothing
.Use Ok to test your knowledge with the following "What Would Scheme Display?" questions:
python3 ok -q wwsd-macros -u
scm> +
______#[+]
scm> list
______#[list]
scm> (define-macro (f x) (car x))
______f
scm> (f (2 3 4)) ; type SchemeError for error, or Nothing for nothing
______2
scm> (f (+ 2 3))
______#[+]
scm> (define x 2000)
______x
scm> (f (x y z))
______2000
scm> (f (list 2 3 4))
______#[list]
scm> (f (quote (2 3 4)))
______SchemeError
scm> (define quote 7000)
______quote
scm> (f (quote (2 3 4)))
______7000
scm> (define-macro (g x) (+ x 2))
______g
scm> (g 2)
______4
scm> (g (+ 2 3))
______SchemeError
scm> (define-macro (h x) (list '+ x 2))
______h
scm> (h (+ 2 3))
______7
scm> (define-macro (if-else-5 condition consequent) `(if ,condition ,consequent 5))
______if-else-5
scm> (if-else-5 #t 2)
______2
scm> (if-else-5 #f 3)
______5
scm> (if-else-5 #t (/ 1 0))
______SchemeError
scm> (if-else-5 #f (/ 1 0))
______5
scm> (if-else-5 (= 1 1) 2)
______2
Q2: Replicate
Write a tail-recursive function that returns a list with x
repeated n
times.
scm> (tail-replicate 3 10)
(3 3 3 3 3 3 3 3 3 3)
scm> (tail-replicate 5 0)
()
scm> (tail-replicate 100 5)
(100 100 100 100 100)
(define (tail-replicate x n)
'YOUR-CODE-HERE
)
Use Ok to test your code:
python3 ok -q tail-replicate
Q3: Scheme def
Implement def
, which simulates a python def
statement, allowing you to write code like
(def f(x y) (+ x y))
.
The above expression should create a function with parameters x
and y
, and
body (+ x y)
, then bind it to the name f
in the current frame.
Note: the previous is equivalent to
(def f (x y) (+ x y))
.Hint: We strongly suggest doing the WWPD questions on macros first as understanding the rules of macro evaluation is key in writing macros.
(define-macro (def func args body)
'YOUR-CODE-HERE)
Use Ok to test your code:
python3 ok -q scheme-def
Q4: Repeatedly Cube
Implement the following function, which cubes the given value x
some number n
times, based on the given skeleton.
For information on how to use let, see the scheme spec or ask a member of course staff.
(define (repeatedly-cube n x)
(if (zero? n)
x
(let
(_________________________)
(* y y y))))
Use Ok to test your code:
python3 ok -q repeatedly-cube