[Haskell-beginners] tower hanoi problem

[Dudley Brooks]

On 2/17/15 10:56 PM, Dudley Brooks wrote:
> On 2/16/15 7:06 PM, Doug McIlroy wrote:
>>> My way of working is one problem at the time.
>>> So first solve the itterate one and after that I gonna try to solve the
>>> recursion one.
>>> Otherwise I get confused.
>> This is the crux of the matter. You must strive to think those thoughts
>> in the opposite order. Then you won't get confused.
>>
>> Recursion takes a grand simplifying view: "Are there smaller problems of
>> the same kind, from the solution(s) of which we could build a 
>> solution of
>> the problem at hand?" If so, let's just believe we have a solver for the
>> smaller problems and build on them. This is the recursive step.
>>
>> Of course this can't be done when you are faced with the smallest
>> possible problem. Then you have to tell directly how to solve
>> it. This is the base case.
>>
>> [In Hanoi, the base case might be taken as how to move a pile
>> of one disc to another post. Even  more simply, it might be how
>> to move a pile of zero discs--perhaps a curious idea, but no more
>> curious than the idea of 0 as a counting number.]
>>
>> A trivial example: how to copy a list (x:xs) of arbitrary length.
>> We could do that if we knew how to copy the smaller list tail, xs.
>> All we have to do is tack x onto the head of the copy. Lo, the recipe
>>     copy (x:xs) = x : copy xs
>> Final detail: when the list has no elements, there is no smaller
>> list to copy. We need another rule for this base case. A copy
>> of an empty list is empty:
>>     copy [] = []
>>
>> With those two rules, we're done. No iterative reasoning about
>> copying all the elements of the list. We can see that afterward,
>> but that is not how we got to the solution.
>>
>> [It has been suggested that you can understand recursion thus
>>     > Do the first step.  Then (to put it very dramatically)
>>     > do *everything else* in *a single step*!
>> This point of view works for copy, and more generally for
>> "tail recursion", which is trivially transformable to iteration.
>> It doesn't work for Hanoi, which involves a fancier recursion
>> pattern. The "smaller problem(s)" formulation does work.]
>
> I simplified it (or over-dramatized it) to the point of 
> not-quite-correctness.  I was trying to dramatize the point of how 
> surprising the idea of recursion is, when you first encounter it 
> (because I suspected that the OP had not yet "grokked" the elegance of 
> recursion) -- and remembering my own Aha! moment at recursive 
> definitions and proofs by induction in high school algebra, back when 
> the only "high-level" programming language was assembly.  I see that I 
> gave the mistaken impression that that's the *only* kind of recursive 
> structure.  What I should have said, less dramatically, is
>
>     Do the base case(s)
>     Then do the recursion -- whatever steps that might involve, 
> including possibly several recursive steps and possibly even several 
> single steps, interweaved in various possible orders.
>
> You can't *start* with the recursion, or you'll get either an infinite 
> loop or an error.
>
> I wouldn't quite call the conversion of tail-recursion to iteration 
> trivial, exactly ... you still have to *do* it, after all!  And when I 
> did CS in school (a long time ago), the equivalence had only fairly 
> recently been recognized.  (By computer language designers, anyway.  
> Maybe lambda-calculus mathematicians knew it long before that.)  
> Certainly the idea that compilers could do it automatically was pretty 
> new.  If it were *completely* trivial, it would have been recognized 
> long before! ;^)
>
> If you're younger you probably grew up when this idea was already 
> commonplace.  Yesterday's brilliant insight is today's trivia!

BTW, since, as you and several others point out, the recursive solution 
of Towers of Hanoi does *not* involve tail recursion, that's why it's 
all the more surprising that there actually is a very simple iterative 
solution, almost as simple to state as the recursive solution, and 
certainly easier to understand and follow by a novice or non-programmer, 
who doesn't think recursively.
>
>> In many harder problems a surefire way to get confused is to
>> try to think about the sequence of elementary steps in the final
>> solution. Hanoi is a good case in point.
>>
>> Eventually you will come to see recursion as a way to confidently
>> obtain a solution, even though the progress of the computation is
>> too complicated to visualize. It is not just a succinct way to
>> express iteration!
>>
>> Doug McIlroy
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>