# Ultimate Theory of Weiqi Rules

PREFACE

The term “ Ultimate Theory” comes from physics, being a complete description of the laws of physics in a special mathematical form. From Maxwell’s equations at its origin, we can see the methodological differences between it and the classical physics that were based on experiments from Newton to Faraday. When physics is perfectly expressed by mathematics, physics is also perfect and ultimate, which is the meaning of the Ultimate Theory. The starting point of our previous research on weiqi rules has been that weiqi go already exists, and we have tried to describe it as accurately and as reasonably as possible. The research approach instead, imitating the ‘ultimate’ in physics, is to generate weiqi automatically in a mathematical and logical way. That is, we treat weiqi as a unique mathematical system interpreted on a board. Mathematics is based on axioms, and mathematical systems arise under axiomatic deduction. The ultimate study of weiqi rules is to the formation of weiqi via the weiqi axioms, so that the theory of weiqi rules expressed in this way will be perfect and ultimate.

I. AXIOMATIC FRAMING OF THE RULES OF WEIQI An axiom is a self-evident, rationally based and accepted, most fundamental proposition. The rules of weiqi consist of three axioms, one rule corollary, and one example of goal reasoning, in two categories: two established axioms, one weiqi axiom, the one rule corollary, and the one case of goal-reasoning.

(i) Established axioms: 1. Basic description: Weiqi consists of a weiqi board and black and white stones, the board is n lines vertically and horizontally (the standard board is taken to be n = 19). The standard board is 19 x 19 and has 361 intersections; there must be a sufficient number of stones.

2. Rules of procedure: Each player has one __colour__? of stones. They take turns to place one stone on an empty point on the board. Stones placed on the board do not move again. These established elements of weiqi are few and basic, natural and inevitable, and can therefore be called established axioms.

(ii) Weiqi axiom, i.e. the core rules or characteristic rules of weiqi: 3. Rule of capture: the stones with no liberties must be removed from the board. This is the core of weiqi and is what makes it the surrounding game it is.

The axioms, that is, the artificial stipulations, end here, and all the rest of the rules must be inferred from the three axioms, so that no other artificial stipulation need be made.

(iii) Corollary 4. Rule corollary: the ko rule This item is an inference from 2. and 3. This item will be discussed later.

5. Goal corollary: the goal of weiqi The goal of the game is a product of the rules. Since the rules of weiqi are constructed in a system of axioms, the goal must come out of the axioms, i.e. be an axiomatic corollary. Thus the goal of an axiomatisation can only be one of two physical objects in the axioms: the board or the stones in Axiom 1.

(1) With the stones in Axiom 1 as the goal, the board in Axiom 1 is the vehicle for achieving the goal: more stones survive on the board in accordance with Axioms 2, 3 and 4. This is the ancient rule of weiqi, generally known as ‘stone scoring.’ Of course, it can also mean capturing more of the opponent's stones. More dead stones means fewer live stones, and the scoring with live stones or dead stones is actually the same thing.

(2) Taking the board in Axiom 1 as the goal, the stones in Axiom 1 are the tools to achieve the goal. That means using the stones to occupy the points on the board (19 x 19 = 361) in accordance with Axioms 2, 3 and 4, with the goal of taking the larger share.

But what is ‘occupation’? The most axiomatic occupation is to occupy with stones directly, and the result is equivalent to (1). But the eye-positions in (1) do not count, because no stones can live there. But if the goal is points on the board, the points of the eye-positions are part of the occupying stones’ group. Therefore, containing the eye-positions can be considered axiomatic as well, although perhaps to a slightly weaker extent. This is now the Chinese rules, generally known as ‘area scoring.’

Discussion: Can there be other goals then? Axiom 1 is that weiqi consists of only the board and the stones, and there is no third element, so that the only two natural goals under axiomatic inference are these. To set other goals would be an artificial addition outside the axiomatic system. An axiomatic system of weiqi rules with goals set outside the axiomatic system is likely to lead to discord in the system. Japanese territory scoring, including backfill of dead stones, is an artificial design that has nothing to do with axiomatics. Artificially designed rules that are not deduced from axioms are likely to conflict with other axioms and lead to self-logical paradoxes.

The value meaning of the territory only holds when it is used as an inference in the system of goals of (1) or (2). The inference of territory is used because it simplifies calculation of the goal of (1) or (2). The intrinsic axiomatisation of a goal can only be (1) or (2). The failure of Japanese rules is that they are obscured by appearances and so lose their essence.

II. About the ko rule The ko rule is a special rule in weiqi and has always been laid down in the form of "Before a stone can be repeatedly removed, a move must be made elsewhere on the board." This rule is obviously not natural enough. Thus, the idea of prohibiting repetition of the game position has been proposed, which is a good theory, but this is also a new axiom and strictly speaking its axiology is weak. The fact is that we do not need to specify the ko rule, which is already logically implicit in Rules 2 and 3.

Immediate removal of a ko-stone by this action: Taking away a stone the opponent has just played and putting back a stone the opponent has just captured, essentially negates the opponent's previous move, i.e. negates the right of the opponent to place a stone under Axiom 2. Although it has a basis in Axiom 3, there is a rule of placing before there is a rule of capturing, and Axiom 3 arises after Axiom 2, so Axiom 2 takes precedence. Thus the ko-rule can be considered as a corollary of Axioms 2 and 3.

Where the prohibition on negation is extended from the previous move to any of the previous moves: It is forbidden to negate the opponent's moves, that is, to make the opponent face a game situation he has already faced. This means prohibiting repeated game positions. Although extension leads to a reduction in logical necessities, it is nevertheless still necessary to adopt this extension for multiple kos, and in particular for cases like moonshine life.

III. MISCELLANEOUS 1. Void move (pass) Since weiqi aims at living or occupying space, a move is a move to live or to occupy space, so if no move is made it implies that the game is no longer to be played, and if the opponent does not make a move either, both players are no longer playing and so the game is over. Therefore, making no moves is regarded as proposing the end of the game and then agreeing to the end of the game, which can be understood as a corollary to Rule 5. If the opponent continues to play, the game of course continues, and the unplayed move is then called a void move or a pass.

2. Forbidden moves Suicide is of course not forbidden in the axiomatic system.

3. End of the game and agreement as to end of the game These are agreed procedural rules that guarantee the achievement of the outcome of Rule 5 subject to Rules 1, 2, 3 and 4.

4. Compensation for first move komi This is a convention, not a basic rule of weiqi. There is no provision for compensation in any of the current weiqi rules texts, but rather it is set out in the competition rules. Because, in principle, komi compensation is only one of the ways to balance the advantage of playing first, the available options are whether to use this method, how much to compensate, and how to compensate.

Author: Chen Zuyuan

Translation: John Fairbairn