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On the number of Go positions on lattice graphs
Abstract
We use transfer matrix methods to determine bounds for the numbers of legal Go positions for various numbers of players on some planar lattice graphs, including square lattice graphs such as those on which the game is normally played. We also Þnd bounds on limiting constants that describe the behaviour of the number of legal positions on these lattice graphs as the dimensions of the lattices tend to inÞnity. These results amount to giving bounds for some speciÞc evaluations of Go polynomials on these graphs.
... Two Go polynomials based on these concepts were introduced in [31], studied further in [42,36,39], and mentioned by János in his first inventory of the Zoo [69,70]. One of them simply counts legal positions: ...
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.
... A relatively new topic of interest in combinatorial game theory is the enumeration of positions. Early work has focused on specific types of positions, such as Go end positions [6,5,16] and second-player win position for some lesser known games [7,11]. Recently, Domineering positions, as well as specific types of positions, were counted in [9]. ...
Distance games are games played on graphs in which the players alternately colour vertices, and which vertices can be coloured only depends on the distance to previously coloured vertices. The polynomial profile encodes the number of positions with a fixed number of vertices from each player. We extend previous work on finding the polynomial profile of several distance games (Col, Snort, and Cis) played on paths. We give recursions and generating functions for the polynomial profiles of generalizations of these three games when played on paths. We also find the polynomial profile of Cis played on cycles and the total number of positions of Col and Snort on cycles, as well as pose a conjecture about the number of positions when playing Col and Snort on complete bipartite graphs.
... Enumeration of positions has been studied, directly or indirectly, for several combinatorial games. The first few papers on counting game positions considered the problem of enumerating specific types of positions-Go end positions in [7,6,16] and second-player win positions for two games in [8,9]. In the game Node Kayles, played on a graph, the two players alternate choosing vertices not adjacent to any previously chosen ones, thus forming an independent set. ...
Domineering is a two player game played on a checkerboard in which one player places dominoes vertically and the other places them horizontally. We give bivariate generating polynomials enumerating Domineering positions by the number of each player's pieces. We enumerate all positions, maximal positions, and positions where one player has no move. Using these polynomials we count the number of positions that occur during alternating play. Our method extends to enumerating positions from mid-game positions and we include an analysis of a tournament game.
... They also find estimates for the number of legal games of Go. In [FS07], bounds for the number of legal Go positions for various numbers of players on some planar lattice graphs investigated. ...
... The motivation for game polynomials came from Farr [6] in 2003 where the number of end positions and some polynomials of the game Go were considered, and work in this area was continued by Tromp and Farnebäck [10] in 2007 and by Farr and Schmidt [7] in 2008. Even though Go is not a placement game since pieces are removed, it shares many properties with this class of games. ...
Placement games are a subclass of combinatorial games which are played on
graphs. We will demonstrate that one can construct simplicial complexes
corresponding to a placement game, and this game could be considered as a game
played on these simplicial complexes. These complexes are constructed using
square-free monomials.
With increased public interest in the ancient game of Go since 2016, it is an espe- cially good time to use it in teaching. The game is an excellent source of exercises in the theory of computation. We give some exercises developed during our research on Go which were then used when teaching this subject at Monash University. These are based on One-Dimensional Go (1D-Go) which uses a path graph as its board. They are about determining whether or not a position is legal and counting the number of legal positions. Curriculum elements that may be illustrated and practised using 1D-Go include: regular expressions, linear recurrences, proof by induction, nite automata, regular grammars, context-free grammars and languages, pushdown automata, and Turing machines.
We present several results concerning the number of positions and games of Go. We derive recurrences for L(m,n), the number of legal positions on an m ×n board, and develop a dynamic programming algorithm which computes L(m,n) in time O(m
3n
2λ
m
) and space O(m
λ
m
), for some constant λ< 5.4. An implementation of this algorithm enables us to list L(n,n) for n ≤ 17. For larger boards, we prove the existence of a base of liberties
of ~2.9757341920433572493. Based on a conjecture about vanishing error terms, we derive an asymptotic formula for L(m,n), which is shown to be highly accurate.
We also study the Game Tree complexity of Go, proving an upper bound on the number of possible games of (mn)L(m,n) and a lower bound of on n×nboards and on 1 ×n boards, in addition to exact counts for mn ≤ 4 and estimates up to mn = 9. Most proofs and some additional results had to be left out to observe the page limit. They may be found in the full version available at [8].
Computer Go is one of the biggest challenges faced by game programmers. This survey describes the typical components of a Go program, and discusses knowledge representation, search methods and techniques for solving specific subproblems in this domain. Along with a summary of the development of computer Go in recent years, areas for future research are pointed out.
These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics held at Rutgers University. Their aim is to link together algorithmic problems arising in knot theory, statistical physics and classical combinatorics. Apart from the theory of computational complexity concerned with enumeration problems, introductions are given to several of the topics treated, such as combinatorial knot theory, randomised approximation algorithms, percolation and random cluster models. To researchers in discrete mathematics, computer science and statistical physics, this book will be of great interest, but any non-expert should find it an appealing guide to a very active area of research.
This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This definition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less “local” in nature than those used to define more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modifications similar in spirit to the deletion–contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also briefly record some correlation inequalities.
The class of # P-complete problems is a class of computationally eqivalent counting problems (defined by the author in a previous paper) that are at least as difficult as the NP-complete problems. Here we show, for a large number of natural counting problems for which there was no previous indication of intractability, that they belong to this class. The technique used is that of polynomial time reduction with oracles via translations that are of algebraic or arithmetic nature.
It is shown that, given an arbitrary GO position on an n × n board, the problem of determining the winner is Pspace hard. New techniques are exploited to overcome the difficulties arising from the planar nature of board games. In particular, it is proved that GO is Pspace hard by reducing a Pspace-complete set, TQBF, to a game called generalized geography, then to a planar version of that game, and finally to GO.
The Oriental game of Go contains a unique method by which pieces, called stones, are captured and made safe from capture. A group of stones safe from capture is called safe, uncon~tion~y alive, or similar terms. Life or its lack can be determined by lookahead through the game tree, at some expense. We present a graph-theoretic static anaiysis of the board arrangement which determines unconditional life or its lack, together with proofs of its equivalency to look ahead. An algorithm for the static evaluation is given and we argue that it is the preferable method for computer Go play. These results constitute the fust realistic theorems in the theory of Go.
Counting legal positions in Go, BSE Hons disserta-tion, Clayton School of Information Technology, Monash Uni-versity
- J Schmidt
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Go for Beginners, Ishi Press, 1972, and Penguin Books
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Counting legal positions in go
- J Schmidt
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