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

PSPACE-completeness of Bloxorz and of games with 2-Buttons

Bloxorz is an online puzzle game where players move a 1 by 1 by 2 block by tilting it on a subset of the two dimensional grid. Bloxorz features switches that open and close trapdoors. The puzzle is to move the block from its initial position to an upright position on the destination square. We show that the problem of deciding whether a given Bloxorz level is solvable is PSPACE-complete and that this remains so even when all trapdoors are initially closed or all trapdoors are initially open. We also answer an open question of Viglietta, showing that 2-buttons are sufficient for PSPACE-hardness of general puzzle games. We also examine the hardness of some variants of Bloxorz, including variants where the block is a 1 by 1 by 1 cube, and variants with single-use tiles

PSPACE-completeness of Bloxorz and of games with 2-Buttons

Bloxorz is an online puzzle game where players move a 1 by 1 by 2 block by tilting it on a subset of the two dimensional grid. Bloxorz features switches that open and close trapdoors. The puzzle is to move the block from its initial position to an upright position on the destination square. We show that the problem of deciding whether a given Bloxorz level is solvable is PSPACE-complete and that this remains so even when all trapdoors are initially closed or all trapdoors are initially open. We also answer an open question of Viglietta, showing that 2-buttons are sufficient for PSPACE-hardness of general puzzle games. We also examine the hardness of some variants of Bloxorz, including variants where the block is a 1 by 1 by 1 cube, and variants with single-use tiles

PSPACE-completeness of Bloxorz and of games with 2-buttons

Bloxorz is an online puzzle game where players move a 1 × 1 × 2 block by tilting it on a subset of the two dimensional grid, that also features switches that open and close trapdoors. The puzzle is to move the block from its initial position to an upright position on the goal square. We show that the problem of deciding whether a given Bloxorz level is solvable is PSPACE-complete and that this remains so even when all trapdoors are initially closed or all trapdoors are initially open. We also answer an open question of Viglietta [6], showing that 2- buttons are sufficient for PSPACE-hardness of general puzzle games. We also examine the hardness of some variants of Bloxorz, including variants where the block is a 1 × 1 × 1 cube, and variants with single-use tiles

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