Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2011

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Reduce in Mathematica 5 vs Mathematica 8 (2nd problem)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg115269] Re: Reduce in Mathematica 5 vs Mathematica 8 (2nd problem)
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 6 Jan 2011 02:03:01 -0500 (EST)

olfa wrote:
> Hi Mathematica Community,
> 
> First,wish you happy and successfull new year.
> 
> For this 2nd problem in the same subject,I have this system to solve:
> 
> Reduce[Not[
>   ForAll[{aaP, abP, iP, jP, sP, tP, uP, xP, yP, zP},
>    Implies[t == tP && i + x == iP + xP && y == yP &&
>      j t + z == jP tP + zP && t x + z == tP xP + zP &&
>      Floor[Log[j]/Log[2]] == Floor[Log[jP]/Log[2]] &&
>      Floor[Log[x]/Log[2]] == Floor[Log[xP]/Log[2]] && x >= xP,
>     t x == tP xP]]]]
> 
> in mathematica 5 the output is given in a very short time and is "the
> system cannot be solved with the method available to Reduce" this
> suits me (although I wish it to be  the output "True" which is the
> right answer)
> 
> in mathematica 8 the kernel still in running indefinitely and this
> does not suit me at all :(
> 
> so how to deal with that?

One response mentioned using TimeConstrained. This is a useful thing to 
do for such problems.

Another useful thing is to state explicitly all your domain assumptions. 
You show inequalities and also use equalities involving logarithms. Are 
you considering that these might involve negatives e.g. if xP<0? I 
assume not, but of course I am just guessing (something that you want to 
avoid when posing questions to a Usenet forum).

Also you might want to simplify as much as possible. In this case you 
seem to be looking for a counterexample to an implication. This could be 
recast as a problem for FindInstance. And you can remove extraneous 
variables. And insert the ones that are missing. And do explicit rule 
replacements to handle var1==var2 constraints.

Also you could perhaps tackle relaxations of the original problem, to 
see if solutions exist in such cases. For instance, it was pointed out 
in a prior response that Reduce does not seem able to handle Floor[...] 
constructs. You might want to take this to heart because it will save 
you time and effort. So relaxing those Floor equality constraints, you 
could replace e.g.
Floor[Log[x]/Log[2]] == Floor[Log[xP]/Log[2]]
with
x <= 2*xP
(this also uses the constraint that x>=xP, and I also add some 
restrictions of positivity because i did not think you meant for those 
logarithms to get negative). The point here is that this does not force 
the Floors to be equal, but they will now differ by at most 1. hence it 
provides a smallish relaxation of the original problem.

So at long last here is a problem that can be solved.

In[1097]:= vars = {i, j, x, y, z, iP, jP, tP, xP, yP, zP};

In[1127]:=
expr1 = i + x == iP + xP && j t + z == jP tP + zP &&
     t x + z == tP xP + zP && j <= 2*jP && x <= 2*xP && x >= xP + 1 &&
     j >= 1 && jP >= 1 && x >= 1 && xP >= 1 && t >= 1 /. {t -> tP,
     y -> yP};

In[1128]:= res = FindInstance[expr1, vars, Integers]

Out[1128]= {{i -> 0, j -> 6, x -> 6, y -> 0, z -> 0, iP -> 2, jP -> 4,
    tP -> 1, xP -> 4, yP -> 0, zP -> 2}}

Does this give a counterexample to the original implication?

In[1130]:=
Implies[t == tP && i + x == iP + xP && y == yP &&
      j t + z == jP tP + zP && t x + z == tP xP + zP &&
      Floor[Log[j]/Log[2]] == Floor[Log[jP]/Log[2]] &&
      Floor[Log[x]/Log[2]] == Floor[Log[xP]/Log[2]] && x >= xP,
     t x == tP xP] /. t -> tP /. res[[1]] // N

During evaluation of In[1130]:= Floor::meprec: Internal precision limit 
$MaxExtraPrecision = 50.` reached while evaluating Floor[Log[4]/Log[2]]. >>

During evaluation of In[1130]:= Floor::meprec: Internal precision limit 
$MaxExtraPrecision = 50.` reached while evaluating Floor[Log[4]/Log[2]]. >>

Out[1130]= False

So yes, we have a valid counterexample.

Daniel Lichtblau
Wolfram Research


  • Prev by Date: Re: Session Privacy in Mathematica 8 w/ Wolfram|Alpha free-form input?
  • Next by Date: Running Fortran programs from Mathematica
  • Previous by thread: Re: Mathematica 8 & reports / books
  • Next by thread: Running Fortran programs from Mathematica