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Symmetry structure in natural and artificial systems such as crystallography, chemistry, physics, and permutation puzzles, et cetera, can facilitate understanding and manipulation of these systems, as is well-known in the mathematical sciences and from the algebraic theory of groups (the latter now with about two centuries of continuous development). However, until now, these methods have not been fully exploited in Artificial Intelligence (AI). The authors show how AI systems based on computer algebraic methods can make use of such mathematical symmetry structure to automatically generate and manipulate hierarchical coordinate systems for finite systems whose generating symmetries are given.
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