2048 is (PSPACE) Hard, but Sometimes Easy

We prove that a variant of 2048, a popular online puzzle game, is PSPACE-Complete. Our hardness result holds for a version of the problem where the player has oracle access to the computer player's moves. Specifically, we show that for an $n \times n$ game board $\mathcal{G}$, computing a sequence of moves to reach a particular configuration $\mathbb{C}$ from an initial configuration $\mathbb{C}_0$ is PSPACE-Complete. Our reduction is from Nondeterministic Constraint Logic (NCL). We also show that determining whether or not there exists a fixed sequence of moves $\mathcal{S} \in \{\Uparrow, \Downarrow, \Leftarrow, \Rightarrow\}^k$ of length $k$ that results in a winning configuration for an $n \times n$ game board is fixed-parameter tractable (FPT). We describe an algorithm to solve this problem in $O(4^k n^2)$ time.Comment: 13 pages, 11 figure

Push-Pull Block Puzzles are Hard

This paper proves that push-pull block puzzles in 3D are PSPACE-complete to solve, and push-pull block puzzles in 2D with thin walls are NP-hard to solve, settling an open question by Zubaran and Ritt. Push-pull block puzzles are a type of recreational motion planning problem, similar to Sokoban, that involve moving a `robot' on a square grid with $1 \times 1$ obstacles. The obstacles cannot be traversed by the robot, but some can be pushed and pulled by the robot into adjacent squares. Thin walls prevent movement between two adjacent squares. This work follows in a long line of algorithms and complexity work on similar problems. The 2D push-pull block puzzle shows up in the video games Pukoban as well as The Legend of Zelda: A Link to the Past, giving another proof of hardness for the latter. This variant of block-pushing puzzles is of particular interest because of its connections to reversibility, since any action (e.g., push or pull) can be inverted by another valid action (e.g., pull or push).Comment: Full version of CIAC 2017 paper. 17 page

PSPACE-completeness of Pulling Blocks to Reach a Goal

We prove PSPACE-completeness of all but one problem in a large space of pulling-block problems where the goal is for the agent to reach a target destination. The problems are parameterized by whether pulling is optional, the number of blocks which can be pulled simultaneously, whether there are fixed blocks or thin walls, and whether there is gravity. We show NP-hardness for the remaining problem, Pull?-1FG (optional pulling, strength 1, fixed blocks, with gravity).Comment: Full version of JCDCGGG2019 paper, 22 pages, 25 figure

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

Games, puzzles, and computation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 147-153).There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games. In the first half of this thesis, I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis. In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Kohane, Cross Purposes, Tip over, and others.(cont.) Some of these have been well-known open problems for some time. For other games, including Minesweeper, the Warehouseman's Problem, Sokoban, and Rush Hour, I either strengthen existing results, or provide new, simpler hardness proofs than the original proofs.by Robert Aubrey Hearn.Ph.D

Abstract Push-2-F is PSPACE-Complete

We prove PSPACE-completeness of a class of pushingblock puzzles similar to the classic Sokoban, extending several previous results [1, 5, 12]. The puzzles consist of unit square blocks on an integer lattice; some of the blocks are movable. The robot may move horizontally and vertically in order to reach a specified goal position. The puzzle variants differ in the number of blocks that the robot can push at once, ranging from just one (Push-1-F) up to arbitrarily many (Push-*-F). We prove that Push-k-F and Push-*-F are PSPACEcomplete for k ≥ 2 using a reduction from Nondeterministi