[Haskell-beginners] Fwd: tower hanoi problem

Sat Feb 21 15:50:22 UTC 2015

I personally find the assertion more confusing than helpful. Interpreting a
pattern of "x:xs" as "acknowledging the base case" by eliminating it first
reads to me as a double negative: "if it is not the case that this list
does not have a first element" or some such.

I simply prefer to read it as a positive assertion: the list has a head,
and here's what to do with it.

To my eye, this also scales nicely when there are multiple terminal cases.
Here's a tiny example.

Given a list of fractional numbers, produce a smoothed list, where each
value is averaged with its immediate neighbors (except the first and last,
which fail to have both neighbors), as shown below.

[0.0,4.0,2.0,6.0,1.0,2.0]
[    2.0,4.0,3.0,3.0    ]

It seems natural to my eye to express this as, "A list with at least three
elements contributes a value to the result", as in:

smooth :: Fractional n => [n] -> [n]
smooth (a:z@(b:c:_)) = (a + b + c) / 3 : smooth z
smooth _             = []

I prefer this to the alternatives (at least the ones that I came up with)
that either explicitly compute the length to compare it to 3, or that
explicitly fret over the (multiple!) cases that terminate the recursion.

smooth' :: Fractional n => [n] -> [n]
smooth' []            = []
smooth' [_]           = []
smooth' [_,_]         = []
smooth' (a:z@(b:c:_)) = (a + b + c) / 3 : smooth' z

I don't care for having to wade through three terminal cases to get to the
one that interests me, for multiple reasons:

   - It strikes my eye as visual noise that offers no useful information
   compared to the first (shorter) solution.
   - Depending on the intelligence of the compiler, it may waste run time
   by testing for cases that will only be true near the end of a long argument
   list.
   - I can argue termination by observing that the recursive evaluation (smooth
   z) is made on a shorter argument.
   - In a lazy language, termination may not be an issue!

Evaluating

*Main> take 10 $ smooth [0,1..]
[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0]

works just fine without smooth having to arrive at the end of its argument.

Again, tastes may vary.

I haven't thought of a way to do this with a fold or comprehension that
seems as clear to me. I would be interested in seeing such a solution if
someone else sees a way to do it.

-jn-

---------- Forwarded message ----------
From: Dudley Brooks <dbrooks at runforyourlife.org>
Date: Thu, Feb 19, 2015 at 9:48 AM
Subject: Re: [Haskell-beginners] tower hanoi problem
To: Joel Neely <joel.neely at gmail.com>

 And to clarify my point, I would say that mathematically you do always
have to "take care of" (worry about) the base case first ... and you did!
And in the code, not just in your thinking:  Using "x:xs", rather than
"(head xs)", in the first line acknowledges the base case by making sure to
eliminate it first -- "x:xs" works precisely because it doesn't separate
the *concerns*; it contains an implicit "if this is not the base case".
What it does (why it's useful syntactic sugar) is let you separate (and
reorder) the *actions*.  A guard using "x:xs" does not actually have the
very clean SOC which you recommend, with the result that the concept "base
case" is actually represented in *two* places in your code.

Question:  Could you write it without the first line using "x:xs" or some
other construction which has an implicit "if this is not the base case"?
Probably yes ... probably some kind of "where" clause might put it
typographically at the end.  But that would be because Haskell's
interpreter/compiler executes the test in the "where" clause first.
Checking whether we're looking at the base case would still be the first
major execution step.  I suspect that there's no way to escape that ... the
most that can be done is to "look like" you're escaping it.

On 2/19/15 4:47 AM, Joel Neely wrote:

 Just to clarify the point of my earlier comment...

 I suggest that the "separation of concerns" (SOC) principle has many
applications. A common use shows up in the advice to represent each
distinct concept exactly one place in the code, and to do so in a way that
supports orthogonality (the freedom to mix-and-match).

 In this case, I used it to separate the thought process of designing the
code from the lexical layout of the code.

 I have no business legislating the order in which someone else thinks
about the cases (sometimes more than two!) encountered in decomposing a
problem. However, in my experience, the order in which I think about parts
of the code, and the order in which they are laid out in the source file,
are separate concerns. And I have often found it useful to consider them
separately.

 For example, in some problems (and language implementations) it may help
performance to ensure that the most frequent case is considered first,
especially when there are multiple cases to consider or when the
distinguishing conditions are costly to evaluate.

 I find that making my guards (conditions) explicit allows me the freedom
to order the alternatives in whatever way I find useful, without having to
worry about introducing a defect in the code.

 Incidentally, I also find it interesting to see the subtle effects that
our terminology has on the way we approach problems. Thinking of a list as
"it may be empty or not" takes my thoughts in a different direction than if
I think "it may have a head or not".

 By all means, think about your recursive functions any way you wish! Just
please don't tell me that I must place them is a specific order in my code.

 Regards,
-jn-

On Thu, Feb 19, 2015 at 3:02 AM, Dudley Brooks <dbrooks at runforyourlife.org>
wrote:

>  On 2/18/15 5:29 PM, Mike Meyer wrote:
>
>  On Wed, Feb 18, 2015 at 7:16 PM, Dudley Brooks <
> dbrooks at runforyourlife.org> wrote:
>
>>  Hmm.  Well, I'd say that that's a feature of, specifically, Haskell's
>> pattern-matching strategy and list-description syntax, rather than of
>> recursion in general or the structure of this particular problem.  In other
>> languages with recursion you might have no choice except to start with the
>> base case, even for this problem, or else you'd get the same kind of error
>> you mention below (depending on the language).  I think it's good when
>> you're *learning* recursion to always start with the base case(s).
>>
>  I disagree that this is a Haskell-specific feature. Any else-if like
> structure will have this property, no matter what language it's in. That
> Haskell provides a syntax as part of the function declaration is special,
> but that doesn't let you avoid the else-if construct when the problem
> requires it.
>
>  I don't understand.  I don't believe I said anything about avoiding
> else-if, or about not avoiding it.  But I'm not quite sure what you mean.
> Are you referring to
>
> if condition1
> then instruction1
> elseif condition2
>       then instruction2
>
> ?
>
> But what is condition1?  Wouldn't it probably be the base case, and
> instruction1 the procedure on the base case?
>
> Is there something special about "elseif" that guarantees that
> instruction1 *before* it won't crash if condition1 isn't the base case???
> I'm probably totally missing your intention here.
>
> But anyway, isn't it actually just Haskell's syntax "x:xs" that lets the
> pattern be tested and bypassed without crashing on an empty list, so that
> it *can* fall through to the base case at the end?  If Haskell only had the
> syntax "(head xs), then that *would* crash on the empty list if the empty
> list had not previously been taken care of as a base case, as Joel Neely
> pointed out.
>
> I didn't mean that *no* other language might have such a syntactical
> construction.  (I didn't mean "specifically Haskell", I meant "specifically
> the pattern matching".  Sorry about the ambiguity.)  So if some other
> language has such a construction, then it's still the *syntax* that lets
> you cheat on the base case; it's not the structure of the problem itself
> nor the basic underlying notion of recursion.
>
> I would also argue that in Mr Neely's first example, while the *explicit*
> base case [] is at the end, nevertheless the first line still *implicitly*
> refers to the base case:  pattern matching on "x:xs" says "*if* the data
> has the structure x:xs", i.e. "if it is not a bunch of other stuff ...
> including *not the empty list*!)".  Certainly you couldn't merely do the
> recursive step first without a condition like this particular one.  The
> reason this syntax *seems* to let you avoid thinking about the base case
> first is because secretly it says "only try this step if we're not looking
> at the base case"!
>
>  It may be my fondness for proof by induction, but I think doing the base
> case first is a good idea for another reason. The code for the recursive
> cases assumes that you can correctly handle all the "smaller" cases. If
> that's wrong because some assumption about the base case turns out to be
> false when you actually write it, then you have to rewrite the recursive
> cases for the correct base case. So it's better to make sure your base case
> is going to work before you start writing the code that's going to use it.
>
>  I was a math major, not a CS major, so I'm also prejudiced in favor of
> base case first.  And, as stated above, I think we *are* actually
> *considering* the base case first!  (We're merely putting off telling what
> to *do* with that base case.)  I think that the "syntactic sugar" of some
> languages obscures (intentionally, for purposes of convenience) what's
> really happening mathematically.
>
>
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 --
Beauty of style and harmony and grace and good rhythm depend on simplicity.
- Plato

-- 
Beauty of style and harmony and grace and good rhythm depend on simplicity.
- Plato
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