Computational Understanding and Manipulation of Symmetries

Details Computational Understanding and Manipulation of Symmetries Computational Understanding and Manipulation of Symmetries

2009-08-21

arxiv.org/abs/0908.3091v1

Computer Science

Artificial Intelligence

8 pages, 4 figures

Scientific paper

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). We 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. Such hierarchical coordinate systems correspond to subgroup chains in the group structure determined by generating symmetries of the system. We demonstrate how, for any finite-state symmetry system, these coordinate systems (1) can be generated automatically, i.e. deriving formal models for understanding the finite-state symmetry system, and (2) can be deployed in manipulating the system automatically. Thus, without learning, manipulation of such systems is reduced via algebra to sequential computation in a hierarchy of simple (or simpler) group coordinates. This general, implemented method is illustrated with examples from coordinate systems on permutation puzzles such as Rubik's cube.

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