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From: sullivan@indra.com (Steve Sullivan)
Subject: Lattice constraint problem
Message-ID: <DzuB8B.Dus.0.net@indra.com>
Organization: Mathcom, Inc.
Date: Fri, 25 Oct 1996 16:31:22 GMT
Lines: 63

Hi -

I need to solve a system of inequality constraints on a complete lattice.
Do you have any recommendations for papers, books, or people
who have published in this area?

Let A = {a1, a2, ...} be a complete lattice.

Find values of xi on the lattice satisfying constraints of the form:
  xi <= xj
  xi <= aj

We also have constraints of the form:
     ( xi1 <= xj1  &  xi2 <= xj2  &  ... & xim <= xjm)
  OR ( xi1 <= xk1  &  xi2 <= xk2  &  ... & xim <= xkm)
  OR  ...


FOR EXAMPLE:

Suppose we have lattice L = {bottom, a1, a2, a3, top}
where
a1 # a2          (meaning a1 and a2 are not comparable)
bottom < a1 < a3 < top, 
bottom < a2 < a3 < top,

Picture:

   top
    |
    a3
   /  \
  a1  a2
   \  /
   bottom

Find x1, x2, x3, x4 on L subject to the constraints:

x3 <= x4
(x4 <= a1 & x1 <= a1 & x2 <= a1)  OR  (x4 <= a2 & x1 <= a2 & x2 <= a2)
x2 <= a1
all xi > bottom

This has the unique solution:

x1 = a1
x2 = a1
x3 = a1

Is there any reasonable solution method for this system?
Or will I have to use some sort of trial and error with backtracking?
Can you recommend any references, texts, people, or other
resources?

Many thanks -

Steve

=========================================================================

Steve Sullivan    sullivan@mathcom.com

=========================================================================
