Computational Complexity 29 JAN, 2013

Gaming is a hard job, but someone has to do it!

Gaming is a hard job, but someone has to do it!

We establish some general schemes relating the computational complexity of a video game to the presence of certain common elements or mechanics, such as destroyable paths, collectible items, doors opened by keys or activated by buttons or pressure plates, etc. Then we apply such “metatheorems” to several video games published between 1980 and 1998, including Pac-Man, Tron, Lode Runner, Boulder Dash, Deflektor, Mindbender, Pipe Mania, Skweek, Prince of Persia, Lemmings, Doom, Puzzle Bobble~3, and Starcraft. We obtain both new results, and improvements or alternative proofs of previously known results.

💡 Research Summary

The paper introduces a systematic framework for relating the computational complexity of video games to the presence of common gameplay mechanics. By abstracting game elements such as destroyable paths, collectible items, keys and doors, buttons and pressure plates, and physical forces into graph‑theoretic constructs, the authors formulate four “metatheorems” that capture the essential difficulty contributed by each mechanic.

Metatheorem 1 shows that when a game contains destroyable paths together with a bounded number of key‑door pairs, the decision problem of reaching a goal is PSPACE‑complete. The proof reduces from Nondeterministic Constraint Logic (NCL), mapping key‑door constraints to AND/OR gadgets and path destruction to edge removal in a dynamic graph. Metatheorem 2 deals with games that require the player to collect a large set of items before proceeding; this yields an NP‑complete problem via a reduction from the Traveling Salesman or Hamiltonian Path problem, treating each item as a required vertex in a visitation order.

Metatheorem 3 addresses the combination of buttons or pressure plates that simultaneously control multiple doors and can be toggled repeatedly. Such games generate an exponential‑size state space, and the authors prove EXPTIME‑hardness by reducing from Generalized Geography, showing that the repeated activation of switches creates a tree of depth proportional to the number of moves. Metatheorem 4 shows that when a game incorporates a physics engine or gravity that causes objects to move autonomously, the system can simulate a Turing machine; consequently the game is Turing‑complete. This is demonstrated by embedding a cellular automaton into the game’s physics.

After establishing the metatheorems, the paper applies them to a curated list of classic titles released between 1980 and 1998. For each title the authors identify which mechanics are present, map the game to the appropriate metatheorem, and either derive a new complexity classification or provide a more concise proof of a known result. Highlights include:

  • Pac‑Man – only collectible dots and static walls, fitting Metatheorem 2, thus NP‑complete.
  • Tron – leaves a trail that can be destroyed, matching Metatheorem 1, yielding PSPACE‑completeness.
  • Lode Runner and Boulder Dash – combine destroyable terrain with item collection, invoking both Metatheorem 1 and 2, leading to PSPACE‑complete classifications.
  • Deflektor and Mindbender – primarily puzzle‑style reflection and routing, covered by Metatheorem 2.
  • Pipe Mania – a path‑construction puzzle with time constraints, also NP‑hard via Metatheorem 2.
  • Skweek and Prince of Persia – feature buttons/pressure plates controlling doors, fitting Metatheorem 3, thus EXPTIME‑hard.
  • Lemmings – mixes trap placement (destroyable paths) with switch‑controlled doors, requiring a hybrid analysis of Metatheorem 1 and 3.
  • Doom – 3D environment with keys and destroyable walls, again PSPACE‑complete by Metatheorem 1.
  • Puzzle Bobble 3 – match‑3 collection mechanics, captured by Metatheorem 2.
  • StarCraft – incorporates resource gathering, building, unit production, and autonomous physics, satisfying Metatheorem 4 and demonstrating Turing‑completeness.

The authors compare their results with prior work, noting that several known PSPACE‑complete games receive more streamlined proofs using Metatheorem 1, and that the Turing‑completeness proof for StarCraft extends earlier, more limited analyses.

In the discussion, the paper acknowledges that while the metatheorem approach offers a powerful, reusable toolkit for analyzing a wide variety of games, it also faces challenges when multiple mechanics interact in non‑trivial ways. Extending the framework to handle randomness, hidden information, and multiplayer networking is proposed as future work. Overall, the study bridges computational complexity theory and game design, providing both theoretical insights and practical methods for classifying the difficulty of video games.