Playing games with algorithms: Algorithmic combinatorial game theory

Demaine, Erik D.; Hearn, Robert A.

doi:10.1017/cbo9780511807251.002

Cited by 37 publications

(30 citation statements)

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“…They have a long history. Hearn [36] and Demaine et al [18] showed that tiles can be arranged to create logic gates and used this technique to prove pspace-completeness for a variety of sliding-block puzzles. Hearn expressed the idea of building computers from the sliding blocks-many of the logic gates could be connected together, and the user could propagate a signal from one gate to the next by sliding intermediate tiles.…”

Section: Underactuated Controlmentioning

confidence: 99%

“…Many variations of block-pushing puzzles have been explored from a computational complexity viewpoint with a seminal paper proving NP-hardness by Gordon Wilfong in 1991 [64]. The general case of motion-planning when each command moves particles a single unit in a world composed of even a single robot and both fixed and moveable squares is in the complexity class pspace-complete [18,20,38].…”

Section: Manipulationmentioning

confidence: 99%

See 1 more Smart Citation

Particle computation: complexity, algorithms, and logic

Becker

Demaine²,

Fekete

et al. 2017

Nat Comput

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We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). Upon activation of the field, each particle moves maximally in the same direction until forward progress is blocked by a stationary obstacle or another stationary particle. In an open workspace, this system model is of limited use because it has only two controllable degrees of freedom-all particles receive the same inputs and move uniformly. We show that adding a maze of obstacles to the environment can make the system drastically more complex but also more useful.We provide a wide range of results for a wide range of questions. These can be subdivided into external algorithmic problems, in which particle configurations serve as input for computations that are performed elsewhere, and internal logic problems, in which the particle configurations themselves are used for carrying out computations.For external algorithms, we give both negative and positive results. If we are given a set of stationary obstacles, we prove that it is NP-hard to decide whether a given initial configuration of unit-sized particles can be transformed into a desired target configuration. Moreover, we show that finding a control sequence of minimum length is PSPACE-complete. We also work on the inverse problem, providing constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations.For internal logic, we investigate how arbitrary computations can be implemented. We demonstrate how to encode dual-rail logic to build a universal logic gate that concurrently evaluates and, nand, nor, and or operations. Using many of these gates and appropriate interconnects, we can evaluate any logical expression. However, we establish that simulating the full range of complex interactions present in arbitrary digital circuits encounters a fundamental difficulty: a fan-out gate cannot be generated. We resolve this missing component with the help of 2×1 particles, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we provide rules for replicating arbitrary digital circuits.

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Section: Underactuated Controlmentioning

confidence: 99%

Section: Manipulationmentioning

confidence: 99%

Particle computation: complexity, algorithms, and logic

Becker

Demaine²,

Fekete

et al. 2017

Nat Comput

Self Cite

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“…The Find and Verify Solution level solves the puzzle starting from the setup generated in the previous level and using one of the many game solving techniques to evolve from setup to the solution state; for reviews of these techniques we refer to [22][23][24]. Depending on the amenability of the puzzle to different solving strategies, this level may employ algorithms from brute force search to genetic algorithms, game-tree search, and SAT-solving.…”

Section: Workflow Structurementioning

confidence: 99%

POGGI: generating puzzle instances for online games on grid infrastructures

Iosup

2010

Concurrency and Computation

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key words: game content generation, grid computing, puzzle, MMOG, resource management. SUMMARYThe developers of Massively Multiplayer Online Games (MMOGs) rely on attractive content such as logical challenges (puzzles) to entertain and generate revenue from millions of concurrent players. While a large part of the content is currently generated by hand, the exponentially increasing number of players and their new demand for player-customized content have made manual content generation undesirable. Thus, in this work we investigate the problem of automated, player-customized game content generation at MMOG scale; to make our investigation tractable, we focus on puzzle game content generation. In this context, we design POGGI, an architecture for generating player-customized game content using on-demand, grid resources. POGGI focuses on the large-scale generation of puzzle instances that match well the solving ability of the players, and that lead to fresh playing experience. Using our reference implementation of POGGI, we show through experiments in a real resource pool of over 1,600 nodes that POGGI can generate commercial-quality content at MMOG scale. IntroductionMassively Multiplayer Online Games (MMOGs) have emerged in the past decade as a new type of large-scale distributed application: real-time virtual world simulations entertaining at the same time millions of players located around the world. In real deployments, the operation and maintenance of these applications includes two main components, one running the large-scale virtual world simulation, the other populating the virtual world with content that keeps the players engaged (and paying). So far, content has been provided exclusively by human content designers, but the growth of the player population, the lack of scalability of the production pipeline, and the increase in the price ratio between human work and computation make this situation undesirable for the future. Extending our previous work [14], here we investigate the problem of automated, player-customized content generation for MMOGs using grids. * Correspondence to: Email: a.iosup@tudelft.nl. We thank Dr. Dick Epema and Dr. Miron Livny for support. There are many types of content present in MMOGs, from 3D objects populating the virtual worlds, to abstract challenges facing the players. Since content generation is timeconsuming and costly, MMOG developers seek the content types that entertain and challenge players for the longest time. Puzzle games, that is, games in which the player is entertained by solving a logical challenge, are a much sought-after MMOG content type; for instance, players may spend hours on a chess puzzle [17,24] that enables exiting a labyrinth. Today, puzzle game content is generated by teams of human designers, which poses scalability problems to the production pipeline. While several commercial games have made use of content generation [1,5,7,29], they have all been small-scale games in terms of number of players, and did not consider the generation of puzzle game...

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“…Following the success of Sudoku, generalizations such as Mojidoku which uses letters instead of digits, and other similar logic puzzles like Hitori, Masyu, Futoshiki, Hashiwokakero, or Nurikabe were developed. Many of them have been proved to be NP-complete [12,5].Interactive ZKPs were introduced by Goldwasser et al [8], and it was then shown that for any NP-complete problem there exists an interactive ZKP protocol [7]. An extension by Ben-Or et al [1] showed that every provable statement can be proved in zero-knowledge.…”

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confidence: 99%

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Physical Zero-Knowledge Proof for Makaro

Bultel

Dreier

Dumas

et al. 2018

Lecture Notes in Computer Science

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Akari, Takuzu, Kakuro and KenKen are logic games similar to Sudoku. In Akari, a labyrinth on a grid has to be lit by placing lanterns, respecting various constraints. In Takuzu a grid has to be filled with 0's and 1's, while respecting certain constraints. In Kakuro a grid has to be filled with numbers such that the sums per row and column match given values; similarly in KenKen a grid has to be filled with numbers such that in given areas the product, sum, difference or quotient equals a given value. We give physical algorithms to realize zero-knowledge proofs for these games which allow a player to show that he knows a solution without revealing it. These interactive proofs can be realized with simple office material as they only rely on cards and envelopes. Moreover, we formalize our algorithms and prove their security.We also prove the security of our constructions.Related Work: Sudoku, introduced under this name in 1986 by the Japanese puzzle company Nikoli, and similar games such as Akari, Takuzu, Kakuro and Ken-Ken have gained immense popularity in recent years. Following the success of Sudoku, generalizations such as Mojidoku which uses letters instead of digits, and other similar logic puzzles like Hitori, Masyu, Futoshiki, Hashiwokakero, or Nurikabe were developed. Many of them have been proved to be NP-complete [12,5].Interactive ZKPs were introduced by Goldwasser et al. [8], and it was then shown that for any NP-complete problem there exists an interactive ZKP protocol [7]. An extension by Ben-Or et al. [1] showed that every provable statement can be proved in zero-knowledge. They also showed that physical protocols using envelopes exist, yet their construction is -due to its generality -rather involved an often impractical for real problem instances. Proofs can also be non-interactive in the sense that the prover and verifier do not need to interact during the protocol [3]. For more background on ZKPs see for example [15].As ZKPs have always been difficult to explain, there are works on how to explain the concepts to non experts, partly using physical protocols as illustrations. For example, in their famous paper [18], Quisquater et al. propose "Ali Bababa's cave" as a tool to explain Zero-Knowledge Proof to kids. In [20], ZKP's are illustrated using a magician that can count the number of sand grains in a heap of sand. Naor et al. used the well-known "Where's Waldo?" cartoons to explain the concept of ZKP to kids, and also proposed an efficient physical protocol for the problem in [17].In 2007, the same authors proposed a ZKP for Sudoku using cards [9], which partly inspired Cobra's solution in our paper. This was later extended for Hanjie [4].As in [9], we here also assume an abstract shuffle functionality which is essentially an indistinguishable shuffle of a set of sealed envelopes or of face down cards. This functionality is necessary to prevent information leakage, and cannot be realized neither by the verifier nor the prover. The verifier cannot perform this action, as otherwise he could be pe...

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Playing games with algorithms: Algorithmic combinatorial game theory

Cited by 37 publications

References 75 publications

Particle computation: complexity, algorithms, and logic

Particle computation: complexity, algorithms, and logic

POGGI: generating puzzle instances for online games on grid infrastructures

Physical Zero-Knowledge Proof for Makaro

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