Influence in go

Bill Spight · #1

This is a place for talking about influence in go. I have submitted a presentation for the go symposium this weekend. Any questions or comments that people have can go here.

I am also adding some material here about influence.

Bill

hyperpape · #2

Bill Spight wrote:

I am also adding some material here about influence.

Where is my watch-checking emoticon? ;-)

Bill Spight · #3

The presentation is Saturday at 1:00 p. m. EDT. Check back then. (I wanted to start this topic so the organizers would have the URL.)

I may post some things in advance.

Meanwhile, see this topic: viewtopic.php?f=15&t=1341

EdLee · #4

Bill Spight · #5

In the decades since Zobrist introduced an influence function in his go program, go programmers and others interested in influence have not come to a consensus about how to estimate it. I don't know, either.

However, I have come up with a first approximation, which looks promising.

This diagram shows part of the hypothesized influence of the Black stone. The four adjacent points each have influence of 1, the points two steps away have influence of ½, those three steps away have influence of ¼, those four steps away have influence of 1/8, etc. (I did not show the influence past three steps.)

This calculation must be modified to reflect the strength of the stones. The estimate for the tengen stone is almost 16 pts., but that is probably too high. It is a good approximation for a 4-4 stone, however, yielding a komi of 8, which is pretty close.

Bill Spight · #6

Here is an example of it in use.

Click Here To Show Diagram Code: [go]$$W Yasui Chitoku (W) vs. Honinbo Genjo, after move 34 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . O . . . X . . X . . . . . . . . | $$ | . . O , . . . . . , . . . . X , X . . | $$ | . . X . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . O . . | $$ | . . O , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O . . . . . . . . . . . . . . . . | $$ | . . . . X . . . . . . . . . . . . . . | $$ | . . . X . . . O . . . . . X O O . . . | $$ | . . X , . O . O X , . . X X X O O . . | $$ | . . . X O . O X X . . . X O O . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Estimated count: 0.7 for White.

Click Here To Show Diagram Code: [go]$$W Boundaries of likely areas $$ --------------------------------------- $$ | C C V V . . . A . . A C C C A C C C C | $$ | C V . . V . A . . . . A A A . A A C C | $$ | V . . O V . A X . . X . . . . . . A C | $$ | V V O V V . A . . , . . . . X , X . A | $$ | . . X . . . . A A A A A A A . . . A . | $$ | . V . V . . . . . . . . . . A A A . . | $$ | V . O . V . . . . . . . . . . . . . . | $$ | . . . . V . . . . . . . . . . V V V . | $$ | . . . . V . . . . . . . . . V . O . V | $$ | . . O , V . . . . , . . . . . V . V C | $$ | V . . V . . . . . . . . . . . V V C C | $$ | C V V . . . . . . . . . . . . V V C C | $$ | V V O . . . V V . . . . . . V . . V C | $$ | . . . A X . . V . . . A A . V . . V C | $$ | A A A X . . . O . A A . . X O O . . V | $$ | . . X , . O V O X , . . X X X O O . . | $$ | A . . X O V O X X . . . X O O . . . V | $$ | C A A . . . . . A . A A . V . . . V C | $$ | C C A . V V . A . A . . . V . . V C C | $$ ---------------------------------------[/go]

indicates a neutral point.

Here I considered a point likely area if its net influence was 1/3 or greater in favor of one player. Among the open points, there are 82 likely Black points and 89 likely White points.

This function does not handle edges or corners well. The 3 neutral points in the bottom right, bottom left, and top left corners probably belong to the surrounding group. The points in the top right corner and top side are another matter. There is room for an invasion.

Click Here To Show Diagram Code: [go]$$W After move 56 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . O . O . O X X . X X . . . . . . . . | $$ | . . O , O . O . . O . . . . X , X . . | $$ | . X X X O X O . . . . . . . . . . . . | $$ | . . . . X . . . . . . . . . X . O . . | $$ | . . O X . X . . . . . . . . . . . . . | $$ | . . O . O . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . X . O . . | $$ | . . O , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O . . . . . . . . . . . . . . . . | $$ | . . . . X . . O . . . . . . . . . . . | $$ | . . . X . . . O X . . . . X O O . . . | $$ | . . X , . O . O X , . . X X X O O . . | $$ | . . . X O . O X X . . . X O O . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Estimated net count: 4.1 for Black.

Click Here To Show Diagram Code: [go]$$W Boundaries of likely areas $$ --------------------------------------- $$ | C V . . . V A . . . . A C C A C C C C | $$ | V . . . V . . A A . . . A A . A A C C | $$ | V O . O . O X X . X X . . . . . . A C | $$ | V . O , O . O . . O . A . . X , X A A | $$ | . X X X O X O . . . . . A . . A . . . | $$ | . . . A X . . . . . . A A . X . O V V | $$ | V . O X . X . A . . . . A . A . V V C | $$ | . V O . O . . . . . . . A . A . V V C | $$ | . . V . V . . . . . . . A . X . O . V | $$ | . . O V V . . . . , . . . A A , V V C | $$ | V . . V . . . . . . . . . . . . V C C | $$ | C V V . . . V V . . . . . . . V V C C | $$ | V V O . . . V V . . . . A . V . . V C | $$ | . . . . X . . O . A A A A . . V . V C | $$ | A A A X . . . O X . . . . X O O . . V | $$ | . . X , . O V O X , . . X X X O O . . | $$ | A . A X O V O X X . . . X O O . . . V | $$ | C A A . . . . . A . A A . V . . . V C | $$ | C C A . V V . A . A . . . V . . V C C | $$ ---------------------------------------[/go]

82 likely Black points, 79 likely White points.

Click Here To Show Diagram Code: [go]$$B Chitoku (W) vs Genjo, after move 102, 1 White prisoner $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . O . X X . . . . . . . . . . | $$ | . O . O . O X X O X X X . . . . . X . | $$ | . . O , O O O O O O O . . . X X X O . | $$ | . X X X O X O . . . . . . . . . O . . | $$ | . O O X X X . . X . . . . . X . O . . | $$ | . . O X . X . . . . . . . . . . . . . | $$ | . . O . O . X . . . . . . . . . . . . | $$ | . . . . O . . X . . . . . . X . O . . | $$ | . . O , X . O X . , . . . . . , . . . | $$ | . . . . . X X O . . . . . . . . O . . | $$ | . O . O . . . X . . . . . . O . . . . | $$ | . X O . . . . . . . . . . . . . . . . | $$ | . X O . X . . O . . . . . . . . . . . | $$ | . X . X . O . O X . . . . X O O . . . | $$ | . . X , . O . O X , . . X X X O O . . | $$ | . . . X O . O X X . . . X O O . . . . | $$ | . . . X . O . O X . . X . X O . . . . | $$ | . . . . . . . . . . . . X . . . . . . | $$ ---------------------------------------[/go]

Net territory estimate: 5.4 (for Black).

Click Here To Show Diagram Code: [go]$$B Likely area boundaries $$ --------------------------------------- $$ | . . . . . V . A . . . . . . . . . A C | $$ | . . . . . O . X X . . . . . . . . . A | $$ | . O . O . O X X O X X X A . . . . X . | $$ | V . O , O O O O O O O . . A X X X O V | $$ | . X X X O X O . . . . . . A . . O . . | $$ | V O O X X X . . X . . A A . X . O . . | $$ | . . O X . X . . A . . . A . A . V . . | $$ | . . O . O . X A . A . . A . A . V . . | $$ | . . V . O . . X A . A A A A X . O . . | $$ | . . O , X . O X . A A . . . . V . . . | $$ | V V . . . X X O . . . . . V V . O . . | $$ | . O V O . . . X . A . . . V O . . . . | $$ | . X O . . . . . . A . . . . V . . . . | $$ | A X O . X . . O . . A A A . . V . . . | $$ | . X . X . O V O X A . . . X O O . . . | $$ | . . X , . O . O X , . . X X X O O . . | $$ | . . . X O . O X X . . . X O O . . . . | $$ | A . . X . O . O X . . X . X O . . V V | $$ | C A . A . V V . . A . . X A . V V C C | $$ ---------------------------------------[/go]

Final diagram:

Click Here To Show Diagram Code: [go]$$B After move 231, 1 Black prisoner, 4 White prisoners, Black wins by 9 $$ --------------------------------------- $$ | . . . . . . O O X . . . . . . . X . X | $$ | . . . . . O . X X O X . X . . . X X O | $$ | O O . O . O X X O X X X O . . . . X O | $$ | X O O , O O O O O O O O X X X X X O O | $$ | X X X X O X O . . . . O O X . X O O . | $$ | X O O X X X . . X X X X X . X . O . . | $$ | O . O X . X O O X O X O . . . X X O . | $$ | . O O X O X X . . O X X O . . O X O . | $$ | O . . O O X . X . . O O X X X X O . . | $$ | . O O O X . O X . O . X O . . , O . . | $$ | O . O X X X X O O . X X O O X X O . . | $$ | X O . O O X . X O . X . X O O O . . . | $$ | X X O . O X X X . O X X X O X O . . . | $$ | . X O O X O X O O O O X . X X O . . . | $$ | . X X X X O O O X O X . . X O O . . . | $$ | . . X , O O . O X X . . X X X O O . . | $$ | . . . X O . O X X . . . X O O . . . . | $$ | . . . X X O O O X . . X . X O O O . . | $$ | . . . X O O . X . . . . X X X X O . . | $$ ---------------------------------------[/go]

Click Here To Show Diagram Code: [go]$$B Points of opposite color than projected $$ --------------------------------------- $$ | . . . . . . O W X . . . . . . . X . X | $$ | . . . . . O . X X O X . X . . . X X W | $$ | O O . O . O X X O X X X O . . . . X O | $$ | B O O , O O O O O O O O X X X X X O O | $$ | X X X X O X O . . . . O O X . X O O . | $$ | B O O X X X . . X X X X X . X . O . . | $$ | O . O X . X O O X O X O . . . X B O . | $$ | . O O X O X X . . W X X O . . O B O . | $$ | O . . O O X . X . W W W X X X X O . . | $$ | . O O O X . B X . W . X O . . , O . . | $$ | O . O X X X X O O . X X O O B B O . . | $$ | X O . O O X . X O . X . X O O O . . . | $$ | X X O . O X X X . W X X X O B O . . . | $$ | . X O O X O X O O O W X . X X O . . . | $$ | . X X X X O O O X W X . . X O O . . . | $$ | . . X , O O . O X X . . X X X O O . . | $$ | . . . X O . O X X . . . X O O . . . . | $$ | . . . X X O O O X . . X . X O O O . . | $$ | . . . X O O . X . . . . X X X B O . . | $$ ---------------------------------------[/go]

I chose the ranges {-1, -1/3} for White influence, {-1/3, 1/3} for neutral points, and {1/3, 1} for Black influence as a simple tripartite division. A division into points with net Black influence and net White influence seems too precise to me.

Even though the overall net influence is 5.4 for Black, there are 84 open points in the Black range and 84 open points in the White range. (There are 92 points in the neutral range.) 19 points are in the opposite area at the end of the game, so that's an error rate for specific points of about 11%.

The count estimate of 5.4 for Black seems reasonable, given the final score of +9 and the fact that Black has sente.

Game Record:

cyclops · #7

Bill Spight wrote:

.... This diagram shows part of the hypothesized influence of the Black stone. The four adjacent points each have influence of 1, the points two steps away have influence of ½, those three steps away have influence of ¼, those four steps away have influence of 1/8, etc. (I did not show the influence past three steps.) ....

This Spight series: 4, 8, 11 ... how does it continue. If n is the number of terms then Spight Influence = 16 - 8*(n+2)/2^n . As Bill stated this grows quickly to 16 for large n.

RobertJasiek · #8

What is the purpose of still pursuing influence as a visual / light / lattice distance model? If calculation time complexity is important, then even Wolf's stone proximity model seems a slightly better alternative but also has too little relation to reality.

However, all those naive models fail to describe what influence actually is: degrees of connectivity, life and territory (see Joseki 2 Strategy). For this precise model, the calculation time complexity is high (at least, if exact values for the degrees are sought).

For the sake of having good purposes, a small calculation time complexity and a close relation to reality, there is the mobility difference concept (see in the aforementioned source; variations of the concept are straightforward).

So let me ask again: What are the purposes of Bill's model? Why would they not be better fulfilled by my models? How to overcome the missing relation between Bill's model and the basic influence aspects connection, life, territory? Could a model not inherently relying on those basic aspects have justification (other than as a pure thought experiment) in comparison to models relying on them?

Considering the mighty threat title, let me make it clear again: Influence in go is (in its precise and general form) defined in terms of either player's degrees of n-connectivity, m-life and t-territory, per intersection and due to stones as sources of influence.

Bill Spight · #9

RobertJasiek wrote:

So let me ask again: What are the purposes of Bill's model?

The main purpose is to estimate the current count. A secondary purpose is to estimate the degree to which each point currently "belongs" to either player. (Note that these are not probability estimates.)

Quote:

Why would they not be better fulfilled by my models?

No reason at all.

RobertJasiek · **#10**

Bill Spight wrote:

The main purpose is to estimate the current count.

I recommend t-territory for that purpose.

Quote:

A secondary purpose is to estimate the degree to which each point currently "belongs" to either player.

This can mean connected, alive (if a stone is placed there) or territory. I recommend my model:) Some intersections will be neutral for some degree(s). E.g., an intersection can be neutral with respect to connectivity if either player's stone played there would be 0-connected to some friendly stones.

Bill Spight · **#11**

Example 2.

Click Here To Show Diagram Code: [go]$$B Honinbo Sansa (W) - Koshio Rigen, after move 30 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . X . . | $$ | . . . . X . . . . . . . X . . X O . . | $$ | . . . , . . . . . , . . . . . , O . . | $$ | . . . . . . . . . . . . . . . O . . . | $$ | . . X . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . O . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . O . . . . . . . . . . . . . . . . | $$ | . X O . . . . . . . . . . . . . . . . | $$ | . X O . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . X X X . . . . . . . . . . . . . . . | $$ | . O O , X . . . . , . . . . O , . . . | $$ | . . . O X . X . . . O . . . . . . . . | $$ | . . X O O O X . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Estimated count: 0.2 for Black.

Click Here To Show Diagram Code: [go]$$B After move 100 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . X . | $$ | . . . . . . . . . . . . . . X . X O O | $$ | . . O . X X O . O O O X X . . X O . . | $$ | . . . , . . O . X X X O . . . , O . . | $$ | . . O O X . . . . . O . O . . O . . . | $$ | . . X X X . O . X X X X . O O . . O . | $$ | . . . . . . . O . O . . O O X . O X . | $$ | . . . . . . . . X . X X O X X . O X . | $$ | . . O . . . O . . . . O X . . . X X . | $$ | . . . , . . . X . , . O X . . , X O . | $$ | . . O . . . . . . O . O X . . . X O . | $$ | . X O . . . . . . . X X . . . X O O . | $$ | . X O . . . . . . . . . . . . . X O . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . X X X . . . . . . . . . . . . O . . | $$ | . O O , X . . . . , . . . . O , . . . | $$ | . . . O X X X . . . O . . . . . . . . | $$ | . . X O O O X . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Estimated net count: 9.3 for White

Click Here To Show Diagram Code: [go]$$B Boundaries of projections $$ --------------------------------------- $$ | C C V . A . V V . . V A A A . . . X . | $$ | C V V . . . . . V V . . . . X . X O O | $$ | V . O . X X O . O O O X X . . X O . . | $$ | . V . , . . O . X X X O . . . , O . . | $$ | V . O O X . . . . . O . O V V O . . V | $$ | . . X X X . O . X X X X . O O . V O . | $$ | . . . . . . . O . O . . O O X . O X . | $$ | . . . . . . . . X . X X O X X . O X . | $$ | V V O . . . O . . . . O X . A . X X . | $$ | . . V , . . . X . , . O X . . A X O . | $$ | . . O V V V . . . O . O X . . . X O V | $$ | A X O . V . . A . . X X . . . X O O . | $$ | A X O . . . . A . . A . . A A A X O . | $$ | . . . . A A A A . . A A A . . . . . V | $$ | . X X X . . . A . . . . . V V V O V . | $$ | . O O , X . . . A , V V V . O , . . V | $$ | V . . O X X X . A V O . V . . . . V C | $$ | . . X O O O X . A . V V C V V V V C C | $$ | . V . V V . . A . . V C C C C C C C C | $$ ---------------------------------------[/go]

Click Here To Show Diagram Code: [go]$$W Final board, Jigo $$ --------------------------------------- $$ | . O O X . . X O O O O X . . . . . X . | $$ | . O X X . . X X . O X X . X X . X O O | $$ | O . O . X X O O O O O X X O X X O . . | $$ | O O O X X O O . X X X O O O O , O . . | $$ | X X O O X . . O X . O X O . . O . O O | $$ | . X X X X . O . X X X X . O O O X O X | $$ | X X O X . X O O O O . . O O X X . X X | $$ | X O O X O X O X X X X X O X X X . X . | $$ | O . O X X . O X . . X . X . . . X X X | $$ | . . O X O O O X O O X . X . X , X O X | $$ | O O O O X O . O O O X . X X O X X O O | $$ | O X O . X O X O X . X X . O O X O O . | $$ | O X O O O X O O . . . O O O X X X O . | $$ | . X . . X X O O X X X X O O X . O . . | $$ | . X X X . . X X X O X X O . O O O . . | $$ | X O O , X . . . X O X O X O O X O . . | $$ | X X O O X X X . X O O O X X O X O O . | $$ | X O . O O O X X O X O . X . X X O . . | $$ | . O O . O X X . O . O O X X X O O . . | $$ ---------------------------------------[/go]

Given the final result, the estimation of -9.3 seems too big for White.

Game Record:

{More to come

}

topazg · **#12**

Bill Spight wrote:

Given the final result, the estimation of -9.3 seems too big for White.

I'm following this with some interest, but I do have an observation of concern on this statement (and others like it) - Without access to the discussions between professionals on the game in question, how do we know that this isn't an accurate assumption, where White's theoretical advantage was slowly removed by suboptimal play?

I think we'd need a considerable sample size before its possible to start evaluating the accuracy of the estimations - even then, there are lots of other factors - if your model overestimates certain situations and underestimate others, you could end up predicting jigo correctly, to feel that the model performed well, when it actually performed equally badly in both directions. I'm not saying the model _did_ perform badly, but it can be hard to separate the possibilities without really extensive examples putting models against games and observing the way the model changed over time in the game. It would be quite interesting to see how the evaluation changed every 6 moves from move 60 to the end for example, and comparing moyo games to territorial games where neither side made any clear blunder, so that the behaviour of your model over time in different situations can be compared.

RobertJasiek · **#13**

topazg wrote:

how do we know that this isn't an accurate assumption

By doing positional judgements...! :)

topazg · **#14**

RobertJasiek wrote:

topazg wrote:

how do we know that this isn't an accurate assumption

By doing positional judgements...!

Wasn't that rather my point?

Bill Spight · **#15**

topazg wrote:

Bill Spight wrote:

Given the final result, the estimation of -9.3 seems too big for White.

I'm following this with some interest, but I do have an observation of concern on this statement (and others like it) - Without access to the discussions between professionals on the game in question, how do we know that this isn't an accurate assumption, where White's theoretical advantage was slowly removed by suboptimal play?

I think that we can trust pro play more than we can trust the estimates.

Quote:

I think we'd need a considerable sample size before its possible to start evaluating the accuracy of the estimations - even then, there are lots of other factors - if your model overestimates certain situations and underestimate others, you could end up predicting jigo correctly, to feel that the model performed well, when it actually performed equally badly in both directions.

My purpose here is not statistical. The estimates are both crude and biased.

Quote:

I'm not saying the model _did_ perform badly,

I am.

There are known problems with it.

Influence in this approach is proportional to the strength of the stones. Dead stones exert no influence, half-dead stones exert half influence, etc. In general, we have no way of determining the strength of stones.

Influence needs to be corrected for overconcentration. The whole board calculations that I actually use do so by capping Black or White influence for each point at 1. In theory, that is not enough of a correction. However, since stones strengthen each other, fully correcting for overconcentration without correcting for strengthening would yield poor results. Besides, it is not like either correction is known.

Interaction with the edge and corner is complex. You can clearly see in both games how the corner points are considered neutral. Normally, we expect the edge to increase the influence of a stone. OTOH, if the opponent gets an effective play near the same edge point, its influence is enhanced, as well.

This influence function is also a gote function. Sente has an effect. In a sense, it collapses the influence function, but its effect is not huge. Still, to apply the function, you should correct for sente, if possible. (For instance, in the second game, Black has sente against the top left corner. I know that, and you know that, but does the average 5 kyu know that? Does a computer program know that?)

Quote:

but it can be hard to separate the possibilities without really extensive examples putting models against games and observing the way the model changed over time in the game. It would be quite interesting to see how the evaluation changed every 6 moves from move 60 to the end for example, and comparing moyo games to territorial games where neither side made any clear blunder, so that the behaviour of your model over time in different situations can be compared.

The idea of making evaluations at different points in the game is a good one. I chose around move 100 in these games because of Jowa's advice to evaluate the position at moves 30, 50, and 100. I think he stopped there because later in the game it is more difficult to overcome an inferior position against an equal opponent.

I'll go back and add some estimates at earlier points in the example games.

oren · **#16**

Thanks, Bill! This has been interesting to follow.

RobertJasiek · **#17**

Bill Spight wrote:

Influence needs to be corrected for overconcentration.

This can be done by considering both influence and efficiency.

topazg · **#18**

Bill Spight wrote:

topazg wrote:

Bill Spight wrote:

Given the final result, the estimation of -9.3 seems too big for White.

I'm following this with some interest, but I do have an observation of concern on this statement (and others like it) - Without access to the discussions between professionals on the game in question, how do we know that this isn't an accurate assumption, where White's theoretical advantage was slowly removed by suboptimal play?

I think that we can trust pro play more than we can trust the estimates.

True, but slightly off the point

My point is that your estimation was not at the end of the game. Pros would be the first to admit that their play isn't perfect, and that dropping back from an even position to a 10 point loss is by no means unreasonable, even if it was over a single poor decision between your template point and the end. There are lots of games with pro commentaries, and it may be that this game was considered dead even at the point you predicted it would be, so to make the assumption that the end of the game being a 10 point difference means your mid-game estimate was off seems unjustified, that's all

david_gfe_games · **#19**

Bill,

You mentioned your paper (Influence in Go) at the Symposium in August. All I can find is the link back to this thread. Are you papers available for download?

msgreg · **#20**

david_gfe_games wrote:

Bill,

You mentioned your paper (Influence in Go) at the Symposium in August. All I can find is the link back to this thread. Are you papers available for download?

If you found links back to this thread, you may have already found the link to the video at the Go Symposium website by clicking on the video link in the menu. Bill's is session 5.

Influence in go

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