Theory of Combinatorial Games

Aim: To present a systematic development of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to Chess and Go in small strides at a gradient that is not too steep. Presentation: Mostly informal; examples of combinatorial games sampled from various strategic viewing points along scenic mountain trails illustrate the theory. Add-on:Atasteof constraint logic, a new tool to prove intractabilities of games.

[1]  Aviezri S. Fraenkel,et al.  The generalized Sprague-Grundy function and its invariance under certain mappings , 1986, J. Comb. Theory, Ser. A.

[2]  C. L. Bouton.  Nim, A Game with a Complete Mathematical Theory , 1901 .

[3]  Aviezri S. Fraenkel,et al.  Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete , 1981, Discret. Appl. Math..

[4]  Aviezri S. Fraenkel,et al.  Complexity, appeal and challenges of combinatorial games , 2004, Theor. Comput. Sci..

[5]  Aaron N. Siegel.  Combinatorial Game Theory , 2013 .

[6]  Cedric A. B. Smith.  Graphs and composite games , 1966 .

[7]  Aviezri S. Fraenkel,et al.  Strategy for a class of games with dynamic ties , 1975 .

[8]  A. Fraenkel,et al.  Theory of annihilation games—I☆ , 1982 .

[9]  Robert M. Hierons,et al.  Reaching and Distinguishing States of Distributed Systems , 2010, SIAM J. Comput..

[10]  A. Blokhuis.  Winning ways for your mathematical plays , 1984 .

[11]  John Michael Robson,et al.  Combinatorial Games with Exponential Space Complete Decision Problems , 1984, MFCS.

[12]  John Michael Robson,et al.  N by N Checkers is Exptime Complete , 1984, SIAM J. Comput..

[13]  Erik D. Demaine,et al.  PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation , 2002, Theor. Comput. Sci..

[14]  Aviezri S. Fraenkel,et al.  Theory of annihilation games , 1976 .

[15]  Michael Hoffmann,et al.  Push-2-f is pspace-complete , 2002, CCCG.

[16]  Erik D. Demaine,et al.  Constraint Logic: A Uniform Framework for Modeling Computation as Games , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[17]  Noam D. Elkies.  New Directions in Enumerative Chess Problems , 2005, Electron. J. Comb..

[18]  F. L. Morris.  Playing disjunctive sums is polynomial space complete , 1981 .

[19]  Aviezri S. Fraenkel,et al.  Computing a Perfect Strategy for n*n Chess Requires Time Exponential in N , 1981, ICALP.

[20]  Julian West.  Championship-Level Play of Domineering , 1996 .

[21]  R. Sprague.  Über mathematische Kampfspiele , 1935 .

[22]  Jaroslav Nesetril,et al.  Epidemiography II. Games with a dozing yet winning player , 1988, J. Comb. Theory, Ser. A.

[23]  Aviezri S. Fraenkel,et al.  PSPACE-hardness of some combinatorial games , 1987, J. Comb. Theory, Ser. A.

[24]  Aviezri S. Fraenkel,et al.  Three Annihilation Games , 1978 .

[25]  Thomas S. Ferguson,et al.  Misère Annihilation Games , 1984, J. Comb. Theory, Ser. A.

[26]  L. Yedwab.  ON PLAYING WELL IN A SUM OF GAMES , 1985 .

[27]  Aviezri S. Fraenkel,et al.  Computing a Perfect Strategy for n x n Chess Requires Time Exponential in n , 1981, J. Comb. Theory, Ser. A.

[28]  F. L. Morris,et al.  Prohibiting repetitions makes playing games substantially harder , 1984 .

[29]  Aaron N. Siegel,et al.  Misère quotients for impartial games , 2006, J. Comb. Theory, Ser. A.

[30]  John H. Reif,et al.  Multiple-person alternation , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[31]  Aviezri S. Fraenkel,et al.  Complexity of problems in games, graphs and algebraic equations , 1979, Discret. Appl. Math..

[32]  Thomas J. Schaefer,et al.  On the Complexity of Some Two-Person Perfect-Information Games , 1978, J. Comput. Syst. Sci..

[33]  John Michael Robson,et al.  The Complexity of Go , 1983, IFIP Congress.

[34]  Robert A. Hearn.  Tipover is NP-complete , 2006 .

[35]  Robert A. Hearn.  Amazons, Konane, and Cross Purposes are PSPACE-complete , 2009 .

[36]  Michael O. Rabin,et al.  6. EFFECTIVE COMPUTABILITY OF WINNING STRATEGIES , 1958 .

[37]  Aviezri S. Fraenkel,et al.  Combinatorial games with an annihilation rule , 1974 .

[38]  Thane E. Plambeck,et al.  Taming the wild in impartial combinatorial games , 2005 .

[39]  Erik D. Demaine,et al.  Games, puzzles and computation , 2009 .

[40]  Stefan Reisch,et al.  Hex ist PSPACE-vollständig , 1981, Acta Informatica.

[41]  Larry J. Stockmeyer,et al.  Provably Difficult Combinatorial Games , 1979, SIAM J. Comput..

[42]  J. Conway.  On Numbers and Games , 1976 .

[43]  Edward M. Reingold,et al.  The Complexity of Pursuit on a Graph , 1995, Theor. Comput. Sci..

[44]  Richard J. Nowakowski,et al.  Lessons in Play: An Introduction to Combinatorial Game Theory , 2007 .

[45]  Noam D. Elkies.  On Numbers and Endgames: combinatorial Game Theory in Chess Endgames , 1996 .