An Introduction To Combinatorial Games - Sciencesconf
An introduction to combinatorial gamesEric Duchêne - Aline ParreauLIRIS - Université Lyon 1 - CNRSEcole Jeunes Chercheurs en Informatique MathématiqueLyon - 23 janvier 20171/13
IntroductionGames with:What 2 players playing alternately; perfect information.WhyChessCard gamesOthelloDraughtsWhoTic Tac ToeWhenPachisiGoMain question:Who is winning and how? Exact and approximate resolutions2/13
IntroductionCSMathsWhatCombinatoricsLogicNumber enNim gameby BoutonSpragueGrundyConwayTheoryDeep Games and graphs2/13
Reference books19761981200720133/13
Pure combinatorial games - a definitionBerlekamp, Conway, Guy (Winning Ways, 1981) 2 players: Left and Right, that play alternately and cannot pass theirturn; Perfect information, no chance; Finite number of moves, no draw, always a winner; Winner determined according to the last move (no scoring)ChessCard gamesTic Tac ToeOthelloPachisiDraughtsGo4/13
Game tree.5/13
Game treeNNPPPLRLPRNLP.PRRRNRRLLRR.P5/13
Computing the outcome of Domineering Unknown complexity on a n m board. When n and m are fixed:[Bullcock’s website]6/13
Decomposing Domineering into a sum of games7/13
How to play on a big Domineering game ?8/13
How to play on a big Domineering game ?8/13
Values of positions of Wythoff9/13
The PSPACE classDefinition: a decision problem is PSPACE if it can be solve in polynomialspace by a Turing Machine.The standard PSPACE-complete problem :Quantified Boolean Formula Input : A quantified boolean formula:Q1 x1 Q2 x2 .Qn xn , ϕ(x1 , ., xn )with Qi { , }, xi {0, 1} Output : Is the formula true ?An equivalent problem : QBF-Game10/13
ED-Geography is PSPACE-complete[Schaeffer 1978]Reduction from QBF-Game :(x1 x3 ) (x2 x3 ) (x1 x2 x3 )x1x1x2x2x3x3C1C2C311/13
Some nimbers sequencesArcKayles on a path00112031103322405223301130211045274 0 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 2 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7.Period 34 with some finite exceptions up to 52James Bond Game0 0 0 1 1 1 2 2 0 3 3 1 1 1 0 4.228 known values, periodicity conjectured0.106 Game0 1 0 0 0 1 2 2 2 1 4 4 0 1 0 6.Period 328226140474, with preperiod 465384263797.Guy’s conjecture: all finite octal games have periodic nimber sequence.12/13
ConclusionCurrent research questions ? Graphs and Games: combinatorial games version on graphs Metatheory: Misère, scoring games, loopy games Link with other fields:IIIArtificial Intelligence for generic gamesGame versions of parameters of graphsLogic, automata theory.Merci !13/13
Graphs and Games: combinatorial games version on graphs Metatheory: Misère, scoring games, loopy games Link with other elds: I Arti cial Intelligencefor generic games I Game versions ofparameters of graphs I Logic, automata theory. Merci ! 13/13
strong Combinatorial Chemistry /strong in Drug Research strong Combinatorial chemistry /strong and rational drug design Structure-based and computer-assisted design and virtual screening (LUDI, FlexX et al.) of protein ligands supplement strong combinatorial chemistry /strong . strong Combinatorial /strong design of drugs The necessary tools are already available but scoring functions have to be improved.
1 Introduction to Combinatorial Algebraic Topology Basic Topology Graphs to Simplicial Complexes Posets to Simplicial Complexes 2 Permutation Patterns Introduction and Motivation Applying Combinatorial Algebraic Topology Kozlov, Dimitry. Combinatorial algebraic topology. Vol. 21. Springer Science & Business Media, 2008.
topology and di erential geometry to the study of combinatorial spaces. Per-haps surprisingly, many of the standard ingredients of di erential topology and di erential geometry have combinatorial analogues. The combinatorial theories This work was partially supported by the National Science Foundation and the National Se-curity Agency. 177
Integrating natural product synthesis and combinatorial chemistry*,† A.Ganesan Department of Chemistry, University of Southampton, Southampton SO17 1BJ, United Kingdom Abstract: The fields of natural product total synthesis and combinatorial chemistry have major differences as well as much in common. Unique to combinatorial chemistry is the need
The Games organised at Olympia led to the development of the Panhellenic Games. These included: - The Games at Olympia (Olympic Games): every four years - The Games at Delphi (Pythian Games), 582 B.C.: every four years (third year of each Olympiad) - The Games at the Isthmus of Corinth (Isthmian Games), from 580 B.C.:
Section 3: Playground Markings Games 16 Section 4: Skipping, Hula Hoop & Elastics 25 Section 5: Catching games 32 Section 6: Relay games 41 Section 7: Ball games 48 Section 8: Fun games 59 Section 9: Frisbee games 66 Section 10: Parachute games 70 Section 11: Clapping and rhyming games 74 Useful websites 79
Olympic Winter Games medals Olympic Winter Games posters Olympic Summer Games posters Olympic Summer Games mascots Olympic Winter Games mascots The sports pictograms of the Olympic Summer Games The sports pictograms of the Olympic Winter Games The IOC, the Olympic Movement and the Olympic Games The Olympic programme evolution Torches and torch .
THE GUIDE SPRING BREAK CAMPS 2O2O MARCH 16–27 AGES 5–13. 2 2020 Spring Break Camp Guide WELCOME Build Your COCA Camp Day 2 March 16–20 Camps 3–4 March 23–27 Camps 5–6 Camp Basics 7 Registration Form 8–9 Registration Guidelines/Policies 10 Summer’s coming early this year! Join us over Spring Break for unique and fun arts learning experiences. You’ll find favorites from .
