Abstract
Planning is defined as the problem of synthesizing a desired behavior from given basic operations, and learning is defined as the dual problem of analyzing a given behavior to determine the unknown basic operations. Algorithms for solving these problems in the context of invertible operations on finite-state environments are developed. In addition to their obvious artificial intelligence applications, the algorithms can efficiently find the shortest way to solve Rubik's cube, test ping-pong protocols, and solve systems of equations over permutation groups.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
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