Life In 19x19

I guess my point can be resumed by : how do you know they are erroneous path if you don't explore them?

Tryss wrote:I guess my point can be resumed by : how do you know they are erroneous path if you don't explore them?

With iterative deepening, you know when you find the solution at a certain depth.

Edit: I.e., you do explore them, but only to the depth of the shallowest solution. And, like in this problem, you may be able to disprove a play at a shallower depth. All first moves but one were disproved at depth 4, although the solution has depth 5.

That's for problems, anyway. For actual games, where you may never know the right path, then the general arguments for iterative deepening apply. Not that humans should use it in actual play, OC.

Iterative broadening

Aha! It was Knotwilg's topic, Complexity as width x depth ( https://www.lifein19x19.com/viewtopic.php?f=15&t=17682 ) that got me thinking, if you can iteratively increase the depth, why not iteratively increase the width?

OC, there is literature on that, first by Matt Ginsberg three decades ago, with his paper, Iterative broadening ( https://www.aaai.org/Papers/AAAI/1990/AAAI90-033.pdf ), and also by Tristan Cazenave, with Iterative widening ( http://www.lamsade.dauphine.fr/~cazenav ... s/iw00.pdf ), which he developed further in Generalized widening ( http://www.lamsade.dauphine.fr/~cazenav ... openld.pdf ). Now, both are concerned with computerized search — Ginsberg says that iterative broadening is effective with more than 40,000 nodes —, but maybe either approach can be effectively adapted to human use.

Ginsberg's basic idea is to start with a narrow search, like width 2, to variable depth, and if you don't find a solution or good answer, increase the width. Cazenave's basic idea is to start with a minimal set of moves and do an iterative depth first search with them, widening the set if the search is unsuccessful. He has successfully applied it to open life and death problems in go with as many as 20 possible moves.

I haven't really studied Cazenave's papers, as his ideas are rather more elaborate than I have indicated. But I think I get Ginsberg's idea well enough to apply it to our little problem.

Ginsberg makes the following analogy. You are trying to buy a birthday present for a friend, a woman who likes horses. So you decide to buy her a book about Tennessee walking horses. After going to a few book stores and even a tack shop or two, without finding such a book. At this point you may decide to stop looking for that book and try something else.

His point is this. Suppose that the solution set is evenly distributed over the leaves of the search tree. Then depth first search (with backtracking) is as good as anything. But if the set is unevenly distributed, then not finding a solution after looking in a few leaves of a certain branch of the tree, even though it has not been exhaustively searched, is evidence that it is in a different branch. So you search narrowly, and if fail to find a solution, you broaden the search. Iterative broadening.

Ginsberg makes the point that if there is only one solution, which has an equal chance of being on any leaf, iterative broadening is no help. However, if you have heuristics that increases the probability that the solution lies in a certain branch, then failing to find the solution in that branch reduces the probability, and may justify switching to another branch.

Next I shall apply that idea to our problem.

Can someone please remind me of the very old tsumego heuristic where you assume one player can have two moves on a row. I've never actually tried using it, but from what I recall (very vaguely) it should work as a tool in iterative broadening, no?

John Fairbairn wrote:Can someone please remind me of the very old tsumego heuristic where you assume one player can have two moves on a row. I've never actually tried using it, but from what I recall (very vaguely) it should work as a tool in iterative broadening, no?

Null move heuristic?

Null move heuristic?

I haven't heard that term in the context I'm talking about (tsumego), but in any case it's not the term I'm after, it's the actual application. Is it for killing, or living, or both, is it a first-move or any-move technique? I've never actually used so I have no in-built memory of it.

Edit: Found it here! viewtopic.php?f=15&t=16066

Thanks for the reminder, John.

Progressive narrowing

I have taken another look at the topic John referenced. viewtopic.php?f=15&t=16066

It now seems to me that this heuristic, when it works, is a way of narrowing the search, not broadening it. In a problem or position of any appreciable difficulty, there will normally be more than three possibly relevant moves.

And then I thought about iterative deepening at depth 1. Really? Who does that? Well, who does that consciously?

Harkening back to Kotov's method in Think Like a Grandmaster, he started with identifying candidate moves. How did he do that? Among all of the possible moves on the chess board, some of them stood out, based upon knowledge, experience, and intuition. He had probably also learned some conscious heuristics, like look at checks or look at sacrifices.

It now strikes me that looking over the whole board and letting moves stand out to us is a way of unconsciously doing a depth 1 search that would take too much time, effort, and working memory to do consciously. And sometimes you see deeper than 1 move.

Then you follow up with a narrower conscious search.

A look at the heuristic

The heuristic, copied from that topic.

zermelo wrote:The 3-move tsumego rule: Assume that the problem is black to kill. Now let A, B, C be such intersections that if white would start at A, the only way for black to kill with 2 moves would be playing both B and C. Now the solution to the original problem has to start with black play at A, B, or C.

Click Here To Show Diagram Code

[go]$$Bc Case 1
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | B O O . X . .
$$ | B . W O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

If , both of are necessary to kill. So must be one of these.

Click Here To Show Diagram Code

[go]$$Bc Case 2
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | B O O B X . .
$$ | . W . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Ditto. The only common point is A-17, so must play there.

Click Here To Show Diagram Code

[go]$$Bc Case 3?
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O B X . .
$$ | W . B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This is not actually a case, because . . .

Click Here To Show Diagram Code

[go]$$Bc Case 3x
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | B O O . X . .
$$ | W . B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This also kills, because of White's shortage of liberties.

Click Here To Show Diagram Code

[go]$$Bc Case 3??
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | B O O W X . .
$$ | . B . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This is also not a case.

Click Here To Show Diagram Code

[go]$$Bc Case 3xx
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O W X . .
$$ | B B . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This also kills.

Finally.

Click Here To Show Diagram Code

[go]$$Bc White life
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | W O O . X . .
$$ | . . . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

If , White lives, even if Black plays two moves in a row. must be at A-17.

All of this information is available after depth first search at depth 3, except the necessity of both Black stones, given . We can check that, however, with another White play, i.e., with a depth 4 sequence. To do so we do not have to do a full depth 4 search. (If neither is necessary, then White will die at depth 3, given and two different Black plays.)

Click Here To Show Diagram Code

[go]$$Bc White dies at depth 3
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O 3 X . .
$$ | . 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Assuming that we recognize that White is dead without further search, we can check whether is necessary, given , by the following sequence.

Click Here To Show Diagram Code

[go]$$Bc White lives at depth 4
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O 4 X . .
$$ | 3 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

And we can check whether is necessary, given with the following sequences.

Click Here To Show Diagram Code

[go]$$Bc White lives at depth 4
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 4 O O 1 X . .
$$ | 3 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives at depth 4
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 4 O O 1 X . .
$$ | . 2 3 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Edit: I realized that we have to check both plays for .

Neither of these depth 4 sequences fits neatly into the category of iterative deepening or iterative broadening, although the verification sequence is one step deeper and one step wider.

Click Here To Show Diagram Code

[go]$$Bc White dies at depth 3, II
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O . X . .
$$ | 3 . 2 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Now to check whether is necessary, given .

Click Here To Show Diagram Code

[go]$$Bc White lives at depth 4, II
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 4 O O 3 X . .
$$ | 1 . 2 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives at depth 4, II
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 4 O O . X . .
$$ | 1 3 2 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Edit: Added later.

is necessary, given .

The only point common to both triplets is A-17, which must then be Black's first move. We discovered that with only five (Edit: I can't count. ) sequences at depth 4. When this happens it is a more efficient way to discover Black's first move than iterative deepening, which requires several depth 4 sequences.

Edit: Note that we cannot check necessity simply by changing the color of the Black stones. For instance:

Click Here To Show Diagram Code

[go]$$Bc White dies
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O B X . .
$$ | W . B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O W X . .
$$ | W . B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O B X . .
$$ | W . W O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Generalizing the heuristic

We can generalize the heuristic to any goal. For instance, suppose that White to play can live. Then:

Let A, B, C be such intersections that if Black would start at A, the only way for White to live with 2 moves would be playing both B and C. Now White to live has to start with a play at A, B, or C.

Now let's apply that to the position after at A-17 in our little problem.

Click Here To Show Diagram Code

[go]$$Wc White to play and live
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O . X . .
$$ | . . . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Wc White lives, I
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O W X . .
$$ | B W . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Only way to make a second eye, given .

Click Here To Show Diagram Code

[go]$$Wc White lives, II
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O . X . .
$$ | W B W O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Only way to make a second eye, given .

Click Here To Show Diagram Code

[go]$$Wc Proof of necessity
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O W X . .
$$ | W B 4 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This way kills.

Click Here To Show Diagram Code

[go]$$Wc White lives, III
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O W X . .
$$ | . W B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Only way to make a second eye, given .

Click Here To Show Diagram Code

[go]$$Wc Proof of necessity
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O W X . .
$$ | W 4 B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This way kills.

Click Here To Show Diagram Code

[go]$$Wc White lives, IV
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O B X . .
$$ | W . W O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Only way to make a second eye, given .

By inspection, no play belongs to all four triplets. Therefore there is no play by which White lives. (Also by inspection, there is no ko.)

White is dead. QED.

What?! What do you mean, QED? We already know (assume) that Black to play can prevent White from living without ko. The question is how does Black play to do that?

OK. Calm down. The information is there. For instance:

Click Here To Show Diagram Code

[go]$$Bc Black to play and kill
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O . X . .
$$ | 2 . . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Suppose that White plays . What is the triplet to live that does not contain ?

Click Here To Show Diagram Code

[go]$$Wc White lives, III
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O W X . .
$$ | . W B O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This is it. The Black stone tells Black where to play.

Click Here To Show Diagram Code

[go]$$Bc Black to play and kill
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O . X . .
$$ | 2 . 3 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Bingo!

Likewise:

Click Here To Show Diagram Code

[go]$$Bc Black to play and kill
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O 2 X . .
$$ | . . . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Where should Black play?

Click Here To Show Diagram Code

[go]$$Wc White lives, II
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | X O O . X . .
$$ | W B W O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

This triplet does not contain . Play at .

Click Here To Show Diagram Code

[go]$$Bc Black to play and kill
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O 2 X . .
$$ | . 3 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Bingo!

The Black replies to the other two White plays are obvious, but the same method works.

Is this superior to iterative deepening?

Well, maybe. It performs less search, because the two necessary stones do not dictate move order. And, OC, the second move in each sequence remains the same. However, the comparison of triplets is not something that the search algorithm does. This 'n' that.

More thoughts on the heuristic

Here is the problem, again.

Click Here To Show Diagram Code

[go]$$Bc Black to play
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O . X . .
$$ | . . . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Now suppose that White announces a strategy to play at B-16 (if possible), whatever Black may play.

Click Here To Show Diagram Code

[go]$$Bc White strategy
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O . X . .
$$ | . 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

OC, this strategy will fail, because this is a problem.

Click Here To Show Diagram Code

[go]$$Bc Black kills
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 1 O O 3 X . .
$$ | . 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Black thwarts White's strategy with and . And, we believe, and , in either order, is the only way to kill White, given . That means that succeeds against every other choice for .

Click Here To Show Diagram Code

[go]$$Bc White lives.
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 3 O O 4 X . .
$$ | 1 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 4 O O 3 X . .
$$ | 1 2 . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 3 O O 4 X . .
$$ | . 2 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 4 O O 3 X . .
$$ | . 2 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | . O O . X . .
$$ | 3 2 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

We know from a previous variation that because of the capture, can pass and still live.

What this method does, then, is to branch the game tree, not on , but on . Also, the proof that at A-16 or C-16 fails, which is the point of the heuristic, is the demonstration that White's strategy succeeds against those plays.

This simplifies the question from one of finding one White play and then two Black plays that are necessary for Black to succeed to finding a White strategy for that effectively prunes the game tree by eliminating options for .

Consider, then, this strategy for .

Click Here To Show Diagram Code

[go]$$Bc White strategy
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O . X . .
$$ | . . . O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

How effective is this strategy?

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O 1 X . .
$$ | . 4 3 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O 1 X . .
$$ | . 3 4 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O 1 X . .
$$ | 3 . 4 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

at D-17 fails.

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[go]$$Bc White dies
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O 4 X . .
$$ | 5 3 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

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[go]$$Bc Variation for :w4:
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O . X . .
$$ | 4 3 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

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[go]$$Bc Variation continued, White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | O O O 5 X . .
$$ | O . 6 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

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[go]$$Bc Variation for :b5:, White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | O O O 6 X . .
$$ | O . 5 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

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[go]$$Bc White dies
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O 4 X . .
$$ | 3 5 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

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[go]$$Bc Variation for :w4:, White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O . X . .
$$ | 3 4 1 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

at C-16 fails.

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[go]$$Bc White dies
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O 4 X . .
$$ | 3 1 5 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

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[go]$$Bc Variation for :w4:, White lives
$$ --------------
$$ | . . . . . . .
$$ | X X X X . . .
$$ | 2 O O . X . .
$$ | 3 1 4 O X . .
$$ | O O O X X . .
$$ | . O X . . . .
$$ | O O X . . . .
$$ | . X X . . . .
$$ | . . . . . . .
$$ | . . . , . . .[/go]

at B-16 fails. And, by transposition, at A-16 fails.

Note that Black plays and in order, to avoid having to read transpositions.

This demonstrates that White's strategy of playing at A-17 succeeds against all other plays for . must be at A-17.

OC, in real life a solver would not read out all of these variations, but would know or see that White is alive. I just posted them for completeness.

This is a different way of exploring the game tree than just reading the moves in order. If you can find a good enough strategy for White you can prune moves for Black quite effectively.

Here is an easy tsumego problem from Michael Redmond's channel.

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[go]$$Bc Black to play
$$ ----------------
$$ | . . . . O . . .
$$ | O . . . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

OK. What is a good strategy for ?

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[go]$$Bc White strategy
$$ ----------------
$$ | . 2 . . O . . .
$$ | O . . . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

Does that work?

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[go]$$Bc White lives
$$ ----------------
$$ | C 2 . . O 3 . .
$$ | O C 4 1 O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

Not the best yose, but lives.

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[go]$$Bc Variation for :b3:, White lives
$$ ----------------
$$ | . 2 4 6 O 5 . .
$$ | O M 3 1 O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

prevents the one point eye at , but lives this way, too. (By transposition, succeeds against at 3.)

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[go]$$Bc Black's first play
$$ ----------------
$$ | . 1 . . O . . .
$$ | O . . . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

What is a good strategy for ?

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[go]$$Bc Black strategy
$$ ----------------
$$ | . X 3 . O . . .
$$ | O . . . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

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[go]$$Bc White dies
$$ ----------------
$$ | . X 3 4 O 5 . .
$$ | O . 2 . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

White's only chance to make a second eye fails after . Any other pair of White stones leaves White with one eye and a dead shape. The same is true if simply connects at D-18.

BTW, using two White moves to escape is not kosher.

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[go]$$Bc First two plays
$$ ----------------
$$ | . 1 2 . O . . .
$$ | O . . . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

What's a good strategy for ?

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[go]$$Bc White strategy
$$ ----------------
$$ | . 1 2 . O . . .
$$ | O . 4 . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

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[go]$$Bc White lives
$$ ----------------
$$ | . 1 2 6 O 5 . .
$$ | O . 4 3 O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$Bc First three plays
$$ ----------------
$$ | . 1 2 . O . . .
$$ | O . 3 . O X . .
$$ | . O O O X X . .
$$ | O X X X . . . .
$$ | . . . . . X . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

The rest is easy.

The method of finding a good second move strategy works well for this problem.

Let's try out this method on Xuanxuan Qijing problem 118.

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[go]$$Bc Black to play
$$ --------------
$$ | . . . . X X .
$$ | X X O O O X .
$$ | O X . . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

What's a good strategy for ?

How about this?

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[go]$$Bc Make an eye.
$$ --------------
$$ | . . . . X X .
$$ | X X O O O X .
$$ | O X 2 . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

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[go]$$Bc Eye vs. eye
$$ --------------
$$ | . 1 4 3 X X .
$$ | X X O O O X .
$$ | O X 2 . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

It does no good for Black to make an eye. White wins the semeai.

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[go]$$Bc Crawl twice
$$ --------------
$$ | . . 3 1 X X .
$$ | X X O O O X .
$$ | O X 2 . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

This looks like the best Black can do. The rest I know/see.

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[go]$$Bc Throw-in
$$ --------------
$$ | 5 4 3 1 X X .
$$ | X X O O O X .
$$ | O X 2 . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

at 4

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[go]$$Bc Ko
$$ --------------
$$ | 9 O X X X X .
$$ | 8 7 O O O X .
$$ | O . 2 . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

That seems to be the best that Black can do against White's strategy. That means that should be at 1, 2, or 3 in this variation. Since this is a problem, White should not be able to live outright and avoid the ko.

For instance:

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[go]$$Bc White dies
$$ --------------
$$ | . 4 2 1 X X .
$$ | X X O O O X .
$$ | O X 3 . O X .
$$ | O O O O 5 X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

If blocks to avoid the ko, White dies. takes away White's potential eye, and then takes away Black's potential eye. This variation also shows that is not a good reply if is at 3.

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[go]$$Bc Trap
$$ --------------
$$ | . 2 3 4 X X .
$$ | X X O O O X .
$$ | O X 1 . O X .
$$ | O O O O 5 X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

If takes away White's potential eye, takes away Black's potential eye. lays a trap. If captures, White loses a liberty and wins the semeai.

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[go]$$Bc White lives
$$ --------------
$$ | 6 2 3 5 X X .
$$ | X X O O O X .
$$ | O X 1 4 O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

plays atari. The rest I know/see.

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[go]$$Bc White lives
$$ --------------
$$ | W W X X X X .
$$ | . 7 O O O X .
$$ | O 8 . O O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

makes two eyes, despite the fact that the stones are in atari. This is a result of the special properties of the corner. Something to know/see.

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[go]$$Bc One eye
$$ --------------
$$ | O W X X X X .
$$ | 9 X O O O X .
$$ | O O . O O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

If captures two stones, throws in at . leaving the next diagram.

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[go]$$Bc One eye in snapback
$$ --------------
$$ | . O X X X X .
$$ | X X O O O X .
$$ | O O . O O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

Solution:

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[go]$$Bc
$$ --------------
$$ | . 2 1 3 X X .
$$ | X X O O O X .
$$ | O X 4 . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

OC, could also be at 3 and could also be at 4. lays a trap for Black to play at 4. The cook did not bother problem composers of yore.

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[go]$$Bc Black to play
$$ --------------
$$ | . . a b X X .
$$ | X X O O O X .
$$ | O X c . O X .
$$ | O O O O . X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

The method works for this position, telling us that should be at a, b, or c. But that's fairly obvious, anyway.

The board as a storage device

Tryfon Gavriel, who, under the name, Kingscrusher, runs chessworld.net and who, over many years, has posted and commented on a prodigious number of chess games on youtube, posted this game, https://www.youtube.com/watch?v=nlYbDeophsU , between Leela Chess Zero (LC0) and Stockfish earlier this year, in which he talks about the chess board as a storage device. Kingscrusher quotes Steinitz:

Wilhelm Steinitz wrote:The task of the positional player is systematically to accumulate slight advantages and try to convert temporary advantages into permanent ones

LC0 is a positional player, as, I gather, are all or nearly all neural network chess players. In his commentary on the game, Kingscrusher points out longlived advantages that Leela accumulates and holds on to, finally winning after around 180 moves. (BTW, it was not clear to me from his commentary that those early advantages were enough to win without later mistakes by Stockfish.)

What about the go board as a storage device which preserves slight permanent advantages? Well, ko stones aside, despite sacrifices and furikawari, most stones played on the go board are alive at the end of the game and, if they are well placed, represent persistent advantages. OC, these advantages are not always secure territory, especially in the opening, although they may be represented as expected territory. Nothing special needs to be done to create persistent advantages on the go board, and a slight advantage is all that it takes to win the game.

When I was learning go, it had the reputation of being a more strategical game than chess, and I think that the fact that go stones do not move has a lot to do with that. Since go stones tend to persist on the board, so do any advantages that they have gained. Of course, among top players there are those who have a more positional style, such as Takagawa, and those who have a more tactical style, such as Sakata. For Takagawa, in fact, the ideal was to win without fighting. He may well have gone too far in that direction, but it seems to me that go is basically a positional game.

Bill Spight wrote:The board as a storage device

Tryfon Gavriel, who, under the name, Kingscrusher, runs chessworld.net and who, over many years, has posted and commented on a prodigious number of chess games on youtube, posted this game, https://www.youtube.com/watch?v=nlYbDeophsU , between Leela Chess Zero (LC0) and Stockfish earlier this year, in which he talks about the chess board as a storage device. Kingscrusher quotes Steinitz:

Wilhelm Steinitz wrote:The task of the positional player is systematically to accumulate slight advantages and try to convert temporary advantages into permanent ones

LC0 is a positional player, as, I gather, are all or nearly all neural network chess players. In his commentary on the game, Kingscrusher points out longlived advantages that Leela accumulates and holds on to, finally winning after around 180 moves. (BTW, it was not clear to me from his commentary that those early advantages were enough to win without later mistakes by Stockfish.)

What about the go board as a storage device which preserves slight permanent advantages? Well, ko stones aside, despite sacrifices and furikawari, most stones played on the go board are alive at the end of the game and, if they are well placed, represent persistent advantages. OC, these advantages are not always secure territory, especially in the opening, although they may be represented as expected territory. Nothing special needs to be done to create persistent advantages on the go board, and a slight advantage is all that it takes to win the game.

When I was learning go, it had the reputation of being a more strategical game than chess, and I think that the fact that go stones do not move has a lot to do with that. Since go stones tend to persist on the board, so do any advantages that they have gained. Of course, among top players there are those who have a more positional style, such as Takagawa, and those who have a more tactical style, such as Sakata. For Takagawa, in fact, the ideal was to win without fighting. He may well have gone too far in that direction, but it seems to me that go is basically a positional game.

I don't play chess much now but 20 years before I became a go player, when I was a teenager, I was an avid chess player. Now I play only with my nephew. The games usually follow a pattern of my getting a space advantage, which leads to a material advantage and then I trade off until I can convert a pawn into a queen, the material advantage results in a direct mating attack. Reading your post, it occurred to me that in chess a space advantage is positional. In go maybe overconcentration would be the analog of space in chess.