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Re: Re: unable to FullSimplify

On 24 Apr 2006, at 19:01, Vladimir wrote:

> Andrzej Kozlowski wrote:
>>> In[]:= subdivide[a_ + b_] := FullSimplify[a] + FullSimplify[b];
>>>          FullSimplify[Expand[x + (x + x^2)^4],
>>>            TransformationFunctions -> {Automatic, subdivide}]
>>> Out[]= x + x^4*(1 + x)^4
>>
>> Well, yes, it works nicely here but the question is, if you make this
>> a default transformation transformation for FullSimplify how will it
>> effect complexity?
>
> Some option to FullSimplify would be enough for this to be used only
> when needed. Anyway, it was just one quick and arbitrary example
> to show that FullSimplify can be improved without too much effort
>
>
>> In fact it even seems hard to see how you would avoid numerous
>> unnecessary attempts at simplifying the same expression...
>
> That's quite normal it seems. For example, to simplify just a+b,
> ComplexityFunction is called 28 times(!) Here's a proof:
>
> reveal[expr_] := (Print[expr]; LeafCount[expr])
> FullSimplify[a + b, ComplexityFunction -> reveal]
>
>
>> Some more thoughts: if your subdivide were really a default
>> transformation in FullSimplify it may well lead to an infinite loop,
>> since it would be calling on itself.
>
> Are you sure? This subdivide reduces its argument on each
> invocation so I don't see how infinite loop can be possible.
> I couldn't find any examples with such a problem.
>
>
>> Note however also the folowing interesting feature:
>>
>> u = Expand[x + (x + x^2)^4];
>>
>> FullSimplify[f[u] == f[x + (x + x^2)^4]]
>>
>> True
>
> Well, since f[a]==f[a] evaluates to True, Mathematica simplifies
> the above equation because it is able to prove equality of f's
> arguments.
>
> --
> Vladimir
>

As far as I can tell your last statement is vacuous; it essentially  
says : it happens because it happens. The true reason is actually  
quite complicated, as was explained in Adam Strzebonski's post. I am  
amazed you missed or ignored his reply to this thread since he posted  
it to the list and also sent to you personally. Also, you can trust  
him in this matter much more than me or anyone else, since not only  
is he the programmer of FullSimplify, but also a very good  
mathematician and an expert in exactly the kind of mathematics that  
FullSimlify relies on. Since he also did not choose to contradict my  
other points I think they are basically correct, except perhaps the  
one about the possibility of an infinite loop. (I thought that your  
transformation together with some other transformation used by  
FullSimplify might result in an infinite loop, but I have not been  
able to  find any obvious example).

Of course some things are not a matter of mathematics or programming  
but simply design decisions. One of these is that FullSimplify has  
Automatic as the only built in value for the option  
TransformationFunctions with other values being left to the user. It  
might be that one could add one or two collections of particularly  
useful but high time complexity transformations for some specific  
purposes. I am sure however that not doing so was a carefully  
considered decision.
Finally, I must say I am not a little shocked that anyone can believe  
that FullSimplify could be improved so easily, but as I do not get  
paid to argue about this, I will leave it at that ...

Andrzej Kozlowski
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