Reference: https://www.lifein19x19.com/viewtopic.php?f=15&t=18732

dhu163, you try to reinvent the wheel. I have already provided suitable definitions for thickness:


The gap between a wall and extension:
See Joseki 1 - Fundamentals, p. 233.

First determine the farthest extension that is also a connection, which might be direct or indirect. This must be verified by reading. Then consider also alternatives.

Note that 'indirect connection' is a term defined on p. 120: An indirect connection does not ensure a 100% connection but makes cutting unattractive or at best equal for the opponent.


Definition of thickness by numbers:
See Joseki 2 - Strategy, p. 90.

Thickness of a player's group of stones has these properties:
- The stones are m-connected with each other.
- The stones are n-alive.
- The territory potential is estimated by letting the player make a small number of i = 0..t successive plays using the group's presence to get the greatest territory. For i = 0, count the territory in favour of the player. For every greater i, count the territory increment in favour of the player and divide by 2^i. Form the sum of the values calculated in all i steps.


Definition of influence by numbers:
See Joseki 2 - Strategy, p. 71.

In relation to a player's group of stones, an intersection's influence has all of these properties:
- The player's stone situated or else, if possible, placed on the intersection is m-connected to the group's stones.
- The opponent's stone situated or else, if possible, placed on the intersection is n-connected to at least one live string of his.
- The player's stone situated or else, if possible, placed on the intersection is p-alive.
- The opponent's stone situated or else, if possible, placed on the intersection is q-alive.
- The intersection is the player's t-territory.


From the number-dependent definitions, I have derived the following best informal definitions, where "connection" is "direct or indirect connection":
See Positional Judgement 2 - Dynamics, p. 206:

- Thickness: the group is connected and alive.
- Great thickness: the group retains its thickness even if the defender ignores at least one attacking move.
- Ordinary thickness: the group has thickness but not great thickness.

See also there for a definition of "ordinary influence stones", which are used for my important concept of influence stone difference.

We'll see.

Thanks for your work, though I was aware of it from SL. I think it can be generalised further than your conservative theorems. Perhaps me or someone else will find a clever way to apply it by combining it with influence functions.

These are definitions rather than theorems. My books also contain quite a few principles for applying thickness etc. but they are not theorems. (From the POV of mathematics, my thickness theory is far from the quality of my endgame theory, which has many theorems.) My definition of influence is(!) an influence mapping! When you mention influence functions, it sounds like outdated light models, does it?

Deleted but reposted on the "proper" thread.

In my concept of 'numbers of influence stones', which is the difference of such black and white stones, the actual numbers are counted very quickly and very useful in practice. It occurs in principles in my books but I also frequently apply it in my games.

As to groups constituting thickness, it is important to identify ordinary versus great thickness. The numbers for representing degrees of connection or life are 0 or 1+. Trivial numbers! In practice, it is unnecessary to mention these numbers explicitly but one can also express them implicitly by "is connected", "is alive", "can play elsewhere [once]" etc. Such actual numbers are more interesting for purely theoretical study, e.g., to explain why some thickness is greater than other thickness and in which aspect. Therefore, (also professional) players can avoid such numbers while playing.

Nevertheless, many more players would benefit from calculating numbers of influence stones. It is much easier and faster than counting safe territories but similarly important.

For the purpose of potential of thickness or using it, the number representing possible new points of territory would be very useful but its calculation is hard so that, to exaggerate a little, only Takagawa Kaku did it. I would like to do it but often have too little time during a game because the results of best played fights have to be evaluated. This is the high art of positional judgement. AI emulates that well.

Citation reference: https://www.lifein19x19.com/viewtopic.p ... 35#p273235



I mean my books, in particular,

Positional Judgement 2 - Dynamics

Joseki 2 - Strategy

Joseki 3 - Dictionary

Fighting Fundamentals