Cooperating in Video Games? Impossible! Undecidability of Team Multiplayer Games

We show the undecidability of whether a team has a forced win in a number of well known video games including: Team Fortress 2, Super Smash Brothers: Brawl, and Mario Kart.To do so, we give a simplification of the Team Computation Game [Hearn and Demaine, 2009] and use that to give an undecidable abstract game on graphs. This graph game framework better captures the geometry and common constraints in many games and is thus a powerful tool for showing their computational complexity

Recursed Is Not Recursive: A Jarring Result

Recursed is a 2D puzzle platform video game featuring treasure chests that, when jumped into, instantiate a room that can later be exited (similar to function calls), optionally generating a jar that returns back to that room (similar to continuations). We prove that Recursed is RE-complete and thus undecidable (not recursive) by a reduction from the Post Correspondence Problem. Our reduction is "practical": the reduction from PCP results in fully playable levels that abide by all constraints governing levels (including the 15x20 room size) designed for the main game. Our reduction is also "efficient": a Turing machine can be simulated by a Recursed level whose size is linear in the encoding size of the Turing machine and whose solution length is polynomial in the running time of the Turing machine.Comment: Submitted to MFCS2020, 21 page

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

The Computational Complexity of Angry Birds

The physics-based simulation game Angry Birds has been heavily researched by the AI community over the past five years, and has been the subject of a popular AI competition that is currently held annually as part of a leading AI conference. Developing intelligent agents that can play this game effectively has been an incredibly complex and challenging problem for traditional AI techniques to solve, even though the game is simple enough that any human player could learn and master it within a short time. In this paper we analyse how hard the problem really is, presenting several proofs for the computational complexity of Angry Birds. By using a combination of several gadgets within this game's environment, we are able to demonstrate that the decision problem of solving general levels for different versions of Angry Birds is either NP-hard, PSPACE-hard, PSPACE-complete or EXPTIME-hard. Proof of NP-hardness is by reduction from 3-SAT, whilst proof of PSPACE-hardness is by reduction from True Quantified Boolean Formula (TQBF). Proof of EXPTIME-hardness is by reduction from G2, a known EXPTIME-complete problem similar to that used for many previous games such as Chess, Go and Checkers. To the best of our knowledge, this is the first time that a single-player game has been proven EXPTIME-hard. This is achieved by using stochastic game engine dynamics to effectively model the real world, or in our case the physics simulator, as the opponent against which we are playing. These proofs can also be extended to other physics-based games with similar mechanics.Comment: 55 Pages, 39 Figure

Generation and Analysis of Content for Physics-Based Video Games

The development of artificial intelligence (AI) techniques that can assist with the creation and analysis of digital content is a broad and challenging task for researchers. This topic has been most prevalent in the field of game AI research, where games are used as a testbed for solving more complex real-world problems. One of the major issues with prior AI-assisted content creation methods for games has been a lack of direct comparability to real-world environments, particularly those with realistic physical properties to consider. Creating content for such environments typically requires physics-based reasoning, which imposes many additional complications and restrictions that must be considered. Addressing and developing methods that can deal with these physical constraints, even if they are only within simulated game environments, is an important and challenging task for AI techniques that intend to be used in real-world situations. The research presented in this thesis describes several approaches to creating and analysing levels for the physics-based puzzle game Angry Birds, which features a realistic 2D environment. This research was multidisciplinary in nature and covers a wide variety of different AI fields, leading to this thesis being presented as a compilation of published work. The central part of this thesis consists of procedurally generating levels for physics-based games similar to those in Angry Birds. This predominantly involves creating and placing stable structures made up of many smaller blocks, as well as other level elements. Multiple approaches are presented, including both fully autonomous and human-AI collaborative methodologies. In addition, several analyses of Angry Birds levels were carried out using current state-of-the-art agents. A hyper-agent was developed that uses machine learning to estimate the performance of each agent in a portfolio for an unknown level, allowing it to select the one most likely to succeed. Agent performance on levels that contain deceptive or creative properties was also investigated, allowing determination of the current strengths and weaknesses of different AI techniques. The observed variability in performance across levels for different AI techniques led to the development of an adaptive level generation system, allowing for the dynamic creation of increasingly challenging levels over time based on agent performance analysis. An additional study also investigated the theoretical complexity of Angry Birds levels from a computational perspective. While this research is predominately applied to video games with physics-based simulated environments, the challenges and problems solved by the proposed methods also have significant real-world potential and applications