A Precise and Nearly Complete
Description of the SST Ko Rules

Wilfred J. Hansen <wjh@cmu.edu>
Andrew Consortium
Carnegie-Mellon University
http://www.cs.cmu.edu/~wjh

Copyright Wilfred J. Hansen, 1996. Not to be reproduced in hardcopy or on portable disk without the author's written permission. Created with the Andrew system: www.cs.cmu.edu/~AUIS.

Abstract: A "ko" rule helps ensure termination of the game of go (weiqi, baduk) by preventing games from collapsing into infinite loops. The SST ko rule, introduced by Ing Chang-ki in 1990, attempts to avoid certain deficiencies of earlier ko rules, but is described in an imprecise manner, relying on diagrams. This paper presents two precisely defined rules--cycle sets and disallowed captures--which together describe most of the SST ko rule. In conjunction with a description of the differences from the SST rules, these rules can contribute to wider understanding of, and eventually more precise descriptions for, the SST rules.

Without some rule, most games of go (weiqi in Chinese, baduk in Korean) would end in an infinite loop with each player capturing the stone his opponent had just played; see Figure 1. There is general agreement about most of the rules; but none of the proposed anti-loop rules has met universal acceptance.

Figure 1. Simple ko. If Black plays at 1 to capture the triangled white stone, all the various ko rules bar White from playing immediately at the emptied point to recapture the black stone.

"Ko" is the Japanese term for the repetition in Figure 1 and has come to denote any loop in play. The Japanese rules [1] outlaw the simple repetition of Figure 1, but if a more complex infinite loop occurs, the game is declared to have no result. (This is stronger than a draw: the players must start over.) Other ko rules try to prevent draws.

The "super-ko" rule, adopted by many associations, including the American Go Association [2], declares that repetition of the whole board pattern is not allowed. This rule seems reasonable in most practical cases, but has been criticized as being hard to administer and giving some "surprising" results.

The "SST" rules [3,5,6] avoid the putative difficulties of super-ko, but have been specified in language that is not always clear to an amateur go player. As specified, the rule requires players to recognize certain situations based on example diagrams in the rules.

This note presents two rules which cover the majority of the cases in the SST ko rules. The rules do not refer to patterns or concepts that are only apparent to experienced players. They do not even define the term "ko," although the usual ko rule for Figure 1 is a consequence. Experienced go players may wish to skim quickly through the first section below, which presents some background. The second section defines a few basic terms in preparation for the next two sections which define the basic rules. Subsequent sections compare them to the SST rules.

The rules themselves are in paragraphs preceded by a number and a right parenthesis. Various informal additions to the rules have an "A" in front of the number. These additions eliminate tedious plays necessitated by the other rules and bring the rules into alignment with SST.

1. Background

More complex repetitive situations do occur and the introduction presented three approaches to resolving them. The Japanese rules simply void the game, the super-ko rule dictates non-repetition of the pattern of stones on the board, and the SST rules distinguish two kinds of kos.

Voiding the game as in the Japanese rules is a viable approach since complex repetitions are rare. However, in many amateur tournaments time is limited to something like a single weekend and there may be no good way to play the necessary extra game. Some observers also find the notion of a draw as contrary to the spirit of the simple ko rule of Figure 1.

Super-ko solves all problems of repetition, but is deemed by some experts to have two principal defects. First, it can be difficult to administer, especially by the sort of non-professional, volunteer referee found at most non-Asian tournaments. Detection of a board repetition is not too difficult, but it can be difficult to determine exactly which move began the repetition. For example, when two groups share three kos, the board exhibits six different patterns before it repeats. Even if the players have been keeping a record of the moves, determining which is the first illegal move can be daunting; if there is no record, resolution may try the talents of the best referee. Second, there arise a few obscure situations in which the super-ko rule can lead to results which some think are surprising. In Figure 2, for example, Black can capture one of the two one-eyed white groups, both of which some observers believe "ought" to be alive by virtue of having an eye while the enclosed black group does not. Black attacks first in the upper left group and then in the lower right. At the moment when the board is about to repeat the configuration in Figure 2, it is White's move, one of the white groups has only a single liberty, and an additional liberty can be gotten only by repeating the original board configuration. (Another supposedly surprising outcome is the possibility of life for the moonshine life group on the upper left in the diagram in section 5.4. This life depends on there being absolutely NO playable points on the board, so the situation is highly unlikely, at best.)

Figure 2. A "surprising" result of the super-ko rule. By a sequence of captures Black can capture one of the two white groups, even though they both have an eye and the black groups do not.

Reviewing the deficiencies of existing ko rules, Ing Chang-ki came to the conclusion that a new ko rule was required. After a decade of discussion he came up with the SST rule promulgated in English with document [3]. This rule divides ko battles into two classes, "fighting ko" and "disturbing ko." A fighting ko is one where winning the ko makes a difference to the life or death of a string (a string possibly just as small as the one stone that is being repeatedly captured, but possibly much bigger). In a disturbing ko battle, the safety of groups can be restored after one capture by another capture elsewhere. For these latter situations peaceful coexistence is possible and there is no need for a fight. One simple example of a disturbing ko battle, Figure 3, has two groups sharing four kos.

Figure 3. Disturbing a ko situation. Both the black and white groups have two liberties. If either player captures one of the kos, the opponent can capture another to restore the equality. In SST terminology, the first capture is said to "disturb" the situation.

Classifying kos as either "fighting" or "disturbing" has not met universal enthusiasm. It is undeniably less simple to describe than the super-ko rule and many amateur players are not convinced that the defects of super-ko are sufficiently serious to warrant its abandonment. The defects arise so seldom that there is little experience with problems. It must be noted however, that these defects will become immediately noticed whenever they first affect the outcome of a tournament with a large prize at stake; this has caused crisis in earlier times and led to the first written form of the Japanese rules.

The description of the SST rules also deters its acceptance. Westerners, the most vocal critics, are accustomed to rules expressed as decision procedures in terms of simple concepts that do not require deep understanding of the game. The super-ko rule is such a decision procedure: we need only remember board patterns and detect repetition; there is no need to understand the game at all. The SST rule, however, is stated in terms of situations that can be recognized only by understanding the game and recognizing patterns. The concepts used in the rules include poorly defined terms; for instance "invariation," which may mean the super-ko rule or may mean something slightly different. The word "ko" is used somewhat ambiguously and may refer to a portion of a game diagram, a set of adjacent strings, or even the whole board diagram if it includes a ko. At the very heart of the rules is the distinction between fighting and disturbing kos, which is said to depend on whether "life and death are decided," even though deciding life and death is difficult and the rules elsewhere state that life and death are to be decided only by the play of the stones. Attempts to write a computer program to implement the SST rules as stated have not yet succeeded, from which it would seem that they are not specified clearly or precisely enough.

Despite the above comments, no one has brought forth a game diagram which convincingly demonstrates that SST's distinction between fighting and disturbing ko is indeterminate. Nor has anyone really shown that the SST rule is as hard to administer as the super-ko rule admittedly is. With this in mind, I believe it is appropriate to try to more precisely define the SST ko rule, or at least something close. That way further discussion can be based on fact rather than conjecture.

2. Notation

1) Color of a board point. The color of a board point is empty, black, or white, depending on whether it is unoccupied, occupied by a black stone, or occupied by a white stone, respectively.

2) Adjacent. Two points are adjacent if an arc connects them. Two stones are adjacent if they occupy adjacent points. A stone is adjacent to an empty point if the stone occupies a point adjacent to that empty point.

3) String. A set of stones of a single color is a string if there is a sequence of adjacent like-colored stones from every stone of the set to every other and there is no such sequence to any like-colored stone not in the set.

4) Liberty. A liberty of a string is any empty point adjacent to a stone in that string. The liberty count of a string is its number of liberties.

5) Capture. A string is captured if it has only one liberty and the opponent places a stone on that liberty point. Captured stones are removed from the board, leaving their point empty.

Figure 4. Examples of notation.
i) The colors of points a, b, c, and d are empty, empty, black, and white, respectively.
ii) Points a and b are adjacent. The black stone on point c is adjacent to an empty point at b and a white stone at d.
iii) Points marked s constitute a string. The stones at c and d are not in string s, but are each a string of one stone.
iv) Stone c has 3 liberties, d has 1, and the string s has 7.
v) The string e will be captured by a play at f.

3. Cycle Sets

6) Same board configuration. A board configuration is the board's set of points together with the color of each. Two board configurations are the same if every point in one has the same color as the corresponding point in the other. {Usually we are comparing the configurations on one physical board at two different times.}

7) Same capture. Two capture moves are the same if the capturing stones are played at identical board points and the set of stones captured by the first capture occupied the same board points as the stones captured by the second. {See Figure 5e.}

8) Cycle sets. If a capture is the same as a capture earlier in the game and both leave the same board configuration, all intersections at which stones were played by intervening moves are added to the cycle set for the capture. The cycle set for any capture that has not been repeated is empty.

9) Rule: Cycle sets. If a play makes a capture that has or creates a non-empty cycle set, the opponent's next move must not be at any board point in that set.

Figure 5. Example of a Cycle Set. These six diagrams are all of one portion of a board at different times.
a. Original configuration. Imagine that a white stone at a will divide Black's group in two so that this ko is quite serious.
b. White plays at b to create the ko.
c. Black captures.
d. Black's previous move did not repeat a capture, so it did not create a cycle set. White recaptures at d legally. This recreates the earlier board configuration, but that earlier configuration was not reached by the capture at d. Therefore this capture does not create a cycle set.
e. Black recaptures at e. This capture is the same as c and results in the same board configuration. The two inner points are now in the cycle set for a capture at e. Any time Black makes this capture, White cannot immediately recapture.
f, g. White makes a ko threat elsewhere and Black responds.
h. White captures. This does not create a cycle set because the board differs from when d was played. Black could immediately recapture, and then White would recapture again so the capture at h would now have a cycle set.

Figure 5 demonstrates that the Cycle Sets rule reduces to the ordinary ko rule. The sequence illustrated could be followed every time a ko arises, which would be pointless. To prevent this we need an informal rule:

A1) The etiquette of obvious cycle sets: When a capture could obviously create a cycle set the players should behave as though the set has already been created.

Stones on points in the Cycle Set include those called "hot stones" in SST terminology, but also can include stones which we will later see can be part of disturbing kos. See Figure 6, which shows a cycle set larger than two points.

Figure 6. Cycle Set with six points in it. Black starts the festivities by capturing at 1. There follows the indicated series of captures and then Black recaptures with move 7 at the same point as 1, returning the board configuration to that shown. This adds all six points to the cycle set for the capture at 1, so White must next make a play elsewhere.

The cycle set rule requires that a remote disturbing situation like the upper left in Figure 6 be used as ko threats by both players in succession in order to add those two kos to the cycle set for the capture at 1. It might be nicer to accept the two ko threats at any time during the ko fight, but it is then not so easy to formulate the rule.

4. Disallowed Captures

10) Constrained move. A move is said to be constrained if the opponent's preceding move was a capture that had or created a non-empty cycle set. Otherwise it is unconstrained. {Intuitively, a constrained move is one played when it is the player's turn to make a ko threat.}

11) Threatened. A string is threatened if it has only one liberty. A move threatens a string if its reduces the string's liberties to one.

12) Potentially disallowed. A move is potentially disallowed if it is unconstrained, it threatens an opponent string, and does not increase the liberties of any threatened string of its own color.

13) Counteracted - A potentially disallowed move is counteracted by a subsequent opponent's move which does not capture the stone played as the potentially disallowed move, but does increase to two or more the liberties of the move's threatened strings.

14) Move signature - A move's signature is the set consisting of the board point at which the stone is played, the board points from which stones were captured, and the board points occupied by stones threatened by the move.

15) Rule: Disallowed capture. If a potentially disallowed move is counteracted, then the player who made that move must never again make an unconstrained move with the same signature.

Figure 7. Examples of definitions 11 to 14.
i) The three triangled white stones are threatened.
ii) A play by White at B would threaten the three black stones.
iii) A White play at C would threaten the string of two black stones.
iv) After a White play at C, a play by Black at either of the positions marked D will restore the two threatened black stones to having two liberties. Since the liberties prior to playing C have been restored, the play at C has been counteracted; therefore, another play by White at C would be disallowed (in the unlikely event that C became empty when the two black stones again had two liberties.)
v) The signature of the move at E is that board point, the point to its left where a capture was made, and the points occupied by the other nine white stones whose liberties were reduced to one.
vi) If black captures at E, that capture is potentially disallowed because it decreases white's string to one liberty and increases the liberties of a string which already had two liberties. Next white captures at F. This counteracts the capture at E by increasing the liberties fo the white string to more than one. Later White might capture the E stone and Black would respond by capturing the F stone. Now Black cannot play at E because that move would have the same signature as the earlier counteracted move at that point.

Figure 8. Example of a disallowed capture. If White captures a black stone by playing at 1, that play is potentially disallowed because it reduces the liberties of the big black string to one (and the white string started with two liberties.) Black can counteract the move at 1 by capturing at 2 and restoring the liberties of the black string to 2. The Black capture at 2 is not potentially disallowed because the original liberty count of the black string (after White has captured at 1) is less than two.

Definition (10) for "constrained moves" is introduced so that moves used as ko threats do not become potentially disallowed and moves already disallowed are valid as ko threats. However, they will soon be entered in a cycle set for the ko. For example, see Figure 6. The cycle from 1 to 7 restores the board to its original configuration and points 2, 3, 5, and 6 are added to the cycle set for 1. The upper left will have been used exactly once as a ko threat by each player. In some cases, one player will not respond to the ko threat, preferring to fill the ko. This is fine; the other player is free to make good on the ko threat.

5. Agreement with the SST rules

5.1 Triple kos

5.2 An eye, two kos, and an outside ko

There are two ways White may try to kill black in this corner, both prevented by the rules above. First, White may try to capture b or c and claim that the Etiquette of Obvious Cycles (i.e., "hot stones") force Black to make a ko threat. This fails because White cannot satisfy the Etiquette by demonstrating that a cycle set will arise when Black captures the other of c or b. Instead, White's captures at b and c become disallowed (rule 15). Second, White may try to use b or c as a ko threat when Black tries to capture White by playing at a. This also fails because Black also uses b and c as ko threats, recreating a board configuration and entering b and c into the cycle set for White's capture of a (rule 9). When White has no ko threats, Black takes at a and then captures the white stones.

5.3. Sending two returning one

Black plays at 1, white captures, and black recaptures at x. The board is the same as an earlier board, but it is the other player's turn. This is legal under most rule sets, even including super-ko as defined by the AGA. A cycle set is created only in the unlikely event that a previous play at x captured a stone in the corner. However, even if there is no cycle set, White will be unlikely to play in this corner. No disallowed captures are created because liberty counts prior to the captures are one.

5.4. Moonshine life

White may try to claim that the upper left group cannot be captured because when Black plays at 1, White makes a ko threat in the lower right. However, Black also uses the lower right for a ko threat, the board repeats, and the four ko points in the lower right are entered into the cycle set for the ko at 1 (rule 9). Thereafter white must make ko threats elsewhere, but black need never make threats, just as in SST. If requested to demonstrate capture, black plays on until white has no ko threats; black captures at 1 and white has no ko threats so white passes and black captures the group.

5.5. Triple ko with an eye

If White captures at 1, Black counteracts by capturing one of the other kos and restoring the black string to two liberties. Thereafter, a capture by White at 1 is disallowed (rule 15). Similarly all possible ko captures here can become disallowed and the situation remains on the board, just as in SST.

5.6 Three kos and an external liberty

Both inner strings have two liberties. Under the Disallowed Capture rule (15), if either party captures one of these three kos, the opponent can capture another ko to return both strings to two liberties. Captures inside thus become disallowed. When the white string's external liberty is filled, the ko fight begins as in a usual triple ko.

Curiously, the external liberty may never be filled and the diagrammed situation could remain on the board. Clearly, White will not fill the external liberty; that would be suicidal. When Black fills the external liberty, White captures a ko, forcing Black to make the first ko threat. If Black's ko threats are too small relative to this situation, Black will not fill the external liberty either. Leaving the liberty unfilled is not a problem for the rules: both sets of stones stay on the board for the scoring phase, giving Black two points and White one.

5.7 Eternal life

The SST rules differ slightly. They also forbid the move at 3 following 2, but the other illegal move is the capture at 2 following the play at 1; the play at 4 following 3 is allowed. Even so, both Black and White have to make one ko threat per iteration. To match SST exactly, we would need to add:

A2) Eternal life: When one stone is added to another to make double ko stones in an eternal life position, they cannot be captured immediately.

Bill Spight (BillSpight@aol.com) has noted that after Black 3 White could get a seki by playing 4 at 2, but that this move might be forbidden by the Cycle Set rule. However, the cycle does not occur until 4 is played, so playing 4 at the position of 2 is acceptable. Even on later iterations White can play at 2 after black captures with 3. By a strict reading of the rules Black 3 does not create a cycle set if White has just made a ko threat, because the board configuration is different (due to the stones played for the ko threat and response).

5.7 Triple ko stones

The SST rules [3] provide this explanation:

White 2 and Black 7 are game-disturbing moves that use ko threats cyclically. This must be prohibited.

This implies that White must make a ko threat before playing 2 and Black must do likewise before playing 7. The entire situation is deemed a ko fight by the SST rules (although neither player has ko threats in the diagram.)

The ko fight is much more direct with rules 9 and 15 above; White makes a ko threat in lieu of 2 and then recaptures 1. The analysis begins wth the reasonable assumption that White's move prior to Black 1 did not capture at Black 1 and therefore left the situation as shown on the left without the numbered stones and without an immediately preceding capture. The first iteration ends when White plays 10 and Black then plays 11 at 1 to initiate the second iteration, with White forced to make a ko threat according to rule 9. When Black responds to White's threat, White captures back with a play at 8, and a simple ko fight ensues.

5.8 Double ko stones and a shared liberty

Double ko stones: coexistence with balanced breaths.
Disturbed life in which either side can be the disturber.

A four step sequence of moves can occur: each player plays a stone to reduce to one the liberties of a triangled string and the next two moves are captures, each capturing a triangled string. The first of these captures reduces an opponent string to a single liberty and is potentially disallowed. This move is immedaitely counteracted by the other capture, so the first capture becomes a disallowed move.

5.9 Molasses ko

Molasses ko. Following move 1, Black plays elsewhere, White plays on line 4 or 5, Black captures 1, White captures two stones, and Black plays on the second line, capturing again. The situation is now reversed and White plays elsewhere.

The disallowed captures rule (15) interrupts the cycle inherent in this diagram. When White plays on line 4 or 5, the black string is reduced to one liberty and the capture is counteracted when Black captures the stone marked 1. On the second cycle White will be able to make the other throwin, but on the third cycle White will have no move.

6. Incompleteness for "disturbed death"

To resolve this difference, the rules can be augmented in several ways. My favorite candidate is:

A3) TKO (Two-Kos-One-liberty). A group with two kos and only one liberty cannot capture either ko, except as a ko threat.

One alternative to the TKO rule is to formulate a rule which specifies that every ko must be filled once it has been captured and before any other ko between the same groups can be captured. This scheme requires that "ko" be defined, but may have the virtue of supplanting all the other rules.

7. Conclusion

The aim here has been an approximate statement of the SST ko rule. Proponents of the latter may now be able to more clearly explain it by describing in detail how it differs from the rules above.

Acknowledgements

This note would not have been possible without help from Janice Kim, Michael Simon <74237.3537@compuserve.com>, Sidney Yuan <http://www.yutopian.com/go>, and Yang Yu-Chia of the Ing Foundation. I also received valuable assistance, encouragement, notes, and reactions from John A. Bate <bate@cs.umanitoba.ca>, Jonathan Buss <jfbuss@math.uwaterloo.ca>, James Davies <davies@twics.co.jp>, Dave Ring <dwr2560@tam2000.tamu.edu>, Bill Taylor <wft@math.canterbury.ac.nz>, and John Tromp <tromp@daisy.uwaterloo.ca>. Interesting thoughts were contributed by Terry Benson <terrybenson@delphi.com>, Keith Crews <keith@belmt3.belmont.com>, Andre Engels <csg419@wing.rug.nl>, Harry Fearnley <Harry.Fearnley@eng.ox.ac.uk>, Stig Hemmer <stig@pvv.unit.no>, Phil Straus <pstraus@holonet.net>, and others to whom I apologize for not having kept better notes. Any errors remain despite their best efforts to keep me straight. Be a hero, get your name here: send me substantive comments about this paper prior to 1996.

References

[1] The Nihon Kiin, The Kansai Kiin, The Japanese Rules of Go, The Nihon Kiin, tr. James Davies, April 10, 1989.

[2] AGA Rules Committee, Official AGA Rules of Go, American Go Association, April 1, 1991.

[3] Ing - Ing Chang-ki, Ing's SST Laws of Wei-Chi, 1991, Trans. James Davies; Ing Chang-Ki Wei-Ch'i Educational Foundation; 4F, No. 35, Kuan Fu S. Rd.; Taipei, Taiwan, R.O.C.; Tel:886-2-7614117; Fax:886-2-7686940

[4] Jasiek, Robert, "Ing Ko Examples," http://www.inx.de/~jasiek/ingkoexa.html, July 7, 1996.

[5] Kim, Janice, Michael Simon, and Phil Straus "Ing's SST Rules of Go". Available from bsdserver.ucsf.edu/Go/aga/ or from AMERICAN GO ASSOCIATION; Box 397, Old Chelsea Station; NY, NY 10113; USA

[6] Yuan, Sidney, American Go Extra, Fall(?), 1995. Yutopian Enterprises; 4964 Adagio Ct; Fremont, CA 94538; USA Tel: +1 (510) 659-0138 (1-800)YUTOGO-3. email: yutopian@netcom.com (Originally distributed via internet newsgroup rec.games.go)

[7] Hall, T. Mark, A Problem of Status, British Go J., 94, Spring 1994, p. 29. Followup under the same title: Hazelden, Paul, British Go J., #95, Summer 1994, p. 31.

Appendix. Rules Summary

1) Color of a board point. The color of a board point is empty, black, or white, depending on whether it is unoccupied, occupied by a black stone, or occupied by a white stone, respectively.
2) Adjacent. Two points are adjacent if an arc connects them. Two stones are adjacent if they occupy adjacent points. A stone is adjacent to an empty point if the stone occupies a point adjacent to that empty point.
3) String. A set of stones of a single color is a string if there is a sequence of adjacent like-colored stones from every stone of the set to every other and there is no such sequence to any like-colored stone not in the set.
4) Liberty. A liberty of a string is any empty point adjacent to a stone in that string. The liberty count of a string is its number of liberties.
5) Capture. A string is captured if it has only one liberty and the opponent places a stone on that liberty point. Captured stones are removed from the board, leaving their point empty.
6) Same board configuration. A board configuration is the board's set of points together with the color of each. Two board configurations are the same if every point in one has the same color as the corresponding point in the other.
7) Same capture. Two capture moves are the same if the capturing stones are played at identical board points and the set of stones captured by the first capture occupied the same board points as the stones captured by the second.
8) Cycle sets. If a capture is the same as a capture earlier in the game and both leave the same board configuration, all intersections at which stones were played by intervening moves are added to the cycle set for the capture. The cycle set for any capture that has not been repeated is empty.
9) Rule: Cycle sets. If a play makes a capture that has or creates a non-empty cycle set, the opponent's next move must not be at any board point in that set.
10) Constrained move. A move is said to be constrained if the opponent's preceding move was a capture that had or created a non-empty cycle set. Otherwise it is unconstrained.
11) Threatened. A string is threatened if it has only one liberty. A move threatens a string if its reduces the string's liberties to one.
12) Potentially disallowed. A move is potentially disallowed if it is unconstrained, it threatens an opponent string, and does not increase the liberties of any threatened string of its own color.
13) Counteracted - A potentially disallowed move is counteracted by a subsequent opponent's move which does not capture the stone played as the potentially disallowed move, but does increase to two or more the liberties of the move's threatened strings.
14) Move signature - A move's signature is the set consisting of the board point at which the stone is played, the board points from which stones were captured, and the board points occupied by stones threatened by the move.
15) Rule: Disallowed capture. If a potentially disallowed move is counteracted, then the player who made that move must never again make an unconstrained move with the same signature.
A1) The etiquette of obvious cycles: When a capture could obviously create a cycle set the players should behave as though the set has already been created.
A2) Eternal life: When one stone is added to another to make double ko stones in an eternal life position, they cannot be captured immediately.
A3) TKO (Two-Kos-One-liberty). A group with two kos and only one liberty cannot capture either ko, except as a ko threat.