# Force (Mathematics)

Difficulty: Expert   Keywords: Theory

This page explains the terms force, prevent, play-force, pass-force, virtual-force, and answer-force. Much of the explainations are adapted from Robert Jasiek's Japanese 2003 Rules, Types of Basic Ko paper, and Ko paper.

Note: On this page, the word "something" is used as a placeholder. For example, "to force something" can be used in the context of "to force a cycle", "to force a white two-eye-formation", "to force a black win", and so on.

# Force

A player can force something if he, moving first in imagined alternating play, has at least one strategy that achieves it regardless of the opponent's replies.

A possible formal definition is shown in the Japanese 2003 Rules.

Another possible formal definition is outlined by the following recursive construction:

For a position, a player can force something if that player can make a legal move that reaches a second position that satisfies one of the following:

1. The second position satisfies that something.
2. Any legal move by the opponent from the second position reaches a third position
• which satisfies that something
• or else in which the player can force that something.

## Example

Can Black force "a black two-eye-formation"? (Informal note: A two eye formation does not just mean a formation with two eyes. It means a formation with two one point eyes.)

Variation 1

= pass

Variation 2

Variation 3

Black succeeds in each variation. Whatever White tries (he does not have further alternatives), Black gets a black two-eye-formation. Hence Black can force a black two-eye-formation.

## Note

By contrast, White cannot force Black to make two eyes in this sense of force, even though in informal sense, he can.

Is forcing?

In ordinary parlance we say that forces , even though Black can choose to pass or play elsewhere.

# Prevent

To prevent something is force to fulfil not the something.

# Play-force

A player can play-force something if he can force it by making only plays (without passing).

Note: The opponent may play or pass whenever he likes.

# Pass-force

A player can pass-force something if he can force it by making at least one play and at least one pass.

Note: The opponent may play or pass whenever he likes.

# Virtual-force

## Definition-like Wording

For a ko, a player can virtual-force something if he can force the something while exceptionally he may always make a ko-capture on this ko immediately after the opponent's ko-capture on this ko unless that ko-capture was preceded by the player's ko-capture on a different ko.

A slightly more formal definition is in the paper Types of Basic Ko.

## Rules-like Wording

A player can virtual-force something if he can force the something while a ko is considered and he abides by the Released Ko Rule. The Released Ko Rule is applied to the considered ko and for him as the particular player.

Released Ko Rule: If a particular player has made a move that was not a ko-capture and if then the opponent has made a ko-capture in a ko, then during his next move the particular player may immediately recapture in that ko.

Notes: For virtual-force, the Released Ko Rule is applied to only one particular ko: the ko considered in the definition of virtual-force. - The Released Ko Rule overrides the given input ko rules used for the definition of virtual-force.

## Example 1

Although Black cannot force winning the ko, Black can virtual-force winning the ko.

(For a player and a ko, the player wins the ko when the ko is no longer a ko and the player has got uncapturable life on what was previously the ko intersections.)

= pass

Virtual-force gives Black the right to recapture immediately with .

Anonymous: Huh, what? That's an illegal move (immediate recapture of ko) and since black has no ko threats, I place 6 there myself.

RobertJasiek: It is a legal move since the definition of virtual-force overrides the basic ko rule by creating an exception to it with the text "[...] exceptionally he may always make a ko-capture on this ko immediately after the opponent's ko-capture on this ko [...]". It is not the purpose of virtual-force to reproduce ordinary go rules but to model behaviour under specifically defined circumstances. The model is an abbreviation of a simulation of a won ko fight even if the opponent has local ko threats or if the player has to make approach plays before he can win the ko.

Here, virtual-force simulates a ko fight that Black will win because of his arbitrary number of virtual ko threats. Thereby the local ko threat is insufficient for White to win the ko. (In other positions, Black can win a ko despite having to make approach plays.)

= , = pass, =

This sequence illustrates another application of strategy where Black can virtual-force winning the ko. By the definition of virtual-force, can take back the ko immediately after .

(Note that passing with is avoided here, otherwise can also pass and depending in the input ruleset, the hypothetical sequence would have ended.)

## Example 2

Suppose that the ko rules allow completion of a cycle, the left ko is studied, and White shall virtual-force a cycle or control the whole board.

Although virtual-force gives White extra rights in the left ko, may not be a ko-capture recapturing immediately because the preceding move has also been a ko-capture. To virtual-force a cycle, White needs to follow another plan:

White could virtual-force a cycle because Black was busy preventing White's control of the whole board.

Virtual-force does not give a player extra rights in a sequence of successive ko-captures.

A player can answer-force something if he, moving first in imagined alternating play, has at least one strategy that achieves it while the opponent is supposed to use an answer-strategy that forces something else.

A possible formal definition is shown in RobertJasiek's paper for a general definition of ko.

## Example

White can answer-force a cycle if Black shall prevent "White's area improvement" (i.e., force "not White's area improvement"):

Variation 1

tries to improve White's area. To prevent this, cannot pass but has to capture. then achieves creation of the cycle from move 1 to 3. Thus White has answer-forced a cycle.

Anonymous: Can't black play at ? I was under the impression that all cycles had to be of even length due to turns.

RobertJasiek: Read cycle. A [positional] cycle can be of even or odd length. In a two-player game, a situational cycle is of even length and it is the same player's turn before and after the cycle. - This example's topic is also about whether "White can answer-force a cycle". Already completes a cycle. Therefore a move is immaterial; it is too late; White has already created a cycle, so afterwards Black can do nothing more to prevent it. Black's task is scheduled until a cycle occurs.

# History

The terminology was coined and defined mainly by Robert Jasiek, partly by Robert Pauli. Only later Jasiek noticed that the basic term "force" had been used also in particular types of logic theories, where it is defined differently but serves the same principal purpose.