Material Symmetry to Partition Endgame Tables

Many games display some kind of material symmetry . That is, some sets of game elements can be exchanged for another set of game elements, so that the resulting position will be equivalent to the original one, no matter how the elements were arranged on the board. Material symmetry is routinely used in card game engines when they normalize their internal representation of the cards. Other games such as chinese dark chess also feature some form of material symmetry, but it is much less clear what the normal form of a position should be. We propose a principled approach to detect material symmetry. Our approach is generic and is based on solving multiple rel- atively small sub-graph isomorphism problems. We show how it can be applied to chinese dark chess , dominoes , and skat . In the latter case, the mappings we obtain are equivalent to the ones resulting from the standard normalization process. In the two former cases, we show that the material symmetry allows for impressive savings in memory requirements when building endgame tables. We also show that those savings are relatively independent of the representation of the tables

KQQKQQ and the Kasparov-World Game

The 1999 Kasparov-World game for the first time enabled anyone to join a team playing against a World Chess Champion via the web. It included a surprise in the opening, complex middle-game strategy and a deep ending. As the game headed for its mysterious finale, the World Team re-quested a KQQKQQ endgame table and was provided with two by the authors. This paper describes their work, compares the methods used, examines the issues raised and summarises the concepts involved for the benefit of future workers in the endgame field. It also notes the contribution of this endgame to chess itself

Varying Complexity in CHINESE DARK CHESS Stochastic Game

International audienceCHINESE DARK CHESS is an interesting stochas-tic game that combines revealing, positioning and capturing moves. As many variants are possible, we propose a general protocol that allows to play this game as puzzle, competitive or cooperative game, by 1 to multiple players. Varying complexity can be addressed by this variations. A short visualisation of 4 different games are presented

Representation of Knowledge for Chess Endgames Towards a Self-Improving System

This thesis describes an investigation of the problems involved in representing knowledge within the task area of elementary Chess endgames. Two major criteria are taken for the choice of a model of & the chessplayer's knowledge : firstly, that algorithms constructed using the model should be natural from the viewpoint of a chessplayer and commensurate with his, view of the complexity of the task, and secondly that the algorithms should be capable of refinement in the light of experience in a manner which preserves the previous property. Elementary chess endgames are studied as a field in which programs based on tree-searching and traditional evaluation functions have achieved poor results and where tree-searching seems to play little or no part for people. It is therefore possible to examine problems of knowledge representation and program refinement largely independently of the tree-searching paradigm. A long term aim of the research is to develop a representation suitable as the basis for a fully automatic system of algorithm refinement, whilst maintaining the criteria given above. A model is proposed and algorithms are given for two endgames, King and Rook against King (KRK) and King and Pawn against King (KPK) using this model. It is argued that both algorithms are reasonably natural and compact representations and experiments in refining these algorithms are described in detail. In both cases, the process of refinement is shown to be a reasonably straightforward one (for people) and one which maintains the properties of naturalness and compactness. The possibility of automating this process is considered

Abstractions and Analyses of Grid Games

In this paper, we define various combinatorial games derived from the NQueens Puzzle and scrutinize them, particularly the Knights Game, using combinatorial game theory and graph theory. The major result of the paper is an original method for determining who wins the Knights Game merely from the board\u27s dimensions. We also inspect the Knights Game\u27s structural similarities to the Knight\u27s Tour and the Bishops Game, and provide some historical background and real-world applications of the material

A novel computer Scrabble engine based on probability that performs at championship leve

The thesis starts by giving an introduction to the game of Scrabble, then mentions state-of-the-art computer Scrabble programs and presents some characteristics of our developed Scrabble engine Heuri. Some brief notions of Game Theory are given, along with history of some games in Artificial Intelligence; the fundamental algorithms for game playing, as well as state-of-the-art engines and the algorithms used by them, are presented. Basic elements of Scrabble, such as the Scrabble rules and the letter distribution, are given. Some history and state-of-the-art of Computer Scrabble are commented. For instance, the generation methods of valid moves based on the data structure DAWG (Directed Acyclic Word Graph) and also the variant GADDAG are recalled. These methods are used by the state-of-the-art Scrabble engines Quackle and Maven. Then, the contributions of this thesis are presented. A Spanish lexicon for playing Scrabble has been built that is used by Heuri engines. From this construction, a detailed study and classification of Spanish irregular verbs has been provided. A novel Scrabble move generator based on anagrams has been designed and implemented, which has been shown to be faster than the GADDAG-based generator used in Quackle engine. This method is similar to the way Scrabble players look for a move, searching for anagrams and a spot to play on the board. Next, we address the evaluation of moves when playing Scrabble; the quality of your game depends on deciding what move should be played given a certain board and a rack with tiles. This decision was made initially by Heuri trying several heuristics which ended up with the construction of several engines. We give the explanation of the heuristics used in these engines, all of them based on probabilities. All these initial heuristic evaluation functions (up to six) do not use forward looking, they are static evaluators. They have shown, after testing, an increasing playing performance, which allow Heuri to beat (top-level) expert human players in Spanish, without the need of using sampling and simulation techniques. These heuristics mainly consider the possibility of achieving a bingo on the actual board, whereas Quackle used pre-calculated values (superleaves) regardless of the latter. Then, in order to improve the quality of play of Heuri even more, some additional engines are presented in which look ahead is employed. The HeuriSamp engine, which evaluates a 2-ply search, permits to obtain a defense value. The HeuriSim engine uses a 3-ply adversarial search tree; it contemplates the best first moves (according to Heuri sixth engine heuristic) from Player 1, then some replies to these moves (Player 2 moves) and then some replies to these replies (Player 1 moves). Finally, to improve these engines, opponent modeling is used; this technique makes predictions on some of the opponents' tiles based on the last play made by the opponent. We present results obtained by playing thousands of Heuri vs Heuri games, collecting important information: general statistics of Scrabble game, like a 16 point handicap of the second player, and word statistics in Spanish, like a list of the most frequently played bingos (words that use all 7 tiles of a player's rack). In addition, we present results of matches played by Heuri against top-level humans in Spanish and results obtained by massive playing of different Heuri engines against the Quackle engine in Spanish, French and English. All these match results demonstrate the championship level performance of the Heuri engines in the three languages, especially of the last developed engine that includes simulation and opponent modeling techniques. From here, conclusions of the thesis are drawn and work for the future is envisaged.La tesi comença introduint el joc del Scrabble, esmentant els programes d’ordinador de l’estat de l’art que juguen Scrabble, i presentant algunes característiques del motor de joc de Scrabble que s’ha desenvolupat anomenat Heuri. Es donen breus nocions de la Teoria de Jocs, junt amb la història d’alguns jocs en Intel·ligència Artificial; es presenten els algorismes fonamentals per jugar, així com els motors de joc de l’estat de l’art en diferents jocs i els algorismes que usen. Es comenta també la història i estat de l’art del Computer Scrabble. Es recorden els mètodes de generació de moviments vàlids basats en l’estructura de dades DAWG (Directed Acyclic Word Graph) i en la variant GADDAG, que són usats pels motors de joc de Scrabble Quackle i Maven. A continuació es presenten les contribucions de la tesi. S’ha construït un diccionari per jugar Scrabble en espanyol, el qual és usat per les diferentes versions del motor de joc Heuri. S’ha fet un estudi detallat i una classificació dels verbs irregulars en espanyol. S’ha dissenyat i implementat un nou generador de moviments de Scrabble basat en anagrames, que ha demostrat ser més ràpid que el generador basat en GADDAG usat al motor Quackle. Aquest mètode és similar a la manera en la que els jugadors de Scrabble cerquen un moviment, buscant anagrames i un lloc del tauler on col·locar-los. Seguidament, es tracta l’evacuació dels moviments quan es juga Scrabble; la qualitat del joc depèn de decidir quin moviment cal jugar donat un cert tauler i un faristol amb fitxes. En Heuri, inicialment, aquesta decisió es va prendre provant diferents heurístiques que van dur a la construcció de diversos motors. Donem l’explicació de les heurístiques usades en aquests motors, totes elles basades en probabilitats. Totes aquestes funcions d’avaluació heurística inicials (fins a sis) no miren cap endavant, fan avaluacions estàtiques. Han mostrat, després de ser provades, un rendiment creixent de nivell de joc, el que ha permès Heuri derrotar a jugadors humans experts de màxim nivell en espanyol, sense necessitat d’usar tècniques de mostreig i de simulació. Aquestes heurístiques consideren principalment la possibilitat d’aconseguir un bingo en el tauler actual, mentre que Quackle usa uns valors pre-calculats (superleaves) que no tenen en compte l’anterior. Amb l’objectiu de millorar la qualitat de joc de Heuri encara més, es presenten uns motors de joc addicionals que sí miren cap endavant. El motor HeuriSamp, que realitza una cerca 2-ply, permet obtenir un valor de defensa. El motor HeuriSim usa un arbre de cerca 3-ply; contempla els millors primers moviments (d’acord al sisè motor heurístic d’Heuri) del Jugador 1, després algunes respostes a aquests moviments (moviments del Jugador 2) i llavors algunes rèpliques a aquestes respostes (moviments del Jugador 1). Finalment, per a millorar aquests motors, es proposa usar modelatge d’oponents; aquesta tècnica realitza prediccions d’algunes de les fitxes de l’oponent basant-se en l’últim moviment jugat per aquest. Es presenten resultats obtinguts de jugar milers de partides d’Heuri contra Heuri, que recullen important informació: estadístiques generals del joc del Scrabble, com un handicap de 16 punts del segon jugador, i estadístiques de paraules en espanyol, com una llista dels bingos (paraules que usen les 7 fitxes del faristol d’un jugador) que es juguen més freqüentment. A més, es presenten resultats de partides jugades per Heuri contra jugadors humans de màxim nivell en espanyol i resultats obtinguts d'un gran nombre d’enfrontaments entre els diferents motors de joc d’Heuri contra el motor Quackle en espanyol, francès i anglès. Tots aquests resultats de partides jugades demostren el rendiment de nivell de campió dels motors d’Heuri en les tres llengües, especialment el de l’últim motor desenvolupat que inclou tècniques de de simulació i modelatge d'oponents. A partir d'aquí s'extreuen les conclusions de la tesi i es preveu treballar de cara al futur.Postprint (published version

Climate Simulations of Hot Jupiters: Developing and Applying an Accurate Radiation Scheme

To date more than 1500 exoplanets have been discovered. A large number of these are hot Jupiters, Jupiter-sized planets orbiting < 0.1 au from their parent stars, due to limitations in observational techniques making them easier to detect than smaller planets in wider orbits. This is also, for the same reasons, the class of exoplanets with the most observational constraints. Due to the very large interaction between these planets and their parent stars they are believed to be tidally locked, causing a large temperature contrast between the permanently hot day side and colder night side. There are still many open questions about these planets. Many are observed to have inflated radii, i.e. the observed radius is larger for a given mass than evolutionary models predict. A mechanism that can transport some of the stellar heating into the interior of the planet may be able to explain this. The presence of hazes or clouds has been inferred on some planets, but their composition and distribution remain unknown. According to chemical equilibrium models TiO and VO should be present on the day side of the hottest of these planets, but these molecules have not yet been detected. Cold traps, where these molecules condense out on the night side, have been suggested to explain this. The efficiency of the heat redistribution from the day side to the night side has been found to vary significantly between different planets; the mechanism behind this is still unknown. To begin to answer many of these questions we need models capturing the three-dimensional nature of the atmospheres of these planets. General circulation models (GCMs) do this by solving the equations of fluid dynamics for the atmosphere coupled to a radiative transfer scheme. GCMs have previously been applied to several exoplanets, but many solve simplified fluid equations (shallow water or primitive equations) or highly parametrised radiation schemes (temperature-forcing, gray or band-averaged opacities). We here present an adaptation of the Met Office Unified Model (UM), a GCM used for weather predictions and climate studies for the Earth, to hot Jupiters. The UM solves the full 3D Euler equations for the fluid, and the radiation scheme uses the two-stream approximation and correlated-k method, which are state of the art for both Earth and exoplanet GCMs. This makes it ideally suited for the study of hot Jupiters. An important part of this work is devoted to the adaptation of the radiation scheme of the UM to hot Jupiters. This includes calculation of opacities for the main absorbers in these atmospheres from state-of-the-art high temperature line lists, the calculation of k-coefficients from these opacities, and making sure all aspects of the scheme perform satisfactorily at high temperatures and pressures. We have tested approximations made in previous works such as the two-stream approximation, use of band-averaged opacities and different treatments of gaseous overlap. Uncertainties in current models, such as the lack of high temperature line broadening parameters for these atmospheres, are discussed. We couple the adapted radiation scheme to the UM dynamical core, which has been tested independently. Our first application is devoted to one of the most well-observed hot Jupiters, HD 209458b. Differences between previous modelling works and our model are discussed, and we compare results from the full coupled model with results obtained using a temperature-forcing scheme. We have also developed a tool to calculate synthetic phase curves, and emission and transmission spectra from the output of our 3D model. This enables us to directly compare our model results to observations and test the effect of various parameters and model choices on observable quantities

Divide and Conquer: A Distributed Approach to Five Point Energy Minimization

This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid is the energy minimizer with respect to the s-power law potential for all s in (0,S) and some pyramid with square base is the unique minimizer for all s in (S,15+512/25]. Taking s=1 gives another solution to Thomson's 5 electron problem from 1904. The work here is a simplification of my monograph from 6 years ago which also proved the main result. The version here is half as long as the original. Also, I wrote it in an experimental style which facilitates the verification process. As I explicitly indicate in the text, the proof can be broken up into 10 independent pieces, all less than 15 pages, which can be checked independently of the other pieces.Comment: 77 pages long. This is a computer assisted proof, and the code is available from my website as indicated in the text. I further shortened the proof (and removed typos). My goal is to make this thing as short and as easily verifiable as possibl