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Re: Resolve[Exists[i,{-1,0,2,-3,4,0}>0],Integers] does not work

• To: mathgroup at smc.vnet.net
• Subject: [mg85437] Re: [mg85409] Resolve[Exists[i,{-1,0,2,-3,4,0}>0],Integers] does not work
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 10 Feb 2008 05:21:24 -0500 (EST)
• References: <200802090919.EAA16976@smc.vnet.net>

On 9 Feb 2008, at 10:19, janos wrote:

> Resolve[Exists[x, a x^2 + b x + c == 0 && x > 0], Reals]
>
> (taken from the Help) works fine. However,
>
> Resolve[Exists[i, Part[{-1, 0, 2, -3, 4, 0}, i] > 0], Integers]
>
> does not work.
> My guess is that Part is evaluated differently from Equal,
> but I do not know how to fix the problem.
>
> Thank you.
>
> Janos
>

The problem is that Quanitifer Elimination , which is the algorithm  
that is being used here, works only with polynomial equations and  
inequalities and only over the real or complex numbers. So obviously  
you can't use Part or any other programming (rather than algebraic)  
construct. That's the main reason why this sort of thing won't work at  
all. But in addition, no known general quantifier elimination  
algorithm works over the integers so only very simple special cases  
have been implmented in Mathematica. For example:

Resolve[Exists[i, Element[i, Integers], i^2 == 2]]
  False

works, but the more complicated:

  Resolve[ForAll[i, Element[i, Integers],
      Exists[j, Element[j, Integers], j^2 == i]]]

ForAll[i, Element[i, Integers],
    Exists[j, Element[j, Integers], j^2 == i]]

doesn't, while over the reals there is no problem:

  Resolve[ForAll[i, Element[i, Reals], Exists[j, Element[j, Reals],
        j^2 == i]]]
False

and

  Resolve[ForAll[i, Element[i, Reals] && i >= 0,
      Exists[j, Element[j, Reals], j^2 == i]]]
True

and so on.


Andrzej Kozlowski

• References:
  • Resolve[Exists[i,{-1,0,2,-3,4,0}>0],Integers] does not work
    • From: janos <janostothmeister@gmail.com>