[PPL-devel] integer versus rational solutions

[P M Hill]

On Wed, 8 Jul 2009, Michael Classen wrote:

>> HTH. Let me know if you have further queries wrt this domain, we will glad
>> to help.
>>
>> Pat
>
> Hi Pat,
>
> I basically just want to know if I can get correct integer results
> when using this combination of datatypes. I want to use operations
> like union, intersection, projection, basically most standard
> operations you want to use on polytopes.

Yes. These are already available in the latest release, but the best 
version of the product domain is in the products branch of the GIT 
repository. I would recommend you to use that if possible.

Sorry, I did not reply sooner to your previous email. I did try and see 
what could be done to help solve the problem you described. (i.e., 
transforming a grid x polyhedron product to one where the grid is the 
integer lattice). In fact, I have a proposal for adding a method to the 
product domain that I hope would be sufficient for what you need while, 
from the point of view of the PPL, fits into the existing structures. In 
particular, I am believe that the implementation work would be small!

That is:
In the product domain, (assuming for this explanation that the first 
domain is a grid and the second a C polyhedron) there would be a method 
such as:

bool
Partially_Reduced_Product<Grid, C_Polyhedron, R>
::affine_lattice_transform(const Grid& gr1)

that assigns to *this = <gr, ph> the product <gr1, ph1> such that there is 
an affine function (ie a sequence of affine image mappings) T, T(gr) = gr1 
and T(ph) = ph1.

If there is no invertible affine function T such that T(gr) = gr1, then 
the method could return false and otherwise true.

If you call the method with gr1 = integral lattice, and gr as a full 
dimensional discrete grid you will get the polyhedron transformed to one 
where the integral points correspond to the grid points and you can use 
whatever tools you like to count the number of integer points. By calling 
the same method with this* = <gr1, ph1> with the argument gr, you will
be able to invert the operation.

I can try and implement something like this next week (while away at a 
conference) as I think this would be useful anyway for other applications.

Best wishes,
   Pat