"A player's group of stones is
- n-alive if the opponent starts and the player can make at most n passes before his first play and defends all the stones as alive,
- -n-alive if the player starts and needs at least n plays before the opponent's first play to defend all the stones as alive."
Besides *-alive refers to arbitrarily many passes of the player.
(Citation from Joseki Volume 2 Strategy, 2011.)
"Possible Omission Number (PON)
Consider a group G of color C.
- (a) Group G is neutral (i.e. life and death depends on the next turn). PON of G is 0.
- (b) Group G is alive. If G becomes neutral after n opponent moves in a row but G is still alive after n - 1 consecutive opponent moves, PON of G is n.
- (c) Group G is dead. If G becomes neutral after C is allowed to make n consecutive moves but G is still dead after n - 1 consecutive moves, PON of G is -n."
(Citation from Estimating the Possible Omission Number for Groups in Go by the Number of n-th Dame, Morihiko Tajima and Noriaki Sanechika, Lecture Notes in Computer Science Volume 1558, 1999, pp 265-281, 12 Mar 1999, http://link.springer.com/chapter/10.1007/3-540-48957-6_18. According to Anonymous, a related earlier reference is: Life and Death Refined. R. Popma & L.V. Allis. Heuristic programming in artificial intelligence 3. 1992.)
- Go players say "unsettled" group instead of "neutral" group.
- *-alive is also known as pass-alive or under other names. This may explain why the PON definition does not mention it, but we can assume that its authors were aware of the case.
- -1-alive = possible omission number 0 = unsettled. The concept n-alive emphasises a strength of life as a non-negative number (or *) while the concept possible omission number emphasises the balance of the unsettled status in between life and death. Both concepts (i.e., the calibration of the value 0) have their preferred applications.
DrStraw: I don't see this as being a particularly useful concept unless it is to illustrate the inefficiency of stones. I can see, perhaps, 1-alive and 2-alive as being okay, but surely 3-alive or higher indicates stones have not be played efficiently.
Robert Jasiek: In practice, it often is sufficient to consider mostly positive n and to distinguish only the values 0, 1 and 2+ (which means '2 or greater incl. *'). Efficiency is one aspect but basic life considerations, thickness, influence and 'can ignore ko threats' are more relevant applications. It also is useful to combine m-connected and n-alive, in particular consider whether both values are 0, 1 or 2+. It is the difference between ordinary thickness, great thickness and ultimate thickness, whose strategic value increases in this order. E.g., a great thickness group in a joseki can explain why the opponent has a greater than usual amount of territory. With groups of great thickness, a player can fight well during the middle game (and win ko fights much more easily). If territory is balanced, usually the player with the greater amount or (for this we can use the values) degree of thickness and influence on the board wins. While informal perception of greater degrees can sometimes be sufficient, the values permit a player to verify whether his perception is right. Otherwise, his wrong perception can lead to bad strategies.
3+-alive can indicate inefficiency or be the result of captures or removals. If such removals have been forced in a capturing race, the resulting very great degrees of life need not indicate inefficiency. Of course, one should not reinforce a 2-alive group to make it 3-alive. Already reinforcing a 1-alive group is an exception, which, however, can rarely be useful as a negative ko threat of second order.
Anonymous: [n-alive] is not a new invention. You may want to read: Life and Death Refined. R. Popma & L.V. Allis. Heuristic programming in artificial intelligence 3. 1992
Robert Jasiek: Very interesting, many thanks for the hint! I could not find this paper online and need to go to a library some time. Did Popma and Allis invent the concept? Is their invention that of "possible omission number" or something different? What about N-connection; is there an earlier related invention?
Jonathan Roy: The term Popma and Allis use is called X-Life per a reference to their paper in https://webdocs.cs.ualberta.ca/~mmueller/ps/goeval.pdf