Dieter's ideas on Go Theory
Table of contents |
Table of diagrams Liberty of a stone A chain Separate chains White to capture White to capture White to capture White to kill Groups with two eyes White to play Solution Black to play Solution Strength Influence Surround |
Introduction
This page is a personal project by Dieter. It is an attempt to accurately describe Go theory. Go Theory starts with rules. I have chosen to start from stone counting, for I currently strongly feel that that is how the game originated.
From the rules we can derive principles. For a deeper understanding of those principles, we develop several concepts. The principles and the related concepts finally give rise to guidelines, or as they are called in Go theory, proverbs. The game has three aspects: technique, tactics and strategy, so proverbs will typically look at one of those three aspects.
Your comments below this document are appreciated.
The purpose of the game
The purpose of the game is to put more stones on the board than your opponent.
The rules of the game
Rule 1: Players take turns. Black starts.
Rule 2: the rule of capture
First concept: liberty of a stone
A stone's liberties are its empty adjacent points.
has four liberties.
has three liberties.
has two liberties.
has three liberties, as has the unmarked black stone.
Second concept: A chain and its liberties.
A chain of stones is a set of stones of the same colour so that between each two stones there is a path of adjacent points occupied by stones of the same colour.
The liberties of a chain is defined as the union of the liberties of all stones that make up the chain.
The two black stones on the left form a chain with 6 liberties. The white stones are also a chain with 6 liberties.
The two black stones do not form a chain. You could say there are two chains of one stone each. The white stones form two chains of two stones.
There is a rule of capture stating that ''after playing a stone, the opponent stones without liberties are taken off the board. [10]
Rule 3: The rule of forbidden repetition
In some situations, board positions can be infinitely repeated. The ko rule forbids this. However, the game principles dealing with ko are advanced and do not affect the basic ideas of Go.
Derived concepts
The concept of life and death
First, we'll make a few exercises
White to play and capture Black's stones.
Where and in what order has White to play in order to capture the Black stones.
Can Black save his stones?
Where and in what order has White to play in order to capture the Black stones.
Can Black save his stones?
A group of stones that cannot avoid capture is said to be dead. Turn a group into a dead group is called to kill.
Here it starts to become interesting: Where has White to play in order to kill the Black stones?
Where can Black play in order to prevent his group from being killed?
A group that is safe from capture is called alive. The most typical alive groups are groups with two eyes.
Both black groups are alive.
The concept of territory
If an area of empty points is surrounded by one player, in a way that he will always be able to fill it up with alive stones and his opponent will never be able to put down stones there without them being killed, such an area is called territory.
Basic principles
The principle of liberties
It is important to have liberties for your stones. So, on a small scale, you will look for moves that
- increase your own stones' liberties
- decrease your opponent's stones' liberties
White's stone has one liberty. After the -
exchange, she has two liberties. Black's chains are separated and have a small number of liberties each.
If instead White plays here, Black captures a stone and his chains are virtually unified, giving many liberties.
In this situation after the -
exchange White still has gone from 2 liberties to 1 and will be caught in a ladder.
Instead, here she extends her number of liberties from 2 to 4, leaving Black to play.
The principle of connection
First, again a small exercise to illustrate the principle
White to play and decide the game by knock-out.
Solution: cuts Black into two groups. Both groups now need to make two eyes with a or b. Black cannot do both, so one of the groups will die, giving White a definite victory.
This is a comparable position to the above. Black must find a way to save his central stones or the game will definitely be over.
After , a and b are available to Black in order to make two eyes.
As the above diagrams suggest, to cut your opponent's groups is a major asset in reaching victory. Likewise, if you keep your groups connected, in general you need less moves to make life for your stones.
- Connect your own stones, cut those of your opponent.
Advanced concepts
The concept of thickness or strength?
A group of stones that is connected and has sufficient possibility to make eyes, is called thick or strong. Strength is the quality of a group to be hardly vulnerable to attack.
The Black group on the outside is strong. It is connected and will easily make eyes at a and somewhere in open space.
The White group can be said to be more than strong, i.e. alive.
The concept of influence
Empty points that lie in the vicinity of a (strong) group of stones of the same colour, say white, are likely to become white territory. We say they are White's influence.
The marked zones are clearly influenced by the strong black stones and likely to become territory. The range of Black's influence can be much wider, depending on the rest of the board.
The area behind the White stones is more than influence, it is territory.
Advanced principles
The principle of equivalence
In the above examples we encountered the notion of equivalence. Moves that leave you with equivalent options, different moves to reach a comparable goal, are generally interesting moves.
- Look for moves that leave you with equivalent options.
The principle of intiative
The player who starts has the advantage. This extends to a balanced position, reached after a sequence of moves. Trying to get the initiative in a balanced position is an important principle.
- Try to keep the initiative
The principle of efficiency
From the fact that stones are played in alternate turns, we can derive a meta-principle, namely that you should place your stones with a maximal effect.
- Stones should be placed with maximum efficiency
The principle of flexibility
If a player sees a sequence giving him undisputed victory, will play out this sequence. This can be called "settling the shape". If settling the shape does NOT give a definite advantage, then it gives more information to both players. In particular, when the shape is settled with one move, the opponent is the first to adopt a way of playing that takes the settled shape into account.
Therefore, as long as it is unclear who is favoured by the board position, both players will keep their options open as much as possible. This applies also to parts of the board.
- (i) Preserve flexibility as long as the position is unclear
- (ii) Settle the shape when the position favours you
The principle of surrounding
A group of stones that is surrounded has two disadvantages
- It has to seek life inside its own space
- It has no influence on the rest of the board
Therefor, although a group may not be immediately killable, or surround a substantial amount of extra territory, it is advisable to
- (i) Surround the enemy's groups
- (ii) Avoid being surrounded by the enemy
In a sense, this principle extends the principle of connection with friendly stones to connection with open space. If the rules speak about capture, the true way of the game is encirclement.
Although a White play at a would completely ensure White's territory in the corner, a play at avoids being surrounded. Thanks to
, White will have influence along the upper side and playing at a may even not be necessary, so she increases her efficiency as well.
Skills
Many skills are needed to play Go. A correct attitude?, good concentration, stamina?, a mixture of creativity and faithfulness to the principles, all fall in the category of gamesmanship.
In Go, you particularly need technique, tactics (reading skills?) and strategy (positional judgment) in order to start, branch and evaluate a sequence of moves. Knowing where to start, to cut off branches and where to evaluate, depends on technique and strategy. Correctly building up the tree depends on reading skills?.
Technique
Good technique is built upon nearly all the above principles. Technique is needed to find good moves that have a high success probability in certain situations and to know where to cut off branches. Also, players with superior technique can read much further than those with arbitrary technique, because they need to branch less.
Reading skill
The skill of reading is needed to calculate future positions. In particular, at a lower level life and death problems and capturing races need a lot of deep reading.
Positional judgment
Sequences of moves are evaluated by looking who has more territory, more prospects for alive stones in general (influence and thickness), who is connected or cut and who has the initiative.
Positional judgment aids in determining one's strategy.
Basic proverbs
Basic technique
- Ladder technique. From liberties and connection.
- Net technique. From liberties and connection.
- Nets are better than ladders. From efficiency.
- Extend from atari, if increasing liberties is possible. From liberties and connection.
- Hane against the contact play. From liberties and surrounding.
- Block the push through. From liberties and connection.
- Block the diagonal. From liberties and surrounding.
- Extend from a diagonal contact play. From liberties and surrounding.
- Connect against the peep. From connection.
- Hane at the head. From liberties and surrounding.
- Stretch from a bump. From liberties and surrounding.
- Extend from a crosscut in the absence of neighbouring stones. From liberties, connection and surrounding.
Basic tactics
- The theory of life and death forms an independent chapter which has been adequately described. The positions called seki can be explained through the principle of liberties, much like the concept of life by two eyes.
- The theory of liberties and capturing races has been documented by Richard Hunter.
Basic strategy
- Play away from thickness. Perhaps the most important strategic idea is not to use safe & strong stones to build territory in the vicinity. It is not efficient: one should invest in open area, because the invested stone near a strong group adds little value to the strength of that group. Strong stones should be used instead to invade the opponent's sphere of influence or to attack his weak stones. To play on an empty point that belongs to one's own influence, is also considered inefficient, but can add to the strength of the influencing group.
- The enemy's key point is yours (from equivalence and flexibility)
Heuristics related to the size of the board
Some of the heuristics are determined by the size of the board. In 19x19? Go, the balance between territory and influence lies on the third and fourth line. This is because a territory built along the whole third line, exceeds a territory built inside a central fourth line, but only by 15 points.
- The third line is the line of territory
- The fourth line is the line of influence
- Gladly accept fourth line territory
- Third line to stabilize, fourth line to build
More concepts and proverbs
Important stones
Not all stones on the board are important. There are basically two kinds of important stones:
- Cutting stones: stones that cut the opponent apart into weak groups.
- Stones in a large, uncontrolled area
- Capture the cutting stones
- Cut the side you don't want
Heavy and light
Stones that are important but weak are called heavy. Stones that are weak but not important are light. Light stones can be sacrificed? for a higher purpose. Heavy stones must be saved or the game will become difficult.
Urgent and big
Urgent moves are moves that make important stones strong. Big moves are moves that create new important stones.
ADAPTED UNTIL HERE - MORE TO FOLLOW
S17 Don't build influence in the direction of stable positions. From development, efficiency and thickness.
S18 Build box shaped territories (efficiency, territory)
S19 First corners then sides then center (efficiency)
S20 Approach from the wider side (development)
S21 Block on the wider side (development)
More concepts
Aji is the future potential of a position (stones) which is (are) temporarily lost. Bad aji gives more options (flexibility) to the player who locally has lost (stones).
Stones are light when they have a lot of flexibility. Heavy is the opposite. Mostly, if light stones will be captured, it will be small.
Shape is a characteristic of a number of stones. Determining principles are efficiency and flexibility. Objectives are eyespace and connection, leading to thickness and territory.
The study of the different shapes can be done independently. A static treatise on shape is a first attempt. It relies amongst others on the liberties stones ratio for efficiency. Subsets of good shape are good shape. Extensions of bad shape are bad shape.
Forcing moves are moves that force the opponent to settle the shape and reduce flexibility so that the forcing player can next play according to his opponent's choice.
More technical proverbs
T13 Capture the ladder as soon as possible. (reduce flexibility for the opponent)
T14 Don't try to cut the one-point jump unless it gives a clear advantage (liberties, flexibility)
T15 Don't cut knight's moves unless it gives a clear advantage (liberties, flexibility)
T16 Don't peep where you can cut (connection, flexibility)
Finally, I need a stronger explanation than the ones I have available for the idea that playing close to (strong) stones strengthens both sides. This is the basis for the theory of leaning attacks and sabaki techniques. It also explains why to attack from a distance. Proverbs yet to be included:
Reducing techniques
- Knight's move against capping play from the third line (3rd line and 4th line significance)
- One space extension against capping play from the fourth line (3rd line and 4th line significance)
- Reduce a framework by playing on the sector line (from attacking techiques and connection principle)
Attack and defense techniques
- Attack from a distance (that's a tough one, because it mixes technique with strategy)
- Make a base first, then jump to the center (eyespace over connection ??)
- Knight's move to attack (development)
- One space jump to defend (no clue, lightness and shape maybe)
- Two space jump to sacrifice (lightness)
- Boshi to attack (development)
Cutting and connecting principles
- Avoid connecting along dame warning (efficiency)
Your comments are highly appreciated.
Deshi comments
[1] Charles I'm not aware of any principle exactly covering this point. But see make both ends strong.
RobertJasiek: TDerz, as below, "principle" need not have the same meaning as "tautology under 2-value logic" but can also be used in pretty informal ways. Nevertheless, strategic go theory can be derived from go rules by means of tautologies. This has become especially possible since the Japanese 2003 Rules. In fact, I have already made first attempts in that direction. E.g., it is pretty straightforward to define "life-and-death-group". In the Logical Rules of Go Robert Pauli has defined "2-eye-group". Under the Japanese 2003 Rules this is a concept already on a strategic level. One of the next things to do will be an attempt to prove when self-atari of a 2-eye-group makes sense. All such is low level strategy. It will take much more effort to reach high level strategy. OTOH, for other go rules, late yose or specialized ko play have been derived formally. That is also low level because it is close to the end of alternation, where the strategic complexity is "low". The same can be said so far about research under the Japanese 2003 Rules. Dieter is more optimistic but should just note that his principles are not tautologies.
TDerz Dieter, generally I would say, that I cannot derive several of the statements from the rules (R) and principles (P).
We both will agree believing that they are true, but this derives more from 10.000 of played games and is usually heuristics.
The term derives from means for me more has to do with, thus not indicating any wrong causality. The best example is for me The principle of two eyes: This group without two eyes is (also) alive.
With our background we know that we speak about seki, moonshine life etc. being this group.
For principles however, it can hardly be acceptable that both the affirmative and its opposite are true [2], there must be more, differentiating stuff to it. That is why they cannot be principles.
BTW, I am not a matematician, hence quite ignorant about the subject. I just feel that if these principles were really derivable from the rules, Go was programmable as TicTacToe?, Mühle or Reversi. This is not the case.
I know that the principles are inter-dependent, hence they are not axioms.
I agree with the principle of efficiency - because it is very vague! Dieter, I especially like the part about flexibility with regard to keeping options open. This part is very well written.
The problems start already with the The principle of liberties. I cannot increase all my stones liberties, I must decide and sacrifice some stones etc. for i-iv. (Interdependence of principles)
Other points: The principle of two eyes can be derived from R1+ R3 (eyes not explained & including two false eyes?). It is easy to agree on that.
The principle of connection only necessarily relates to P1 and P3, P2 is not needed.
Points i to iv I also emphasize often to Go learners, then coming to shape for achieving i-iv.
The principle of development
I think the references are actually different: Increasing liberties (P3), establishing a connection (P4), surround empty points (R2) and creating space to make eyes (P4) are all some kind of development.
Again, well written.
Concepts (C) and proverbs
- Hane against the contact play. From P2 and P3. (not P4)
Strategy (S)
S0 The enemy's key point is yours (from R3) Why?
From the principle of flexibility (P6) (P2?) ... The key concept in strategy is the balance of territory and thickness, which is guided by the principles of efficiency (P1) and development (P5). P6?
C2 Thickness: a group of stones which is firmly connected (P4) P5? and which has sufficient eyespace (P3). P4?
C3 Territory: an area which is surrounded (P4 P5?: no cut possible) by a thick group, and in which the opponent has almost no chance to create a lving group (P3). P4?
The meaning of stones or urgency (U)
U3 Stones that are close to the opponent's thickness, have little room for development (P5). P6? U4 Stones that are close to your own thickness also have little room for development (P5). P6?
The meaning of areas or size (bigness) (B)
B1 Open areas, with a lot of room for development, are big.(P5) P6?
B3 Areas close to opponent thickness, are uninteresting (B2 and P6?, not AP1)
What is the A in AP1?
S12 Build box shaped territories (P1, C2)
I do not understand this term 'box'.
C5 Stones are light when they have a lot of flexibility (P6) P2 ?.
C6 Shape is a characteristic of a number of stones. Determining principles are efficiency (P1) and P2 (P6). Objectives are P4 (P3) and P5 ... C7 Forcing moves are moves that force the opponent to settle the shape and reduce flexibility (__P2 __) so that the forcing player can next play according to his opponent's choice.
More technical proverbs
T13 Capture the ladder as soon as possible. (P6) rather C4
T14 Don't try to cut the one-point jump unless it gives a clear advantage (P2, P6) and P3+P5
T16 Don't peep where you can cut (P4, P6) and P5
[2] TDerz I think this comes from Aristotelian logic with its the excluded third. Either Go needs some kind of special fuzzy logic (I think) weighing all kind of interrelated dogma(ta?) & evaluations against each other, or you could apply logic only retroactivley (playing it out and compare afterwards).
I believe, that even if we could play ALL possible Go games, match them and look for the widest winning path, we had big trouble to explain why a particular move is good or (relatively) bad. Often it was supposed that perfect play would be around 13 dan. Discussing moves of your own strength (= your own moves) is always difficult and the abstractness, therewith the discussion of the game becomes more difficult with higher level.
Dieter: Thanks for your comments Tderz. Just one reply for the moment:
The statements Every group with two eyes is alive and There is a group without two eyes that is alive are not each others' opposite. Groups with two eyes are a subset of alive groups, hence can be complemented with other kinds of groups (like seki). The converse statement would be there is a group with two eyes that is not alive. Now if that would be true, I can't call two eyes a principle but must indeed fall back to a heuristic. Your statement is the opposite of Only groups with two eyes are alive, which is not what my principle says. It's simple logic, really, there is nothing fuzzy going on here. (Maybe in Go, but not in this particular principle).
The fact that every group with two eyes is alive (cannot be taken from the board) does follow logically from the rules:
- the rule of suicide or the logical rules forbid existence of a played stone without liberties after the possible captured stones are removed - a player can only play one stone at the time (alternating play) - give a strict definition to an eye then make recurrent extensions of the definition
Since one cannot play inside one eye AND another at the same time, a group with two eyes is alive.
I a not saying I gave an inclusive definition of eye, neither is the derivation of the principle formulated so that no holes can be found. I am saying such is possible and that from the rules of play/capture/suicide and alternating play, the principle can be logically derived.
TDerz Dieter, thank you for pointing out the real opposite of the statement, using a NOT instead of the wrong, misleading without.
WRT your puzzling reference numbers, where they meant to be as I wrote it, or did you mean something, which I interpreted wrongly?
The principles expressed by the terms flexibility and development for example (seem to) overlap. Moreover, the explanation for
the higher numbered development comprises lower numbered principles.
Is this intentionally? My honest question (no sandbagging, I'm a mathematic/logic layman) is: shouldn't principles be independently defined?
Otherwise it would make sense to formulate the ultimate principles "Win this Go game when starting with black" or "Make a draw when given the appropriate komi", being built up by the Rules R and all lower ranked principles P. How much teaching value would it carry? Most probably as much as the all-comprising Do not make mistakes.
Perhaps my argumentation here, to exaggerate the principles and then negate them, looks like eristic dialectic (quarrel), but I am of the opinion that principles should be rather something more antagonistic (while therewith interdependent) e.g. thickness vs. light etc. This would make clear that in Go one has to chose all the time: for example is it usually not possible to play a light, thick, fast developing, all-connecting single move. The better prioritizing derives from imagined future game positions and their evaluations.
Following this argumentation, I conclude (at least for myself) that Go is played by a top-down approach (look ahead, evaluate and chose moves which most likely lead to the most positive, attractive game position) and not by a bottom-up approach (be efficient, make liberties & eyes, connect and develop flexible). I admit that very fast Haya-Go might be played this way, although it is difficult to say what we think (and this will vary very much!).
In so many presentations of good Go players it is emphasized that Go is a game striving for harmony - a good balance of several(!) mutually excluding concepts. This is not an appeal to authority, rather I think these players express with these words harmony and balance all that what you start to analyze in detail on this page.
In fact, in view of above, the Principles seem Concepts to me (except P4 Two eyes).
I admit that I also believe that it should be theoretically possible to set up a logical system which completely explains Go and leads to the winning game (or a draw of two 13-dans). This logic system - if presented to us here and now - could be so complicated that we would not understand this and(?)/OR it would be another theory which does not help to improve our game [3]. Diter tries hard to fit something in between those two extremes.
Therewith my argumentation is a little bit inconsistent here (Go is not solvable by logic vs. there is a helping logical theory based on rules and axioms, not heuristics), this might be due to the complexity of the subject and/or my laymanship in both Go and logic.
[3] (off topic) Now I am very ignorant and call Combinatorial Game Theory such a theory of the second type (for me). Are there players who could benefit from it? Is it more than a formal way of describing what is going on in Yose?
Dear Tderz, thank you for devoting so much time to the project. It is true that the numbering and the references are confusing and sometimes plain wrong (corrected as of feb 2004 - most of the cross-referencing now undone). Please continue to comment on it. I highly appreciate the effort.
[10] Then, if the played stone has no liberties it is taken off the board or considered an illegal move, depending on whether you allow suicide''.
(the above statement has been modified due to discussion below)
ilan: This last statement doesn't make sense to me. Dieter: It allows for suicide and takes care of beginners' confusion about liberties: "Can I play there? I have no liberties."
ilan: Well, I am still confused. I don't see how you can play a legal move in which the played stone has no liberties even after captured stones are removed. Dieter: It's an algorithm really. First remove opponent stones without liberties (not saying there are any). Then your own (not saying there are any). You can play inside someones eye, but if he has other liberties then your stone is removed.
ilan: OK, thanks, but I still believe that your formulation is confusing. In particular, there is an implicit distinction between "taken off the board" and "removed" which should be made explicit, that is, in the second case, the move is cancelled (this actually follows from forbidden repetition, which appears next), whereas in the first case, the move is completed. In other words, the removed stones do not go to the same places afterwards.
Benjamin: No, he wants to allow suicide (at least with multiple stones), and I think the formulation is fine. Anyways, nice text! A few months ago I wrote sth. down that is very similar to your text, and I'm quite convinced that it's the right aproach.
Dieter: Synchronicity: I was pondering the virtues of stone counting when your text fully convinced me of being on the right track. A few teaching experiments later I am all the more enthusiatic. And Ilan, you're right: allowing suicide but forbidding repeated board positions partly contradict each other. I will correct this. But anyway I did not want to get into a rules argument but rather build the fundamental principles of Go upon the rules. I still feel that the fundamentals of Go are not adequately covered in English literature. Kageyama starts with ladders, but actually ladders and nets are already techniques above ground level. I do not really think that beginners need a thorough approach, built upon rules, but I do think it leads to more understanding and less acceptance. Frustration with Kageyama's lack of giving fundamentals (contrary to his urging to know them) lead me to describe Basic Instinct. Frustration with that article (and another frustrating attempt at a static treatise about shape? led me to make this page. Benjamin?, although I think you have devoted yourself to UGS, your proofreadings will be most welcome.