Modeling the Game of Arimaa with Linguistic Geometry Jos´ e Roberto Mercado Vega and Zvi Retchkiman K¨ onigsberg Abstract— A computer defeated a chess world champion for the first time in 1997. This event inspired Omar Syed to develop the game of Arimaa. He intended to make it difficult to solve under present search approaches. Linguistic Geometry is a technique that offers a formal method based on the expertise of human chess masters, to make the development of complex heuristics easier. This article introduces a Linguistic Geometry based model for the game of Arimaa. It gives implementation for each of the essential components of Linguistic Geometry: trajectories, zones, translations and searches. A test case is given and it is used as input for a software implementation of the proposed model. The results given by the software are compared against the analysis made by a human player. I. I NTRODUCTION To study games, they can be represented through large trees. These trees include every possible evolution of the game. Searching through the game’s tree for a solution of the game is the current approach in computer sciences. To reduce the execution time of game search algorithms, heuristics are used during the analysis of the game’s tree. A heuristic is an estimation mechanism or a rule of thumb. Heuristics were first proposed by Claude Shannon [7]. The most popular search algorithms through general trees are depth-first, breath first and best first, while for search in games trees are minimax and alpha-beta [4]. Games are an interesting area of study because of their complexity, it is believed that techniques used for solving some games can also be used to solve other kind of problems. In particular, the study of chess has excelled. A new game called Arimaa was created by Omar and Aamir Syed [10]. They were inspired by the defeat of Garry Kasparov against the supercomputer Deep Blue. The game of Arimaa is a two player complete information, zero-sum game with no random factors. Arimaa was released in 2002 and it was designed to be complex to play well under traditional game search algorithms. This was done with the purpose of fomenting the development of new, ground breaking techniques. A challenge was published along the rules of the game, it consists of a prize of $10,000 USD for anyone who creates a computer program capable of defeating a human expert in a competition of six games. Many computer programs have participated in the Ari- maa challenge. The vast majority of them are based on conventional alpha-beta search algorithms. An example of an Arimaa design is that of David Fotland: “Building a World Champion Arimaa Program” [2]. Fotland’s program was champion of the Arimaa tournament in 2004, this is a tournament only for software. Fotland’s program was defeated by a human expert (Omar Syed). R. Mercado and Z. Retchkiman are with Instituto Polit´ ecnico Nacional CIC, Mexico, e-mail: dragonslayking@yahoo.co.uk, mzvi@cic.ipn.mx Haizhi Zhong published his master thesis in 2005 under the title: “Building a Strong Arimaa-playing Program” [11]. Zhong’s program is based on alpha-beta reductions with some optimizations on the evaluation function based on some Arimaa tactics and selective generation of the movements. It also uses a transposition table and play ordering. In 2006 Christian-Jan Cox published his master thesis: “Analysis and Implementation of the Game Arimaa” [1]. In this work Cox proposes a program called Coxa. It is based on an alpha-beta search algorithm with some optimizations along a transposition table. Cox tests multiple variations of the evaluation function and registers the results. None of the variants of the heuristics represents notable improvements over the search method. A promising modeling technique that has its own po- tential is Linguistic Geometry (LG). LG was created by Boris Stilman [8], and has had a slow, though constant, development. From its origin LG was proposed in accordance to the way of thinking of some human chess experts. LG was developed around a class of games called abstract board games (ABG’s). Processing along LG is done through a hierarchy of formal grammars. The levels of this hierarchy are: trajectories, webs, translations and searches. This article presents a Linguistic Geometry based model for the game of Arimaa. The tools of LG are used to create the model, with some variations to make them adequate for the game of Arimaa. The model is used to implement one of the most common tactics of the game of Arimaa. To test the model a software implementation of the same is used and applied in some chosen positions of the game of Arimaa. Up to date, there is no published work which mixes Arimaa and Linguistic Geometry. II. ARIMAA A. The rules of Arimaa In order to play Arimaa, only a chess board and a set of chess pieces are needed. However, some changes to the rules have to be made. First off, the 64 cells of the board are equal, except for the cells c3, f3, c6 and f6 (according to chess algebraic notation). In Arimaa these are known as trap cells. Instead of black and white pieces, in Arimaa gold and silver are used respectively. Just as in chess, each player has 16 pieces of 6 different types. In descending weight order these are: elephant, camel, two horses, two dogs, two cats and eight rabbits. Remark 1: By convention, diagrams presented have gold player on the lower rows (1 and 2) and silver on the upper rows (7 and 8). Also, gold pieces are shown facing to the right, while silver pieces are facing to the left. 978-1-4244-4815-9/09/$25.00 ©2009 IEEE 379