EDIT: Thanks to everyone. I'm new to the language(just started using it two days ago), so that's why I'm unfamiliar with conds. I may rewrite it if I have time, but I just wanted to make sure I had the basic logic right. Thanks again!
My assignment is to make a tail-recursive function that removes the nth element from the list, 1 <= n <= listlength, with only two parameters, list x and element n. So, (remove 1 '(a b c d)) will return (b c d). I have written the following, and would like some reassurance that it is indeed tail recursive. The only thing I'm fuzzy on is if the recursive call can be nested inside an IF statement.
(define (remove n x)
; if n is 1, just return the cdr
(if (and (not (list? (car x))) (= n 1))
(cdr x)
; if the car is not a list, make it one, and call recursively
(if (not (list? (car x)))
(remove (- n 1) (cons (list (car x)) (cdr x)))
; if n !=1, insert the cadr into the list at the car.
; Else, append the list at the car with the cddr
(if (not(= n 1))
(remove (- n 1) (cons (append (car x) (list(cadr x))) (cddr x)))
(append (car x) (cddr x))))))
Yes, the procedure is tail-recursive, meaning: wherever a recursive call is performed, it's the last thing that happens in that particular branch of execution, with nothing more to do after the recursion returns - hence, we say that the recursive call is in tail position.
This can be clearly seen if we rewrite the procedure using cond instead of nested ifs, here you'll see that every branch of execution leads to either a base case or a recursive case, and all recursive calls are in tail position:
(define (remove n x)
; base case #1
(cond ((and (not (list? (car x))) (= n 1))
; return from base case, it's not recursive
(cdr x))
; recursive case #1
((not (list? (car x)))
; recursive call is in tail position
(remove (- n 1) (cons (list (car x)) (cdr x))))
; recursive case #2
((not (= n 1))
; recursive call is in tail position
(remove (- n 1) (cons (append (car x) (list(cadr x))) (cddr x))))
; base case #2
(else
; return from base case, it's not recursive
(append (car x) (cddr x)))))
For a more technical explanation of why the consequent/alternative parts of an if special form can be considered tail-recursive, take a look at section 3.5 of the current draft of the Revised^7 Report on the Algorithmic Language Scheme - the language specification, here's a link to the pdf file (in essence the same considerations apply to R5RS, it's just that they're explained in more detail in R7RS). In particular:
If one of the following expressions is in a tail context, then the subexpressions shown as ⟨tail expression⟩ are in a tail context
...
(if ⟨expression⟩ ⟨tail expression⟩ ⟨tail expression⟩)
(if ⟨expression⟩ ⟨tail expression⟩)
Here is the Scheme specification on the tail recursion position for syntactic forms:
Related
For my programming languages class I'm supposed to write a function in Scheme to reverse a list without using the pre-made reverse function. So far what I got was
(define (reverseList lst)
(COND
((NULL? lst) '())
(ELSE (CONS (reverseList(CDR lst)) (CAR lst)))
))
The problem I'm having is that if I input a list, lets say (a b c) it gives me (((() . c) . b) . a).
How am I supposed to get a clean list without multiple sets of parenthesis and the .'s?
The problem with your implementation is that cons isn't receiving a list as its second parameter, so the answer you're building isn't a proper list, remember: a proper list is constructed by consing an element with a list, and the last list is empty.
One possible workaround for this is to use a helper function that builds the answer in an accumulator parameter, consing the elements in reverse - incidentally, this solution is tail recursive:
(define (reverse lst)
(reverse-helper lst '()))
(define (reverse-helper lst acc)
(if (null? lst)
acc
(reverse-helper (cdr lst) (cons (car lst) acc))))
(reverse '(1 2 3 4 5))
=> '(5 4 3 2 1)
You are half way there. The order of the elements in your result is correct, only the structure needs fixing.
What you want is to perform this transformation:
(((() . c) . b) . a) ; input
--------------------
(((() . c) . b) . a) () ; trans-
((() . c) . b) (a) ; for-
(() . c) (b a) ; mation
() (c b a) ; steps
--------------------
(c b a) ; result
This is easy to code. The car and cdr of the interim value are immediately available to us. At each step, the next interim-result is constructed by (cons (cdr interim-value) interim-result), and interim-result starts up as an empty list, because this is what we construct here - a list:
(define (transform-rev input)
(let step ( (interim-value input) ; initial set-up of
(interim-result '() ) ) ; the two loop variables
(if (null? interim-value)
interim-result ; return it in the end, or else
(step (car interim-value) ; go on with the next interim value
(cons ; and the next interim result
(... what goes here? ...)
interim-result )))))
interim-result serves as an accumulator. This is what's known as "accumulator technique". step represents a loop's step coded with "named-let" syntax.
So overall reverse is
(define (my-reverse lst)
(transform-rev
(reverseList lst)))
Can you tweak transform-rev so that it is able to accept the original list as an input, and thus skip the reverseList call? You only need to change the data-access parts, i.e. how you get the next interim value, and what you add into the interim result.
(define (my-reverse L)
(fold cons '() L)) ;;left fold
Step through the list and keep appending the car of the list to the recursive call.
(define (reverseList lst)
(COND
((NULL? lst) '())
(ELSE (APPEND (reverseList(CDR lst)) (LIST (CAR lst))))
))
Instead of using cons, try append
(define (reverseList lst)
(if (null? lst)
'()
(append (reverseList (cdr lst)) (list (car lst)) )
)
)
a sample run would be:
1]=> (reverseList '(a b c 1 2 + -))
>>> (- + 2 1 c b a)
car will give you just one symbol but cdr a list
Always make sure that you provide append with two lists.
If you don't give two lists to the cons it will give you dotted pair (a . b) rather than a list.
See Pairs and Lists for more information.
I wanted to write a code in Scheme that writes the square odd elements in list.For example (list 1 2 3 4 5) for this list it should write 225.For this purpose i write this code:
(define (square x)(* x x))
(define (product-of-square-of-odd-elements sequence)
(cond[(odd? (car sequence)) '() (product-of-square-of-odd-elements (cdr sequence))]
[else ((square (car sequence)) (product-of-square-of-odd-elements (cdr sequence)))]))
For run i write this (product-of-square-of-odd-elements (list 1 2 3 4 5))
and i get error like this:
car: contract violation
expected: pair?
given: '()
What should i do to make this code to run properly? Thank you for your answers.
First of all, you need to do proper formatting:
(define (square x) (* x x))
(define (product-of-square-of-odd-elements sequence)
(cond
[(odd? (car sequence))
'() (product-of-square-of-odd-elements (cdr sequence))]
[else
((square (car sequence)) (product-of-square-of-odd-elements (cdr sequence)))]))
Now there are multiple issues with your code:
You are trying to work recursively on a sequence, but you are missing a termination case: What happens when you pass '() - the empty sequence? This is the source of your error: You cannot access the first element of an empty sequence.
You need to build up your result somehow: Currently you're sending a '() into nirvana in the first branch of your cond and put a value into function call position in the second.
So let's start from scratch:
You process a sequence recursively, so you need to handle two cases:
(define (fn seq)
(if (null? seq)
;; termination case
;; recursive case
))
Let's take the recursive case first: You need to compute the square and multiply it with the rest of the squares (that you'll compute next).
(* (if (odd? (car seq)
(square (car seq))
1)
(fn (cdr seq)))
In the termination case you have no value to square. So you just use the unit value of multiplication: 1
This is not a good solution, as you can transform it into a tail recursive form and use higher order functions to abstract the recursion altogether. But I think that's enough for a start.
With transducers:
(define prod-square-odds
(let ((prod-square-odds
((compose (filtering odd?)
(mapping square)) *)))
(lambda (lst)
(foldl prod-square-odds 1 lst))))
(prod-square-odds '(1 2 3 4 5))
; ==> 225
It uses reusable transducers:
(define (mapping procedure)
(lambda (kons)
(lambda (e acc)
(kons (procedure e) acc))))
(define (filtering predicate?)
(lambda (kons)
(lambda (e acc)
(if (predicate? e)
(kons e acc)
acc))))
You can decompose the problem into, for example:
Skip the even elements
Square each element
take the product of the elements
With this, an implementation is naturally expressed using simpler functions (most of which exist in Scheme) as:
(define product-of-square-of-odd-elements (l)
(reduce * 1 (map square (skip-every-n 1 l))))
and then you implement a helper function or two, like skip-every-n.
I can't seem to figure out how to write this function. What I am trying to write is a function expand that takes a list lst as a parameter of the form '(a (2 b) (3 c)) and is evaluated to '(a b b c c c)
This looks like homework, so I'm not giving you a straight answer. Instead, I'll give you some pointers in the right direction. The most useful hint, is that you should split the problem in two procedures, one for processing the "outer" list and the other for generating the repetitions encoded in the inner sublists.
Notice that both procedures are mutually recursive (e.g., they call each other). The expand procedure recurs over the list, whereas the repeat procedure recurs over the number of repetitions. This is the general structure of the proposed solution, fill-in the blanks:
; input: lst - list to be processed
; output: list in the format requested
(define (expand lst)
(cond ((null? lst) ; if the list is null
'()) ; then return null
((not (pair? (car lst))) ; if the first element of the list is an atom
(cons <???> <???>)) ; cons the atom and advance the recursion
(else ; if the first element of the list is a list
<???>))) ; call `repeat` with the right params
; input: n - number of repetitions for the first element in the list
; lst - list, its first element is of the form (number atom)
; output: n repetitions of the atom in the first element of lst
(define (repeat n lst)
(if (zero? n) ; if the number of repetitions is zero
(expand (cdr lst)) ; continue with expand's recursion
(cons <???> ; else cons the atom in the first element and
<???>))) ; advance the recursion with one less repetition
As this was answered three years ago, I don't think that I am helping with homework. Would just like to point out that the two functions really don't need to be mutually recursive. As replicate is a fairly common function, I would propose:
(define (replicate what n)
(if (zero? n)
(list)
(cons what (replicate what (- n 1)))))
(define (my-expand xs)
(if (empty? xs)
(list)
(let ((x (first xs)))
(if (list? x)
(let ((the-number (first x))
(the-symbol (cadr x)))
(flatten (cons (replicate the-symbol the-number)
(my-expand (rest xs)))))
(cons x (my-expand (rest xs)))))))
Of course it is better to use two lists and perform the flatten at the end, something like this:
(define (my-expand xs)
(define (inner-expander xs ys)
(if (empty? xs) (flatten (reverse ys))
(let ((x (first xs)))
(if (list? x)
(let ((the-number (first x))
(the-symbol (cadr x)))
(inner-expander (rest xs) (cons (replicate the-symbol the-number) ys)))
(inner-expander (rest xs) (cons x ys))))))
(inner-expander xs (list)))
I'm having trouble appending a list to another list. Below is my code. When I run (append '(1 2) '(3 4)) I get '(1 3 2 4).
I want the output to be '(1 2 3 4)
(define (append l m)
(if (null? l) '()
(cons (car l) (append m (cdr l)))))
Thanks
Well by switching the two lists around like that (switching the position of m and l when calling append recursively), you'll get the first item of the first list followed by the first item of the second list etc.
If you don't want that, you should keep l as the first argument and m as the second. So you get:
(define (append l m)
(if (null? l) '()
(cons (car l) (append (cdr l) m))))
Of course this doesn't work as wanted either, because now you only get the first list back and nothing is appended at all. What you need to do is, once the first list is fully appended (i.e. once l is empty), you need to return the second one as the tail, like this:
(define (append l m)
(if (null? l) m
(cons (car l) (append (cdr l) m))))
I came across this while studying myself. #sepp2k's answer is a fine piece of instruction guiding OP to correct their code to achieve a recursive definition of append. Here's an alternate definition of my-append using the higher-order function foldr:
(define (myappend xs ys)
(foldr cons ys xs))
I have been working alongside The Little Schemer to learn Scheme and using PLT-Scheme for my environment.
The Little Schemer has helped me tremendously with recursion (it is straightforward for me now) but I'm stuck on a portion of the book that introduces "collectors" and calls the function as a whole a continuation.
Here is the example code they have used. I understand the recursive elements but I am stuck, in particular on the lambda functions - my mind can't follow the path and how the arguments for that lambda function are set (since their only call is to call them again in recursion, there is no concrete use within the function body).
If someone could more-or-less give me a break down of the path of computation through the recursion of the function into the lambda collectors, that may help me.
;; Build a nested list of even numbers by removing the odd ones from its
;; argument and simultaneously multiply the even numbers and sum the odd
;; numbers that occur in its argument.
(define (even-only-collector l col)
(cond
((null? l)
(col (quote ()) 1 0))
((atom? (car l))
(cond
((even? (car l))
(even-only-collector (cdr l)
(lambda (newl p s)
(col (cons (car l) newl)
(* (car l) p) s))))
(else
(even-only-collector (cdr l)
(lambda (newl p s)
(col newl
p (+ (car l) s)))))))
(else
(even-only-collector (car l)
(lambda (al ap as)
(even-only-collector (cdr l)
(lambda (dl dp ds)
(col (cons al dl)
(* ap dp)
(+ as ds)))))))))
;; The collector function
(define (collector newl product sum)
(cons sum
(cons product newl)))
Thank you in advance!!
Try something simpler to see how this works. For example, here's a version of a list-sum function that receives a continuation argument (which is often called k):
(define (list-sum l k)
(if (null? l)
???
(list-sum (cdr l) ???)))
The basic pattern is there, and the missing parts are where the interesting things happen. The continuation argument is a function that expects to receive the result -- so if the list is null, it's clear that we should send it 0, since that is the sum:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) ???)))
Now, when the list is not null, we call the function recursively with the list's tail (in other words, this is an iteration), but the question is what should the continuation be. Doing this:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) k)))
is clearly wrong -- it means that k will eventually receive the the sum of (cdr l) instead of all of l. Instead, use a new function there, which will sum up the first element of l too along with the value that it receives:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) (lambda (sum) (+ (car l) sum)))))
This is getting closer, but still wrong. But it's a good point to think about how things are working -- we're calling list-sum with a continuation that will itself receive the overall sum, and add the first item we see now to it. The missing part is evident in the fact that we're ignoring k. What we need is to compose k with this function -- so we do the same sum operation, then send the result to k:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) (compose k (lambda (s) (+ s (car l)))))))
which is finally working. (BTW, remember that each of these lambda functions has its own "copy" of l.) You can try this with:
(list-sum '(1 2 3 4) (lambda (x) x))
And finally note that this is the same as:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) (lambda (s) (k (+ s (car l)))))))
if you make the composition explicit.
(You can also use this code in the intermediate+lambda student language, and click the stepper button to see how the evaluation proceeds -- this will take a while to go over, but you'll see how the continuation functions get nested, each with it's own view of the list.)
Here's one way to help you "get a more concrete idea". Imagine if the collector were defined thus:
(define (collector l p s)
(display l)
(newline)
(display p)
(newline)
(display s)
(newline))
You can see in the base case, if you pass in an empty list, it will call your function with arguments '(), 1, and 0. Now, work with a one-element list, and see what it'll call your function with. Keep working up with longer and longer lists, until you figure out what's going on.
Good luck!