Towards Automated Discovery of God-Like Folk Algorithms for Rubik’s Cube

Abstract

We present a multi-objective meta-search procedure that constructs candidate algorithms for state-space search puzzles like Rubik's cube. The candidate algorithms take the form of macro databases, i.e., rule tables that specify sequences of actions to perform in different states. Rules are repeatedly applied until the puzzle is solved. The objectives favor candidates that are god-like (solving the puzzle in fewer steps) and folk-like (having fewer rules in the macro database). We build each candidate with a non-deterministic rule table construction, and then optimize over the non-deterministic choice points to find candidates near the Pareto-optimal trades-offs between godliness and folksiness. We prove that the rule table construction is correct: it always terminates and solves every state at termination. This is verified empirically on the full 2x2x2 "pocket" cube, where correct (but unoptimized) constructions take under one hour and the total number of rules is less than 10% the number of possible states. We also empirically assess the multi-objective optimization on restricted variants of the cube with up to 29K possible states, showing relative improvements in the objectives between 14-20%. Avenues for scaling up the method in future work are discussed.