Discovering Rubik's Cube Subgroups using Coevolutionary GP

Abstract

This work reports on an approach to direct policy discovery (a form of reinforcement learning) using genetic programming (GP) for the 3 × 3 × 3 Rubik's Cube. Specifically, a synthesis of two approaches is proposed: 1) a previous group theoretic formulation is used to suggest a sequence of objectives for developing solutions to different stages of the overall task; and 2) a hierarchical formulation of GP policy search is utilized in which policies adapted for an earlier objective are explicitly transferred to aid the construction of policies for the next objective. The resulting hierarchical organization of policies explicitly demonstrates task decomposition and policy reuse. Algorithmically, the process makes use of a recursive call to a common approach for maintaining a diverse population of GP individuals and then learns how to reuse subsets of programs (policies) developed against the earlier objective. Other than the two objectives, we do not explicitly identify how to decompose the task or mark specific policies for reuse. Moreover, at the end of evolution we return a population solving 100% of 17,675,698 different initial Cubes for the two objectives currently in use. CCS Concepts •Computing methodologies → Sequential decision making; Genetic programming;

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“…The path cost of x s is 0. The heuristic function h(x) is obtained from the learned cost-to-go function shown in equation (2).…”

“…Developing machine learning algorithms to deal with this property of the Rubik's cube might provide insights into learning to solve planning problems with large state spaces. Although machine learning methods have previously been applied to the Rubik's cube, these methods have either failed to reliably solve the cube [1][2][3][4] or have had to rely on specific domain knowledge 5,6 . Outside of machine learning methods, methods based on pattern databases (PDBs) have been effective at solving puzzles such as the Rubik's cube, the 15 puzzle and the 24 puzzle 7,8 , but these methods can be memory-intensive and puzzle-specific.…”

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