A multi-dimensional approach to positional chess

Abstract

This paper describes a development of the ideas which were published in the International Journal of Man-Machine Studies (Atkin, 1972), in which the game of chess was discussed in terms of a mathematical relation between the chess pieces and the squares on the chessboard. It is shown that the structures which represent the state of play in any mode can be represented by a complex of connected polyhedra in E53. The positional features of any mode are described in terms of properties of this abstract geometry, such as eccentricity values and structure vectors. An evaluation of the relative positional strengths of possible moves, in any given mode, is built up by way of certain structural mappings on either the complex Kw(S) or its conjugate Ks,(W)- with a corresponding scheme for Black. A simple example of these mappings, based on the traditional piece valuation, is adopted to illustrate the positional strengths of the moves played in specific games. A program which implements this analysis is described briefly. It uses the sum of seven positional features as an evaluation function. Moves are ranked according to this evaluation, except that a simple material exchange calculation is made for each move to estimate its immediate tactical value, and this takes precedence in the ranking. The paper presents the program's highest-ranking move at each play of three games between expert chess players: significant correlation with the moves actually made is achieved.