user_guide:extend:permutations

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user_guide:extend:permutations [2019/01/25 16:02] – ↷ Page moved from reference:permutations to user_guide:extend:permutations oroehriguser_guide:extend:permutations [2019/05/07 13:37] (current) – [The problem] lkastner
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 ===== The problem ===== ===== The problem =====
-Many properties of "big" objects does not have a unique representation.  For example, a polyhedron can be given by a non-redundant system of inequalities and equations, called FACETS and AFFINE_HULL in application "polytope"; both are encoded as matrices, each row containing a normal vector to the corresponding hyperplane.  If you permute the order of FACET inequalities, or exchange the set of basis vectors of AFFINE_HULL, you will still obtain the description of the same polyhedron, but from the pure technical point of view it would be a different object.  As such, the multitude of representations should not constitute a serious problem as long as the functions, say, comparing two polyhedrons for equality, can deal with it algorithmically.+Many properties of "big" objects do not have a unique representation.  For example, a polyhedron can be given by a non-redundant system of inequalities and equations, called FACETS and AFFINE_HULL in application "polytope"; both are encoded as matrices, each row containing a normal vector to the corresponding hyperplane.  If you permute the order of FACET inequalities, or exchange the set of basis vectors of AFFINE_HULL, you will still obtain the description of the same polyhedron, but from the pure technical point of view it would be a different object.  As such, the multitude of representations should not constitute a serious problem as long as the functions, say, comparing two polyhedrons for equality, can deal with it algorithmically.
  
 The situation becomes more intricate, however, when further properties depend on the exact values of these non-unique ones.  For example, the order of rows of the FACETS matrix implicitly assigns an identity to each facet, knowing which is then crucial for correct interpretation of many other properties referring to FACETS, like DUAL_GRAPH or HASSE_DIAGRAM.  Now, when two production rules would create properties referring to FACETS and assume different orders of them, we'd end up with an inconsistent description of a polyhedron. The situation becomes more intricate, however, when further properties depend on the exact values of these non-unique ones.  For example, the order of rows of the FACETS matrix implicitly assigns an identity to each facet, knowing which is then crucial for correct interpretation of many other properties referring to FACETS, like DUAL_GRAPH or HASSE_DIAGRAM.  Now, when two production rules would create properties referring to FACETS and assume different orders of them, we'd end up with an inconsistent description of a polyhedron.
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