Computational Complexity of Two-Dimensional Platform Games

We analyze the computational complexity of various two-dimensional platform games. We state and prove several meta-theorems that identify a class of these games for which the set of solvable levels is NP-hard, and another class for which the set is even PSPACE-hard. Notably COMMANDERKEEN is shown to be NP-hard, and PRINCEOFPERSIA is shown to be PSPACE-complete. We then analyze the related game Lemmings, where we construct a set of instances which only have exponentially long solutions. This shows that an assumption by Cormode in [3] is false and invalidates the proof that the general version of the LEMMINGS decision problem is in NP. We then augment our construction to only include one entrance, which makes our instances perfectly natural within the context of the original game.