so i am trying to understand this piece of code, and after staring at it for far too long i decided to ask here if anyone could help me understand how and why it works
(define knock-knock
(letrec ([dig (lambda (i)
(cons (* i (list-ref knock-knock (- i 1)))
(dig (+ i 1))))])
(cons 1 (dig 1))))
the function is then called by name with the value:
(list-ref knock-knock 5)
So my main problem is that i can not see where the letrec would end. the other thing is that i am not given a list, so what is the 4th element in the list that i am supposed to reference in line 3?
First, a note: this is not normal Scheme, as it requires lazy evaluation.
In lazy evaluation, values are only computed when they are needed. So, for defining knock-knock, we can just do
(cons 1 <thunk: (dig 1)>)
i.e., we generate a pair, but we don't need the second element, so we defer its evaluation until later.
When we actually want to evaluate the second element, we will already have knock-knock defined, so we can reference it.
The next element is computed by taking the previous (i-1-st) element, and multiplies it by i. So this will generate the series {n!}: 1,1,2,6,24,...
A straightforward translation of this code to the (normally lazy) Haskell language goes like this:
knock :: [Int]
knock = 1 : dig 1
where dig i = (i * knock !! (i-1)) : dig (i+1)
Related
I am currently just starting out working through Structure and Interpretation of Computer Programming, and in a section it is going over the creation of an absolute value function in Scheme (takes a value, and returns its absolute value).
I am following exactly how the book does the function:
(define (abs x)
(cond ((< x 0) (- x))
(else x)))
Unfortunately, this is not providing the desired result as the book says. Instead of returning the absolute value, it just returns the value. For example, I type in -5, the function returns -5.
If it is of any help, I am using the BiwaScheme Interpreter (0.6.4).
Any help is greatly appreciated, thanks.
To call a function, enclose its name in parentheses together with any arguments you wish to supply it with. Example,
> (abs -5)
5
Of course if you just type -5 you are not calling any functions, abs in particular, with it:
> -5
-5
The numeric value is just returned as is, as no function call is indicated.
I am supposed to write a function called (nth-filtered f n), where f is a function of one variable and n is a natural number, which evaluates to the nth natural number such that f applied to that number is #t.
If we called
(nth-filtered even? 1) we would get 2
(nth-filtered prime? 10) we would get 29
How do I make it so that it works for any sequential function? What should I think about when approaching this type of problem?
A variable is a variable and + is also a variable. The main difference between a function and some other data type is that you can wrap a function name in parentheses with arguments and it will become a new value.
eg.
(define (double fun)
(lambda (value)
(fun (fun value))))
(define (add1 v)
(+ 1 v))
(define add2 (double add1))
(add2 1) ; ==> 3
Now the contract doesn't say so you deduct by looking that you do (fun ...) that fun needs to be a function. Imagine this:
(define test (double 5)) ; probably works OK
(test 1)
The last one fails since you get application: 5 is not a procedure or something similar. The error message is not standardized.
How to attack your task is by making a helper that has the same arguments as your function but in addition the current number that I guess starts at 1. As I demonstrated you use the function variable as a function and recurse by always increasing the number and reducing n when the f call was #t. The actual function will just use the helper by passing all the parameters in addition to your state variable.
Your problem requires a fold, which is the standard way to iterate other a list while keeping a record of things done so far.
Here a very rackety method using for/fold:
(define (nth-filtered predicate index)
(for/fold ([count 0]
[current #f] #:result current)
([n (in-naturals 1)]) ; we start at 1 but we could start at 0
#:break (= count index)
(values (if (predicate n) (add1 count) count)
n)))
for/fold takes a list of initial state. Here we define count as the number of times the given predicate returned #t and current as the currently tested value.
Then it takes a list of iterators, in this case we only iterate infinitely over (in-naturals).
To make it stop, we provide a #:break condition, which is "when the number of truthy predicates (count) is equal to the requested amount (index)".
for/fold requests that it's body finishes with a list of values for each "state" variable, in order to update them for the next iteration. Here we provide two values: one is the new count, the other is just the current n.
You can try it out, it works as you requested:
> (nth-filtered even? 1)
2
> (require math/number-theory)
> (nth-filtered prime? 10)
29
> (nth-filtered prime? 5)
11
I was a bit surprised by this racket code printing nay when I expected yeah:
(define five 5)
(case 5
[(five) "yeah"]
[else "nay"])
Looking at the racket documentation for case makes it clearer:
The selected clause is the first one with a datum whose quoted form is equal? to the result of val-expr.
So it's about quotation. I'm pretty sure that I did not yet fully grasp what quotation in lisps can buy me. I understand it in the viewpoint of macros and AST transformation. However I'm confused why is it helpful in the case of case for instance..?
I'm also curious, with this specification of case, can I use it to achieve what I wanted to (compare the actual values, not the quoted value), or should I use another construct for that? (cond, while strictly more powerful, is more verbose for simple cases, since you must repeat the predicate at each condition).
The problem is that case introduces implicit quote forms, which cause your example to work for 'five (whose value is 'five), instead of five (whose value is 5).
I almost never use case because of exactly this problem. Instead I use racket's match form with the == pattern:
(define five 5)
(define (f x)
(match x
[(== five) "yeah"]
[_ "nay"]))
(f 5) ; "yeah"
(f 6) ; "nay"
This produces "yeah" on only the value 5, just like you expected. If you wanted it to return "yeah" when it's equal to either five or six, you can use an or pattern:
(define five 5)
(define six 6)
(define (f x)
(match x
[(or (== five) (== six)) "yeah"]
[_ "nay"]))
(f 5) ; "yeah"
(f 6) ; "yeah"
(f 7) ; "nay"
And if you really want to match against quoted datums, you can do that by writing an explicit quote form.
(define (f x)
(match x
[(or 'five 'six) "yeah"]
[_ "nay"]))
(f 5) ; "nay"
(f 6) ; "nay"
(f 7) ; "nay"
(f 'five) ; "yeah"
(f 'six) ; "yeah"
These quote forms are implicit and invisible when you use case, lurking there waiting to cause confusion.
The Racket documentation gives this grammar:
(case val-expr case-clause ...)
where
case-clause = [(datum ...) then-body ...+]
| [else then-body ...+]
Let's compare to your example:
(define five 5)
(case 5 ; (case val-expr
[(five) "yeah"] ; [(datum) then-body1]
[else "nay"]) ; [else then-body2])
We see that (five) is interpreted as (datum). This means that five is
a piece of data (here a symbol), not an expression (later to be evaluated).
Your example of case is evaluated like this:
First the expression 5 is evaluated. The result is the value 5.
Now we look at a clause at a time. The first clause is [(five) "yeah"].
Is the value 5 equal (in the sense of equal?) to one of the datums in (five)? No, so we look at the next clause: [else "nay"]. It is an else-clause so the expression "nay" is evaluated and the result is the value "nay".
The result of the case-expression is thus the value "nay".
Note 1: The left-hand sides of case-clauses are datums (think: they are implicitly quoted).
Note 2: The result of val-expr is compared to the clause datums using equal?. (This is in contrast to Scheme, which uses eqv?.
UPDATE
Why include case? Let's see how one can write the example using cond:
(define five 5)
(let ([val five])
(cond
[(member val '(five)) "yeah"]
[(member val '(six seven)) "yeah"] ; added
[else "nay"])
This shows that one could do without case and just use cond.
However - which version is easier to read?
For a case expression it is easy to see which datums the value is compared to.
Here one must look closely to find the datums. Also in the example we know beforehand that we are trying to find the value among a few list of datums. In general we need to examine a cond-expression more closely to see that's what's happening.
In short: having a case-expression increases readability of your code.
For the historically interested: https://groups.csail.mit.edu/mac/ftpdir/scheme-mail/HTML/rrrs-1986/msg00080.html disussed whether to use eqv? or equal? for case.
UPDATE 2
I'll attempt to given an answer to:
I'm still not clear on the quotation vs working simply on the values though.
I'm wondering specifically why doing the quotation, why working on datum instead
of working on values. Didn't get that bit yet.
Both approaches make sense.
Let's for the sake of argument look at the case where case uses expressions rather than datums in the left hand side of a clause. Also following the Scheme tradition, let's assume eqv? is used for the comparison. Let's call such a
case-expression for ecase (short for expression-case).
The grammar becomes:
(ecase val-expr ecase-clause ...)
where
ecase-clause = [(expr ...) then-body ...+]
| [else then-body ...+]
Your example now becomes:
(define five 5)
(ecase five
[('five) "yeah"]
[else "nay")
This doesn't look too bad and the result is what we are used to.
However consider this example:
(ecase '(3 4)
[('five (list 3 4) "yeah"]
[else "nay")
The result of this would be "nay". The two lists resulting from evaluating the expressions '(3 4) and (list 3 4) are not equal in the sense of eqv?.
This shows that if one chooses to use eqv? for comparisions, having expressions available on the left hand side won't be helpful. The only values that work with eqv? atomic values - and therefore one could just as well use implicit quotations and restrict the left hand side to datums.
Now if equal? was used it would make much more sense to use expressions on the left hand side. The original Racket version of case was the same as the one in Scheme (i.e. it used eq?) later on it was changed to used equal?. If case was designed from scratch, I think, expressions would be allowed rather than datums.
The only remaining issue: Why did the authors of Scheme choose eqv? over equal? for comparisons? My intuition is that the reason were performance (which back in the day was more important than now). The linked to post from the rrrs-authors mailing list gives two options. If you dig a little further you might be able to find responses.
I can't find a reference right now, but case statements use literal, unevaluated data in their different clauses because it is both a frequent use-case and more easily subject to efficient compilation.
You could probably write your own version of Clojure's condp macro or a custom conditional operator to handle your use case.
I attempted to follow the solution provided in this question, but it simply didn't work.
Essentially, my function works like so:
(define (item-price size normal-addons premium-addons discount)
(define price 0)
(+ price (* normal-addon-cost normal-addons) (* premium-addon-cost premium-addons) size)
(cond
.. some conditions here
[else price]))
However, I am met with the following error:
define: expected only one expression for the function body, but found 2 extra parts
Now, I've tried wrapping the body of the function in a 'begin', however when run it claims that 'begin' is not defined. I am using the Beginner Student language version as oppose to straight-up Racket. Any insight on a workaround?
The problem remains the same: in the language that's being used, we can't write more than one expression inside a function body, we can't use begin to pack more than one expression, and both let and lambda (which would have allowed us to create local bindings) are forbidden. That's a lot of restrictions, but we can get around using a helper function that calculates the price each time:
(define normal-addon-cost 10) ; just an example
(define premium-addon-cost 100) ; just an example
(define (price size normal-addons premium-addons)
(+ (* normal-addon-cost normal-addons)
(* premium-addon-cost premium-addons)
size))
(define (item-price size normal-addons premium-addons discount)
(cond
... some conditions here ...
[else (price size normal-addons premium-addons)]))
Alternatively: if price is used only once, simply in-line the expression that calculates it, there's no need to create a local variable or a helper function.
So basically this code's purpose is to simply print out the first n even numbers.
for (i = 0; i <=n; i+= 2)
{
print i;
}
Thing is though, I don't understand Scheme at all. So, help please.
There are several ways to convert the code in the question to Scheme. The first one I can think of:
(define (print-even n)
(let loop ((i 0))
(if (<= i n)
(begin
(print i)
(newline)
(loop (+ i 2))))))
Notice this:
The solution is written as a recursive procedure
Instead of a for loop, I'm using a construct called a named let, which permits the initialization of some iteration variables (i in this case, initialized to 0) and the repeated execution of a recursive procedure (loop in this case), producing an effect similar to a for, even in performance
The stop condition in the "loop" is handled with essentially the same expression: repeat the body of the iteration as long as (<= i n), when that condition becomes false, the iteration ends
The begin surrounds the body of the "loop", just as the curly braces {} do in the original code
The print procedure performs the expected operation; for readability I added a new line after printing each number
The increment part of the original loop i += 2 is handled by the expression (+ i 2), inside the recursive call
So you see, the process being executed is essentially the same, only the way to write it (the syntax!) is different. Give it a try, type this:
(print-even 6)
... And the following will get printed on the screen:
0
2
4
6
Another possible way to implement the procedure, more similar to the original code, although (this is completely subjective) less idiomatic than the previous one:
(define (print-even n)
(do ((i 0 (+ i 2))) ((> i n))
(print i)
(newline)))
Finally, if you're using Racket this will seem even more familiar to you:
#lang racket
(define (print-even n)
(for ((i (in-range 0 (+ n 1) 2)))
(print i)
(newline)))
The first big difference between Scheme and other languages is this: In Scheme, you do (almost) everything recursively.
To implement a simple loop, for instance, you would define a recursive function. This function would first check to see whether it's time to break out of the loop; if is is, it would return the final value. (There is no final value in this case, so it would just return something like (void) or '().) Otherwise, the function would do whatever it's supposed to do, then call itself again.
Any loop variables (such as i) become arguments to the function.
Hopefully this helps you understand how to do this.
The Scheme way to do something like this is using a recursive function like the one below.
(define (doit value n)
(if (<= value n)
(begin
;;...perform loop body with x...
(display value)(newline)
(doit (+ value 2) n))))
To call this function you call (doit 2 n) where n is your n in the for loop.
With regards to learning Scheme, I recommend the first two links below.
For additional information on Scheme see
SICP
How to Design Programs
Schemers
Related Stackoverflow Question
Scheme Cookbook Looping Constructs