I'm trying to understand the scheme code of make-counter procedure. It's a higher order procedure (a procedure outputs another procedure) and I'm stuck with it.
(define make-counter
(lambda (n)
(lambda ()
(set! n (+ n 1))
n)))
(define ca (make-counter 0))
(ca)
(ca)
This outputs 1 and 2 respectively as expected. Why do we need 2 nested procedures here? What are their functions individually?
I'd be appreciated if someone explains in details. Thanks from now on.
Indented properly, this is:
(define make-counter
(lambda (n)
(lambda ()
(set! n (+ n 1))
n)))
By the way, you can use a different syntax:
(define (make-counter n)
(lambda ()
(set! n (+ n 1))
n))
make-counter is a function that accepts a number n and returns an object called closure, which acts like a function but contains a state. Different invocations of make-counter will produce different closures, even when given the same n in argument. A closure can be called using the function-call syntax, as you experimented.
When you call the closure, the code that is contained within is executed. In your example, the closure accepts zero arguments, and mutates the variable named n. Again, the binding from n to a value is local to the closure and different for all instances of counters. But inside a particular counter, n always refer to the same memory location.
A call to the set! function changes what n evaluates to, and replaces the previous value with (+ n 1), incrementing the local counter variable.
Related
Programming in racket, I am trying to write a function that takes a single integer, and returns a function that increments that integer by another integer. For example:
((incnth 5) 3) --> 8
((incnth 3) -1) --> 2
Unfortunately I don't seem to understand lambda functions still, because my code keeps saying that my lambda is not a function definition. Here is what I wrote.
(define (incnth n)
(lambda (f) (lambda (x) (+ n x))))
You have one more lambda than it's needed. If I understand correctly, the idea is to have a procedure that creates procedures that increment a number with a given number. So you should do this:
(define (incnth n) ; this is a procedure
(lambda (x) (+ n x))) ; that returns a lambda
The returned lambda will "remember" the n value:
(define inc2 (incnth 2))
And the resulting procedure can be used as usual, with the expected results:
(inc2 40)
=> 42
((incnth 5) 3)
=> 8
((incnth 3) -1)
=> 2
I feel that understanding this subtlety might help me to understand how scope works in Scheme.
So why does Scheme error out if you try to do something like:
(define (func n)
(define n (+ 1 n))
n)
It only errors out at runtime when calling the function.
The reason I find it strange is because Scheme does allow re-definitions, even inside functions. For example this gives no error and will always return the value 5 as expected:
(define (func n)
(define n 5)
n)
Additionally, Scheme also seems to support self-re-definition in global space. For instance:
(define a 5)
(define a (+ 1 a))
gives no error and results in "a" displaying "6" as expected.
So why does the same thing give a runtime error when it occurs inside a function (which does support re-definition)? In other words: Why does self-re-definition only not work when inside of a function?
global define
First off, top level programs are handled by a different part of the implementation than in a function and defining an already defined variable is not allowed.
(define a 10)
(define a 20) ; ERROR: duplicate definition for identifier
It might happen that it works in a REPL since it's common to redefine stuff, but when running programs this is absolutely forbidden. In R5RS and before what happened is underspecified and didn't care since be specification by violating it it no longer is a Scheme program and implementers are free to do whatever they want. The result is of course that a lot of underspecified stuff gets implementation specific behaviour which are not portable or stable.
Solution:
set! mutates bindings:
(define a 10)
(set! a 20)
define in a lambda (function, let, ...)
A define in a lambda is something completely different, handled by completely different parts of the implementation. It is handled by the macro/special form lambda so that it is rewritten to a letrec*
A letrec* or letrec is used for making functions recursive so the names need to be available at the time the expressions are evaluated. Because of that when you define n it has already shadowed the n you passed as argument. In addition from R6RS implementers are required to signal an error when a binding is evaluated that is not yet initialized and that is probably what happens. Before R6RS implementers were free to do whatever they wanted:
(define (func n)
(define n (+ n 1)) ; illegal since n hasn't got a value yet!
n)
This actually becomes:
(define func
(lambda (n)
(let ((n 'undefined-blow-up-if-evaluated))
(let ((tmpn (+ n 1)))
(set! n tmpn))
n)))
Now a compiler might see that it violates the spec at compile time, but many implementations doesn't know before it runs.
(func 5) ; ==> 42
Perfectly fine result in R5RS if the implementers have good taste in books.
The difference in the version you said works is that this does not violate the rule of evaluating n before the body:
(define (func n)
(define n 5)
n)
becomes:
(define func
(lambda (n)
(let ((n 'undefined-blow-up-if-evaluated))
(let ((tmpn 5)) ; n is never evaluated here!
(set! n tmpn))
n)))
Solutions
Use a non conflicting name
(define (func n)
(define new-n (+ n 1))
new-n)
Use let. It does not have its own binding when the expression gets evaluated:
(define (func n)
(let ((n (+ n 1)))
n))
I'm having some trouble deciphering this code snippet.
(define (stream n f)
(define (next m)
(cons m (lambda () (next (f m)))))
(next n))
(define even (stream 0 (lambda (n) (+ n 2))))
I understand that 'even' is defined as a variable using the 'stream' function, which contains parameters 0 and '(lambda (n) (+ n 2))'. Inside of 'stream', wouldn't '(next n)' indefinitely create cons nodes with a car of n + 2? Yet when I return 'even', it is a cons of (0 . # < Closure>). Could someone be so kind as to explain why this is? Thanks!
Everytime (lambda ...) is evaluated it becomes a closure. What it means is that f is free and available for both procedures next and for the anonymous procedure it creates. m, which is a bound variable in next is also captured as a free variable in the anonymous procedure. If the language were dynamically bound none of them would exist if you called the resulting procedure in the cdr since the bindings would no longer exist, but since we have closures the variable exists even after the procedures that created them ended.
The pair returned by the stream procedure has references to the closure created when (lambda (n) (+ n 2)) was evaluated though the name f even though the call is finished. Thus if you do:
((cdr even)) ; ==>
(#<closure>) ; ==>
(cons 2 #<new-closure>) ; ==> Here
It's important to know that the evaluation of next becomes a pair with a new closure that has a new free variable m that is 2 this time. For each time you call the cdr it creates a new pair with a new closure. If it was the same each time you wouldn't have different result each iteration.
A lambda on it's own doesn't run the code in it's body. Thus instead of infinite recursion you only get one step and the result (cons 2 #<new-closure>). You need to call the cdr of this again ni order to get one more step. etc. Likewise if I did this:
(define (test a)
(define (helper b)
(+ a b))
helper) ; return the helper that has a as closure variable
Since we actually don't use the name one could just have anonymous lambda there instead of the define + the resulting variable. Anyway, you get something that gives you partially application:
((test 5) 2) ; ==> 7
(define ten-adder (test 10))
(ten-adder 2) ; ==> 12
(ten-adder 5) ; ==> 15
I know this is maybe an oddball idea, but I thought might as well give it a try to ask here.
I was experimenting in Racket about state representation without local variables.
The idea was defining a function that prints it's parameter value and if called again gives me another value. Since pure functions called with the same parameter always produce the same result, my workaround-idea got me the following.
(define (counter n)
(displayln n)
(λ () (counter (add1 n)))) ; unapplied lambda so it doesn't go in a loop
Then I devised a function to call counter and its resulting lambdas a certain number of times.
(define (call proc n)
(unless (zero? n)
(let ([x (proc)])
(call x (sub1 n)))))
Which results in this:
> (call (counter 0) 5)
0
1
2
3
4
5
What is the name for the concept applied here? Propably it's something trivial what you need in real applications all the time, but since I have no experience in that respect yet so I can't pinpoint a name for it. Or maybe I just complicated something very simple, but nonetheless I would appreciate an answer so I can look further into it.
Sorry if my question is not clear enough, but english is not my first language and to ask about things I have no name for makes me feel kinda uncertain.
You're using closures to save state: a lambda form stores the environment in which it was defined, and you keep redefining the procedure (called x in your code), so each time it "remembers" a new value for n.
Another way to do the same would be to let the procedure itself keep track of the value - in other words, counter should remember the current n value between invocations. This is what I mean:
(define (counter initial)
(let ((n (sub1 initial)))
(lambda ()
(set! n (add1 n))
n)))
In the above code, the first invocation of counter returns a new lambda that closes over n, and each invocation of that lambda modifies n and returns its new value. Equivalently, we could use Racket-specific syntax for currying and the begin0 special form to achieve the same effect:
(define ((counter n))
(begin0
n
(set! n (add1 n))))
Either way, notice how the procedure "remembers" its previous value:
(define proc (counter 0))
(proc)
=> 0
(proc)
=> 1
And we would call it like this:
(define (call proc n)
(unless (zero? n)
(displayln (proc))
(call proc (sub1 n))))
(call (counter 0) 5)
=> 0
1
2
3
4
Also notice that the above fixes an off-by-one error originally in your code - the procedure was being called six times (from 0 to 5) and not five times as intended, that happened because call invokes counter five times, but you called counter one more time outside, when evaluating (counter 0).
In an effort to find a simple example of CPS which doesn't give me a headache , I came across this Scheme code (Hand typed, so parens may not match) :
(define fact-cps
(lambda(n k)
(cond
((zero? n) (k 1))
(else
(fact-cps (- n 1)
(lambda(v)
(k (* v n))))))))
(define fact
(lambda(n)
(fact-cps n (lambda(v)v)))) ;; (for giggles try (lambda(v)(* v 2)))
(fact 5) => 120
Great, but Scheme isn't Common Lisp, so I took a shot at it:
(defun not-factorial-cps(n k v)
(declare (notinline not-factorial-cps)) ;; needed in clisp to show the trace
(cond
((zerop n) (k v))
((not-factorial-cps (1- n) ((lambda()(setq v (k (* v n))))) v))))
;; so not that simple...
(defun factorial(n)
(not-factorial-cps n (lambda(v)v) 1))
(setf (symbol-function 'k) (lambda(v)v))
(factorial 5) => 120
As you can see, I'm having some problems, so although this works, this has to be wrong. I think all I've accomplished is a convoluted way to do accumulator passing style. So other than going back to the drawing board with this, I had some questions: Where exactly in the Scheme example is the initial value for v coming from? Is it required that lambda expressions only be used? Wouldn't a named function accomplish more since you could maintain the state of each continuation in a data structure which can be manipulated as needed? Is there in particular style/way of continuation passing style in Common Lisp with or without all the macros? Thanks.
The problem with your code is that you call the anonymous function when recurring instead of passing the continuation like in the Scheme example. The Scheme code can easily be made into Common Lisp:
(defun fact-cps (n &optional (k #'values))
(if (zerop n)
(funcall k 1)
(fact-cps (- n 1)
(lambda (v)
(funcall k (* v n))))))
(fact-cps 10) ; ==> 3628800
Since the code didn't use several terms or the implicit progn i switched to if since I think it's slightly more readable. Other than that and the use of funcall because of the LISP-2 nature of Common Lisp it's the identical code to your Scheme version.
Here's an example of something you cannot do tail recursively without either mutation or CPS:
(defun fmapcar (fun lst &optional (k #'values))
(if (not lst)
(funcall k lst)
(let ((r (funcall fun (car lst))))
(fmapcar fun
(cdr lst)
(lambda (x)
(funcall k (cons r x)))))))
(fmapcar #'fact-cps '(0 1 2 3 4 5)) ; ==> (1 1 2 6 24 120)
EDIT
Where exactly in the Scheme example is the initial value for v coming
from?
For every recursion the function makes a function that calls the previous continuation with the value from this iteration with the value from the next iteration, which comes as an argument v. In my fmapcar if you do (fmapcar #'list '(1 2 3)) it turns into
;; base case calls the stacked lambdas with NIL as argument
((lambda (x) ; third iteration
((lambda (x) ; second iteration
((lambda (x) ; first iteration
(values (cons (list 1) x)))
(cons (list 2) x)))
(cons (list 3) x))
NIL)
Now, in the first iteration the continuation is values and we wrap that in a lambda together with consing the first element with the tail that is not computed yet. The next iteration we make another lambda where we call the previous continuation with this iterations consing with the tail that is not computed yet.. At the end we call this function with the empty list and it calls all the nested functions from end to the beginning making the resulting list in the correct order even though the iterations were in oposite order from how you cons a list together.
Is it required that lambda expressions only be used? Wouldn't a named
function accomplish more since you could maintain the state of each
continuation in a data structure which can be manipulated as needed?
I use a named function (values) to start it off, however every iteration of fact-cps has it's own free variable n and k which is unique for that iteration. That is the data structure used and for it to be a named function you'd need to use flet or labels in the very same scope as the anonymous lambda functions are made. Since you are applying previous continuation in your new closure you need to build a new one every time.
Is there in particular style/way of continuation passing style in
Common Lisp with or without all the macros?
It's the same except for the dual namespace. You need to either funcall or apply. Other than that you do it as in any other language.