Counting Domineering Positions, Which Overlaps with Some of Our Work As Discussed in Section 2
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Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first. -
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game Will Johnson June 9, 2011 Contents 1 To Knot or Not to Knot 4 1.1 Some facts from Knot Theory . 7 1.2 Sums of Knots . 18 I Combinatorial Game Theory 23 2 Introduction 24 2.1 Combinatorial Game Theory in general . 24 2.1.1 Bibliography . 27 2.2 Additive CGT specifically . 28 2.3 Counting moves in Hackenbush . 34 3 Games 40 3.1 Nonstandard Definitions . 40 3.2 The conventional formalism . 45 3.3 Relations on Games . 54 3.4 Simplifying Games . 62 3.5 Some Examples . 66 4 Surreal Numbers 68 4.1 Surreal Numbers . 68 4.2 Short Surreal Numbers . 70 4.3 Numbers and Hackenbush . 77 4.4 The interplay of numbers and non-numbers . 79 4.5 Mean Value . 83 1 5 Games near 0 85 5.1 Infinitesimal and all-small games . 85 5.2 Nimbers and Sprague-Grundy Theory . 90 6 Norton Multiplication and Overheating 97 6.1 Even, Odd, and Well-Tempered Games . 97 6.2 Norton Multiplication . 105 6.3 Even and Odd revisited . 114 7 Bending the Rules 119 7.1 Adapting the theory . 119 7.2 Dots-and-Boxes . 121 7.3 Go . 132 7.4 Changing the theory . 139 7.5 Highlights from Winning Ways Part 2 . 144 7.5.1 Unions of partizan games . 144 7.5.2 Loopy games . 145 7.5.3 Mis`eregames . 146 7.6 Mis`ereIndistinguishability Quotients . 147 7.7 Indistinguishability in General . 148 II Well-tempered Scoring Games 155 8 Introduction 156 8.1 Boolean games . -
Combinatorial Game Theory
Combinatorial Game Theory Aaron N. Siegel Graduate Studies MR1EXLIQEXMGW Volume 146 %QIVMGER1EXLIQEXMGEP7SGMIX] Combinatorial Game Theory https://doi.org/10.1090//gsm/146 Combinatorial Game Theory Aaron N. Siegel Graduate Studies in Mathematics Volume 146 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 91A46. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-146 Library of Congress Cataloging-in-Publication Data Siegel, Aaron N., 1977– Combinatorial game theory / Aaron N. Siegel. pages cm. — (Graduate studies in mathematics ; volume 146) Includes bibliographical references and index. ISBN 978-0-8218-5190-6 (alk. paper) 1. Game theory. 2. Combinatorial analysis. I. Title. QA269.S5735 2013 519.3—dc23 2012043675 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
Combinatorial Game Theory: an Introduction to Tree Topplers
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Fall 2015 Combinatorial Game Theory: An Introduction to Tree Topplers John S. Ryals Jr. Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Ryals, John S. Jr., "Combinatorial Game Theory: An Introduction to Tree Topplers" (2015). Electronic Theses and Dissertations. 1331. https://digitalcommons.georgiasouthern.edu/etd/1331 This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. COMBINATORIAL GAME THEORY: AN INTRODUCTION TO TREE TOPPLERS by JOHN S. RYALS, JR. (Under the Direction of Hua Wang) ABSTRACT The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing. Key Words: combinatorial game theory, tree topplers, domineering, hackenbush 2009 Mathematics Subject Classification: 91A46 COMBINATORIAL GAME THEORY: AN INTRODUCTION TO TREE TOPPLERS by JOHN S. RYALS, JR. B.S. in Applied Mathematics A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 2015 c 2015 JOHN S. -
New Results for Domineering from Combinatorial Game Theory Endgame Databases
New Results for Domineering from Combinatorial Game Theory Endgame Databases Jos W.H.M. Uiterwijka,∗, Michael Bartona aDepartment of Knowledge Engineering (DKE), Maastricht University P.O. Box 616, 6200 MD Maastricht, The Netherlands Abstract We have constructed endgame databases for all single-component positions up to 15 squares for Domineering. These databases have been filled with exact Com- binatorial Game Theory (CGT) values in canonical form. We give an overview of several interesting types and frequencies of CGT values occurring in the databases. The most important findings are as follows. First, as an extension of Conway's [8] famous Bridge Splitting Theorem for Domineering, we state and prove another theorem, dubbed the Bridge Destroy- ing Theorem for Domineering. Together these two theorems prove very powerful in determining the CGT values of large single-component positions as the sum of the values of smaller fragments, but also to compose larger single-component positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist single-component Domineering positions with that value. Second, we investigate Domineering positions with infinitesimal CGT values, in particular ups and downs, tinies and minies, and nimbers. In the databases we find many positions with single or double up and down values, but no ups and downs with higher multitudes. However, we prove that such single-component ups and downs easily can be constructed, in particular that for any multiple up n·" or down n·# there are Domineering positions of size 1+5n with these values. Further, we find Domineering positions with 11 different tinies and minies values. -
Combinatorial Games, Theory and Applications -. [ Tim Wylie
Combinatorial Games, Theory and Applications Brit C. A. Milvang-Jensen February 18, 2000 Abstract Combinatorial games are two-person perfect information zero-sum games, and can in theory be analyzed completely. However, in most cases a com- plete analysis calls for a tremendous amount of computations. Fortunately, some theory has been developed that enables us to find good strategies. Fur- thermore, it has been shown that the whole class of impartial games fairly easy can be analyzed completely. In this thesis we will give an introduction to the theory of combinatorial games, along with some applications on other fields of mathematics. In particular, we will examine the theory of impartial games, and the articles proving how such games can be solved completely. We will also go into the theory of Thermography, a theory that is helpful for finding good strategies in games. Finally, we will take a look at lexicodes that are error correcting codes derived from game theory. Contents 1 Introduction 3 2 Basic Concepts 5 2.1 The Game of Domineering . 8 2.2 Good, Bad and Fuzzy . 10 2.2.1 - and -positions . 11 P N 2.3 Structure on Games . 12 2.4 Dominated and Reversible Options . 17 3 Impartial Games 20 3.1 The Game of Nim . 20 3.1.1 Nimbers . 20 3.1.2 The Analysis of Nim . 21 3.1.3 Bouton's Article . 22 3.1.4 Turning Turtles . 25 3.2 Grundy's and Sprague's conclusions . 26 4 Numbers 32 4.1 Games as Reals and Surreals . 32 4.1.1 Numbers Born on Day 1 . -
Combinatorial Games, Volume 43
http://dx.doi.org/10.1090/psapm/043 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures A Subseries of Proceedings of Symposia in Applied Mathematics Volume 43 COMBINATORIAL GAMES Edited by Richard K Guy [Columbus, Ohio, August 1990) Volume 42 CRYPTOLOGY AND COMPUTATIONAL NUMBER THEORY Edited by C. Pomerance {Boulder, Colorado, August 1989) Volume 41 ROBOTICS Edited by R. W. Brockett (Louisville, Kentucky, January 1990) Volume 40 MATRIX THEORY AND APPLICATIONS Edited by Charles R. Johnson (Phoenix, Arizona, January 1989) Volume 39 CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS Edited by Robert L. Devaney and Linda Keen (Providence, Rhode Island, August 1988) Volume 38 COMPUTATIONAL COMPLEXITY THEORY Edited by Juris Hartmanis (Atlanta, Georgia, January 1988) Volume 37 MOMENTS IN MATHEMATICS Edited by Henry J. Landau (San Antonio, Texas, January 1987) Volume 36 APPROXIMATION THEORY Edited by Carl de Boor (New Orleans, Louisiana, January 1986) Volume 35 ACTUARIAL MATHEMATICS Edited by Harry H. Panjer (Laramie, Wyoming, August 1985) Volume 34 MATHEMATICS OF INFORMATION PROCESSING Edited by Michael Anshel and William Gewirtz (Louisville, Kentucky, January 1984) Volume 33 FAIR ALLOCATION Edited by H. Peyton Young (Anaheim, California, January 1985) Volume 32 ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS Edited by R. W. McKelvey (Eugene, Oregon, August 1984) Volume 31 COMPUTER COMMUNICATIONS Edited by B. Gopinath (Denver, Colorado, January 1983) Volume 30 POPULATION BIOLOGY Edited by Simon A. Levin (Albany, New York, August 1983) Volume 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton (San Francisco, California, January 1981) Volume 28 STATISTICAL DATA ANALYSIS Edited by R. -
Lessons in Play.Pdf
Left Right Louise Richard Positive Negative bLack White bLue Red Vertical Horizontal Female Male Green Gray Symbol Description Page G = GL GR Definition of a game 36, 66 L R G ˘and˛G ¯ Typical left option and right option 36 {A k B | C,˛ D} Notation for a game 36 G + H GL + H, G + HL GR + H, G + HR 68 R L −G ˘−G −G (negative)˛ ¯ 69 ∼ ˛ G = H ˘identical˛ game¯ trees (congruence) 66 ˛ G = H game equivalence 70 ≥, ≤, >, < comparing games 73 G « H G 6≥ H and H 6≥ G (incomparable, confused with) 74 G H G 6≤ H (greater than or incomparable) 74 G ¬ H G 6≥ H (less than or incomparable) 74 G ∼ H far-star equivalence 195 n integers 88 m/2j numbers or dyadic rationals 91 ↑ and ↓ {0 | ∗} “up” and {∗ | 0} “down” 100 ⇑ and ⇓ ↑ + ↑ “double-up” and ↓ + ↓ “double-down” 101 ∗ {0 | 0} “star” 66 ∗n {0, ∗, ∗2,... | 0, ∗, ∗2,...} “star-n” (nimbers) 136 © ¨G; G 0 k 0 |−G “tiny-G” and its negative “miny-G” 108 Á loony 22 far-star 196 LS(G), RS(G) (adorned) left and right stops (161) 123 Gn and gn games born by day n and their number 117 ∨ and ∧ join and meet 127 n n · G G + G + G + · · · + G 103 G · U zNorton product}| { 175 Gn “G-nth” 188 .213 uptimal notation 188 → G n G1 + G2 + · · · + Gn 189 AW(G) atomic weight 198 Lessons in Play Lessons in Play An Introduction to Combinatorial Game Theory Michael H. Albert Richard J. Nowakowski David Wolfe A K Peters, Ltd. -
Combinatorial Games: from Theoretical Solving to AI Algorithms Eric Duchene
Combinatorial games: from theoretical solving to AI algorithms Eric Duchene To cite this version: Eric Duchene. Combinatorial games: from theoretical solving to AI algorithms. SUM, Sep 2016, NIce, France. hal-01883569 HAL Id: hal-01883569 https://hal.archives-ouvertes.fr/hal-01883569 Submitted on 28 Sep 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Combinatorial games: from theoretical solving to AI algorithms Eric Duch^ene? Universit´ede Lyon, CNRS Universit´eLyon 1, LIRIS, UMR5205, F-69622 [email protected] Abstract. Combinatorial game solving is a research field that is fre- quently highlighted each time a program defeats the best human player: Deep Blue (IBM) vs Kasparov for Chess in 1997, and Alpha Go (Google) vs Lee Sedol for the game of Go in 2016. But what is hidden behind these success stories ? First of all, I will consider combinatorial games from a theoretical point of view. We will see how to proceed to properly define and deal with the concepts of outcome, value, and winning strategy. Are there some games for which an exact winning strategy can be expected? Unfortunately, the answer is no in many cases (including some of the most famous ones like Go, Othello, Chess or Checkers), as exact game solving belongs to the problems of highest complexity. -
Positions of Value *2 in Generalized Domineering and Chess
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5 (2005), #G06 POSITIONS OF VALUE *2 IN GENERALIZED DOMINEERING AND CHESS Gabriel C. Drummond-Cole Department of Mathematics, State University of New York, Stony Brook, NY 11794, USA [email protected] Received: 5/29/04, Revised: 7/9/05, Accepted: 7/31/05, Published: 8/1/05 Abstract Richard Guy [5] asks whether the game-theoretic value *2, the value of a nim-heap of size 2, occurs in the games of Domineering or chess. We demonstrate positions of that value in Generalized Domineering and chess. 1. Overview We follow the terminology of Winning Ways [1], considering two-player alternating-turn finite-termination complete-information chance-free games whose loss conditions are ex- actly the inability to move, and which always end in a loss. According to this definition, chess is not a game because it can be tied or drawn and the loss condition is not the inability to move. We will answer these objections. A game position can be expressed recursively as {GL | GR} where GL, called the left-options of G, is a collection of positions to which one player, called Left, can move, if it is her turn, while GR, the right-options of G, is a collection of positions to which the other player, called Right, can move if it is hers. We abuse terminology by also using the symbols GL and GR to refer to generic elements of these collections. We can recursively define a binary operation + on the collection of game positions by G+H = {GL+H, G+HL | GR+H, G+HR} and an involution − by −G = {−GR |−GL}. -
Counting Domineering Positions
1 2 Journal of Integer Sequences, Vol. 24 (2021), 3 Article 21.4.8 47 6 23 11 Counting Domineering Positions Svenja Huntemann Department of Mathematics and Physical Sciences Concordia University of Edmonton Edmonton, AB T5B 4E4 Canada [email protected] Neil Anderson McKay Department of Mathematics and Statistics University of New Brunswick Saint John, NB E2L 4L5 Canada [email protected] Abstract Domineering is a two player game played on a checkerboard in which one player places dominoes vertically and the other places them horizontally. We give bivariate generating polynomials enumerating Domineering positions by the number of each player’s pieces. We enumerate all positions, maximal positions, and positions where one player has no move. Using these polynomials we count the number of positions that occur during alternating play. Our method extends to enumerating positions from mid-game positions and we include an analysis of a tournament game. 1 Introduction Combinatorial games are 2-player games with perfect information and no chance devices, such as Chess or Go. Many combinatorial games have, for a fixed starting position, a finite number of options and the game is guaranteed to end in a finite number of moves; in theory we could determine by computer which player would win if both players play perfectly. In practice game theorists and computer scientists have not determined the outcomes of games under perfect-play because of the complexity of the required search. 1 Enumeration of positions has been studied, directly or indirectly, for several combinatorial games. Papers on counting game positions consider the problem of enumerating specific types of positions — Go end positions [6, 7, 16] and second-player win positions for specific lesser known games [8, 9].