Actually, I was saying that "chain" already emulates laziness, just in a somewhat unsafe way, as demonstrated by weird_take. In Haskell you'd probably just write

ints :: Int -> [Int] ints n = [n..]

and be done with it. -- ryan On Wed, May 20, 2009 at 9:41 AM, michael rice wrote:

OK, I think I understand. You were explaining how ML could be made to emulate Haskell laziness using streams, ala Scheme type delayed evaluation, so it's kind of like you were explaining a question I hadn't quite asked yet, which maybe explains my puzzlement, I hope.

Also, though my understanding of both Haskell and ML syntax is still rudimentary, I did catch an error in your definition of chain_take: the first arg of the cons should be i rather than n.

I'm still going through your code and may have further questions.

Thanks for your input.

Michael

--- On Wed, 5/20/09, Ryan Ingram wrote:

From: Ryan Ingram Subject: Re: [Haskell-cafe] showing a user defined type To: "michael rice" Cc: "Brandon S. Allbery KF8NH" , haskell-cafe@haskell.org Date: Wednesday, May 20, 2009, 4:12 AM

(Apologies for my mutilation of ML syntax, I don't completely know the language)

Consider the ML type int list, and this function to build one:

broken_repeat :: int -> int list broken_repeat n = Cons(n, broken_repeat(n))

This function is recursive, and doesn't terminate; it tries to build an infinite list of ints and your computer runs out of heap and/or stack trying to evaluate it.

But the "chain" type doesn't have this problem; you can see it as an int list that gets evaluated "on demand":

repeat :: int -> chain repeat(n) = Link(n, repeat)

always1 :: int -> chain always1(_) = Link(1, always1)

chain_take :: int * chain -> int list chain_take (0,_) = Nil chain_take (n,Link(i,f)) = Cons(n, chain_take(n-1, f(i)))

But, nothing in the "chain" type stops you from passing a different value to the function in the link:

weird_take :: int * int * chain -> int list weird_take (0,_,_) = Nil weird_take (n,v,Link(i,f)) = Cons(i, weird_take(n-1,v,f(v)))

Now, it's possible that chain_take returns the same list for two different "chain" inputs, but weird_take might return different lists depending on how f is implemented.  For example:     chain_take(5, repeat(1)) = [1,1,1,1,1]     chain_take(5, always1(1) = [1,1,1,1,1]

    weird_take(5, 2, repeat(1)) = [1,2,2,2,2]     weird_take(5, 2, always1(1)) = [1,1,1,1,1]

One way to fix this is to embed the "state" of the chain in the closure itself.

So, in ML, the type    unit -> X is commonly called a "thunk"; it can be used to delay computation until later, until it's demanded, just like any lazy value in Haskell.

f :: unit -> Int f () = 1

This f isn't very useful; it's basically the same as "1".  But consider this type:

datatype stream = Stream of (int * (unit -> stream))

stream1 () = Stream(1, stream1)

stream_take :: int*stream -> int list stream_take(0,_) = Nil stream_take(n,Stream(i,f)) = Cons(i, stream_take(n-1, f()))

Now there is no way to pass a different value like we did in weird_take; there's only ().

The difference is that the state gets embedded in the closure for the thunk:

stream_ints :: int -> (unit -> stream) stream_ints = fun n => fun () => Stream(n, stream_ints(n+1))

What you are doing here is encoding laziness; the Haskell version of this type:

...

data Stream = Stream !Int Stream -- !Int means the Int value is strict stream1 = Stream 1 stream1 stream_ints n = Stream n (stream_ints(n+1))

...

stream_take :: Int -> Stream -> [Int] stream_take 0 _ = [] stream_take n (Stream x xs) = x : stream_take (n-1) xs

No extra (\() -> ...) thunk is required, due to laziness :)

  -- ryan

On Tue, May 19, 2009 at 4:25 PM, michael rice wrote:

...

Hi Ryan,

I'm afraid you've lost me. Maybe if you showed how this would be used in ML I would get the picture.

Michael

--- On Tue, 5/19/09, Ryan Ingram wrote:

From: Ryan Ingram Subject: Re: [Haskell-cafe] showing a user defined type To: "michael rice" Cc: "Brandon S. Allbery KF8NH" , haskell-cafe@haskell.org Date: Tuesday, May 19, 2009, 2:40 PM

On Tue, May 19, 2009 at 7:07 AM, michael rice wrote:

...

A little further along in "The Little MLer" the ints function is replaced by other functions like primes and fibs, which also return Links:

fun primes(n)   = if is_prime(n+1)  then Link(n+1,primes)  else primes(n+1)

fun fibs(n)(m)   = Link(n+m,fibs(m))

which are passed to chain_item:

fun chain_item(n,Link(i,f))   = if eq_int(n,1)   then i   else chain_item(n-1,f(i))

which can be called to request the nth (12th) prime number beginning at 1.

- chain_item(12,primes(1)); GC #0.0.0.1.3.61:   (1 ms) val it = 37 : int -

So I guess the answer to your question about whether the function is ever called with a different value may be, yes.

Actually, it's not calling it with another value; notice that chain_item calls f(i), with i coming directly from the chain. Consider this alternate definition: (I'm not sure the syntax is exactly right, but you get the idea)

datatype chain =   Link of (int * ( unit -> chain ))

fun intsFrom(n) = fun unit => (n, intsFrom (n+1)) fun ints(n) = intsFrom n ()

Now you *can't* call the function embedded in the link with another value.

fun chain_item(n,Link(i,f))   = if eq_int(n,1)   then i   else chain_item(n-1,f unit)

And this type for "chain" is almost the same as [Int] in Haskell, due to laziness.

  -- ryan