I'm new to functional language and I'm doing SICP programming assignments using Racket.
Below is a snippet of code, and Racket informs me that define: expected only one expression for the function body, but found 5 extra parts, in line 5 ((define (y k)):
(define (simpson f a b n)
(define h (/ (- b a) n))
(define (y k)
(f (+ a (* k h))))
(define (factor k)
(cond ((or (= k 0) (= k n))
1)
((odd? k)
4)
(else
2)))
(define (term k)
(* (factor k)
(y k)))
(define (next k)
(+ k 1))
(if (not (even? n))
(error "n can't be odd")
(* (/ h 3)
(sum term (exact->inexact a) next n))))
I guess this problem is related to the language settings, but I already use "advanced" option.
Anybody know how to configure Racket properly, or internal "define" is not supported?
Indeed, it's as you say: internal defines are not supported by the Advanced language. For working with the SICP exercises, I've been told it's best to use the neil/sicp package: instructions for using this are detailed here.
However, even the standard Racket language (#lang racket) will support internal defines without problems.
Related
I am learning ELISP and this function should gather any number of parameters plus N and return the sum. This what I've got so far and I cannot figure out where my error is thanks. Also if a char is in the list it should just skip over and not add that to the value. I am using ELISP
(defun sum-numbers (n &rest L)
(let (a 0)
(if (not L) n
(dolist (x L result)
(if (integerp x)
(setq a (+ x a)))))
(setq a (+ a n))))
You are using let incorrectly.
There might be other errors too, but your code is unreadable due to broken indentation. Hit TAB on every line to get the indentation right.
Corrections:
(defun sum-numbers (n &rest L)
- (let (a 0)
+ (let ((a 0))
(if (not L)
n
- (dolist (x L result)
+ (dolist (x L (+ a n))
(if (integerp x)
- (setq a (+ x a)))))
- (setq a (+ a n))))
+ (setq a (+ x a)))))))
(Outside of learning, you'd just use + which already sums an arbitrary number of arguments.)
This is my first post on StackOverflow. I have been working on Exercise 1.11 from SICP and feel I have a viable solution. In transferring from paper to Emacs I seem to have some syntax error that I am unaware of. I tried my best to double and triple check the parenthesis and solve it but the terminal is still giving me an 'object #t is not applicable' message. Could someone please point me in the right direction of how to fix the code so I can test its output properly?"
Exercise 1.11: A function f
is defined by the rule that:
f(n)=n
if n<3
, and
f(n)=f(n−1)+2f(n−2)+3f(n−3)
if n>=3
Write a procedure that computes f by means of a recursive process.
Write a procedure that computes f by means of an iterative process.
(define (f-recur n)
(if ((< n 3) n)
(+ (f(- n 1))
(* 2 (f(n-2)))
(* 3 (f(n-3)))))
(define (f-iter n)
(define (counter n)
(if (<= n 3) 0)
(- n 3))
(define (d n) (+ n (* 2 n) (* 3 n)))
(define (c n) (+ d (* 2 n) (* 3 n)))
(define (b n) (+ c (* 2 d) (* 3 n)))
(define (a n) (+ b (* 2 c) (* 3 d)))
(define (f a b c d counter)
(if ((> (+ counter 3) n) a)
(f (+ b (* 2 c) (* 3 d)) a b c (+ counter 1)))))
(cond ((= counter 0) d)
((= counter 1) c)
((= counter 2) b)
((= counter 3) a)
(else (f a b c d counter))))
I'm pretty sure SICP is looking for a solution in the manner of this iterative fibonnacci:
(define (fib n)
(define (helper n a b)
(if (zero? n)
a
(helper (- n 1) b (+ a b))))
(helper n 0 1))
Fibonacci is f(n)=f(n−1)+f(n−2) so I guess f(n)=f(n−1)+2f(n−2)+3f(n−3) can be made exactly the same way with one extra variable. Your iterative solution looks more like Fortran than Scheme. Try avoiding set!.
So as far as I understand, the following: let, let*, letrec and letrec* are synthetics sugars used in Scheme/Racket.
Now, if I have a simple program:
(let ((x 1)
(y 2))
(+ x y))
It is translated into:
((lambda (x y) (+ x y)) 1 2)
If I have:
(let* ((x 1)
(y 2))
(+ x y))
It is translated into:
((lambda (x) ((lambda (y) (+ x y))) 2) 1)
Now, for my first question, I understand the meaning of a letrec expression, which enables one to use recursion inside a let, but I do not understand how exactly it is done. What is letrec translated to?
For example, what will
(letrec ((x 1)
(y 2))
(+ x y))
be translated into?
The second question is similar about letrec* - But for letrec* I do not understand how exactly it differs from letrec? And also, what will a letrec* expression be translated into?
See the paper "Fixing Letrec: A Faithful Yet Efficient Implementation
of Scheme’s Recursive Binding Construct" by Oscar Waddell, Dipanwita Sarkar, and,
R. Kent Dybvig.
The paper starts with a simple version and proceeds to explain a more sophisticated expansion:
https://www.cs.indiana.edu/~dyb/pubs/fixing-letrec.pdf
Since you example does not have any procedures and no real expressions I'd say you could implement it as:
(let ((x 1) (y 2))
(+ x y))
But to support the intention of the form a basic implementaiton might do something like this:
(let ((x 'undefined) (y 'undefined))
(let ((tmpx 1) (tmpy 2))
(set! x tmpx)
(set! y tmpy))
(+ x y))
Now. letrec is for lambdas to be able to call themselves by the name. So imagine this:
(let ((fib (lambda (n)
(if (< n 2) n
(+ (fib (- n 1)) (fib (- n 2)))))))
(fib 10))
It's important to understand that this does not work and why. Transforming it into a lambda call makes it so easy to see:
((lambda (fib)
(fib 10))
(lambda (n)
(if (< n 2) n
(+ (fib (- n 1)) (fib (- n 2))))))
Somehow you need to evaluate the lambda after fib is created. Lets imagine we do this:
(let ((fib 'undefined))
(set! fib (lambda (n)
(if (< n 2) n
(+ (fib (- n 1)) (fib (- n 2))))))
(fib 10))
Alternatively you can do it with a Z combinator to make it pure:
(let ((fib (Z (lambda (fib)
(lambda (n)
(if (< n 2) n
(+ (fib (- n 1)) (fib (- n 2)))))))))
(fib 10))
This requires more work by the implementation, but at least you do without mutation. To make it work with several mutual recursive bindings probably takes some more work but I'm convinced its doable.
These are the only two ways to do this properly. I know clojure has a recur which imitates recursion but in reality is a a goto.
For variables that do not make closures from the letrec bindings they work as let and later more like let* since Soegaards answer about fix is backwards compatible and some has adapted it. If you write compatible code though you should not be tempted to assume this.
Here is the Y-combinator in Racket:
#lang lazy
(define Y (λ(f)((λ(x)(f (x x)))(λ(x)(f (x x))))))
(define Fact
(Y (λ(fact) (λ(n) (if (zero? n) 1 (* n (fact (- n 1))))))))
(define Fib
(Y (λ(fib) (λ(n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2))))))))
Here is the Y-combinator in Scheme:
(define Y
(lambda (f)
((lambda (x) (x x))
(lambda (g)
(f (lambda args (apply (g g) args)))))))
(define fac
(Y
(lambda (f)
(lambda (x)
(if (< x 2)
1
(* x (f (- x 1))))))))
(define fib
(Y
(lambda (f)
(lambda (x)
(if (< x 2)
x
(+ (f (- x 1)) (f (- x 2))))))))
(display (fac 6))
(newline)
(display (fib 6))
(newline)
My question is: Why does Scheme require the apply function but Racket does not?
Racket is very close to plain Scheme for most purposes, and for this example, they're the same. But the real difference between the two versions is the need for a delaying wrapper which is needed in a strict language (Scheme and Racket), but not in a lazy one (Lazy Racket, a different language).
That wrapper is put around the (x x) or (g g) -- what we know about this thing is that evaluating it will get you into an infinite loop, and we also know that it's going to be the resulting (recursive) function. Because it's a function, we can delay its evaluation with a lambda: instead of (x x) use (lambda (a) ((x x) a)). This works fine, but it has another assumption -- that the wrapped function takes a single argument. We could just as well wrap it with a function of two arguments: (lambda (a b) ((x x) a b)) but that won't work in other cases too. The solution is to use a rest argument (args) and use apply, therefore making the wrapper accept any number of arguments and pass them along to the recursive function. Strictly speaking, it's not required always, it's "only" required if you want to be able to produce recursive functions of any arity.
On the other hand, you have the Lazy Racket code, which is, as I said above, a different language -- one with call-by-need semantics. Since this language is lazy, there is no need to wrap the infinitely-looping (x x) expression, it's used as-is. And since no wrapper is required, there is no need to deal with the number of arguments, therefore no need for apply. In fact, the lazy version doesn't even need the assumption that you're generating a function value -- it can generate any value. For example, this:
(Y (lambda (ones) (cons 1 ones)))
works fine and returns an infinite list of 1s. To see this, try
(!! (take 20 (Y (lambda (ones) (cons 1 ones)))))
(Note that the !! is needed to "force" the resulting value recursively, since Lazy Racket doesn't evaluate recursively by default. Also, note the use of take -- without it, Racket will try to create that infinite list, which will not get anywhere.)
Scheme does not require apply function. you use apply to accept more than one argument.
in the factorial case, here is my implementation which does not require apply
;;2013/11/29
(define (Fact-maker f)
(lambda (n)
(cond ((= n 0) 1)
(else (* n (f (- n 1)))))))
(define (fib-maker f)
(lambda (n)
(cond ((or (= n 0) (= n 1)) 1)
(else
(+ (f (- n 1))
(f (- n 2)))))))
(define (Y F)
((lambda (procedure)
(F (lambda (x) ((procedure procedure) x))))
(lambda (procedure)
(F (lambda (x) ((procedure procedure) x))))))
I am working through SICP. In exercise 1.28 about the Miller-Rabin test. I had this code, that I know is wrong because it does not follow the instrcuccions of the exercise.
(define (fast-prime? n times)
(define (even? x)
(= (remainder x 2) 0))
(define (miller-rabin-test n)
(try-it (+ 1 (random (- n 1)))))
(define (try-it a)
(= (expmod a (- n 1) n) 1))
(define (expmod base exp m)
(cond ((= exp 0) 1)
((even? exp)
(if (and (not (= exp (- m 1))) (= (remainder (square exp) m) 1))
0
(remainder (square (expmod base (/ exp 2) m)) m)))
(else
(remainder (* base (expmod base (- exp 1) m)) m))))
(cond ((= times 0) true)
((miller-rabin-test n) (fast-prime? n (- times 1)))
(else false)))
In it I test if the square of the exponent is congruent to 1 mod n. Which according
to what I have read, and other correct implementations I have seen is wrong. I should test
the entire number as in:
...
(square
(trivial-test (expmod base (/ exp 2) m) m))
...
The thing is that I have tested this, with many prime numbers and large Carmicheal numbers,
and it seems to give the correct answer, though a bit slower. I don't understand why this
seems to work.
Your version of the function "works" only because you are lucky. Try this experiment: evaluate (fast-prime? 561 3) a hundred times. Depending on the random witnesses that your function chooses, sometimes it will return true and sometimes it will return false. When I did that I got 12 true and 88 false, but you may get different results, depending on your random number generator.
> (let loop ((k 0) (t 0) (f 0))
(if (= k 100) (values t f)
(if (fast-prime? 561 3)
(loop (+ k 1) (+ t 1) f)
(loop (+ k 1) t (+ f 1)))))
12
88
I don't have SICP in front of me -- my copy is at home -- but I can tell you the right way to perform a Miller-Rabin primality test.
Your expmod function is incorrect; there is no reason to square the exponent. Here is a proper function to perform modular exponentiation:
(define (expm b e m) ; modular exponentiation
(let loop ((b b) (e e) (x 1))
(if (zero? e) x
(loop (modulo (* b b) m) (quotient e 2)
(if (odd? e) (modulo (* b x) m) x)))))
Then Gary Miller's strong pseudoprime test, which is a strong version of your try-it test for which there is a witness a that proves the compositeness of every composite n, looks like this:
(define (strong-pseudoprime? n a) ; strong pseudoprime base a
(let loop ((r 0) (s (- n 1)))
(if (even? s) (loop (+ r 1) (/ s 2))
(if (= (expm a s n) 1) #t
(let loop ((r r) (s s))
(cond ((zero? r) #f)
((= (expm a s n) (- n 1)) #t)
(else (loop (- r 1) (* s 2)))))))))
Assuming the Extended Riemann Hypothesis, testing every a from 2 to n-1 will prove (an actual, deterministic proof, not just a probabilistic estimate of primality) the primality of a prime n, or identify at least one a that is a witness to the compositeness of a composite n. Michael Rabin proved that if n is composite, at least three-quarters of the a from 2 to n-1 are witnesses to that compositeness, so testing k random bases demonstrates, but does not prove, the primality of a prime n to a probability of 4−k. Thus, this implementation of the Miller-Rabin primality test:
(define (prime? n k)
(let loop ((k k))
(cond ((zero? k) #t)
((not (strong-pseudoprime? n (random (+ 2 (- n 3))))) #f)
(else (loop (- k 1))))))
That always works properly:
> (let loop ((k 0) (t 0) (f 0))
(if (= k 100) (values t f)
(if (prime? 561 3)
(loop (+ k 1) (+ t 1) f)
(loop (+ k 1) t (+ f 1)))))
0
100
I know your purpose is to study SICP rather than to program primality tests, but if you're interested in programming with prime numbers, I modestly recommend this essay at my blog, which discusses the Miller-Rabin test, among other topics. You should also know there are better (faster, less likely to report erroneous result) primality tests available than randomized Miller-Rabin.
It seems to me, you still got correct answer, because in each iteration of expmod you check conditions for previous iteration. You could try to debug exp value using display function inside expmod. Really, your code is not very different from this one.