Exhaustive search and hardness proofs for games

Abstract

This thesis explores several games from two perspectives: exhaustive search and hardness proofs. First, we present an exhaustive search for hardness proofs: a system for finding motion planning simulations. Second, we prove that the pencil-and-paper puzzle Tatamibari is NP-complete, a proof developed using a Tatamibari solver we wrote based on the Z3 SMT solver. Third, we find by computer search that the board game Push Fight played on a board with one column (four squares) removed is a draw. Then we prove that mate-in-1 in generalized Push Fight is NP-complete and that determining the winner of a game in progress is PSPACE-hard. Fourth, we prove that path puzzles are NP-complete, ASP-complete, and #P-complete. We describe a solver for path puzzles based on depth-first search that solves 14 of 15 puzzles from the last chapter of the path puzzles book. Fifth, we present a nonogram solver based on automaton intersection. Relatedly, we prove that finding an optimal automaton intersection ordering is PSPACE-hard. Sixth, we analyze puzzles from the video game The Witness and obtain NP-completeness for most clue types and [sigma]₂-completeness for puzzles containing antibody clues. Finally, we propose a generic framework for parsing screenshots of grid-based video games.