Forum for Alive
Simple concise textual definition. [#15003]
As a beginner, and a coder who thinks textually, I was looking for a very simple logical, textual definition of a living group, from the point of view of the construction (has these properties) rather than the function (ie to resist capture). I couldn't find one that worked for me.
Eventually through my effort and through checking with language models I came up with this: "all stones in the living group must be connected, directly or indirectly to each other, or to at least two surrounded liberties" This was the simplest complete textual definition I could come up with, and it really helped me understand the living construct. So i'm putting it in this discussion, maybe a more experienced player/editor might know if this is useful to add to the site somewhere.
- edit*
Thanks for the replys. Attempting the definition was helpful to my understanding, and reading more about why it isn't that simple is helping more! :)
The term live or alive can include seki which you missed. Further, alive can also include the cases where the defender's group cannot be captured regardless of how it is being attacked, provided that the defender mounts a perfect defense - see Alive - Introductory for an example. In this aspect, a group that is alive can become dead if the defender fails to defend correctly, or fails to respond to an attacking move at all (perhaps to resolve a ko, in which case the attacking move is a ko threat).
I guess what you are trying to define is pass-alive in a mathematically rigorous manner. However, there are at least two problems:
- You do not define the term indirectly.
- You seem to be describing properties of a pass-alive group, but due to the final or world in your sentence, merely satisfying these properties does not guarantee that a group is pass-alive. However, merely changing the final or to and appears to be insufficient (depending on what you mean by indirectly).
Even if the "or" were changed to "and", following Unkx80's remarks, and something were done to disambiguate the word "indirectly", it still would not be a satisfactory definition. Because it also needs something about non-adjacency of two or more surrounded empty intersections, in order to exclude single eyes larger than one intersection.
If we're talking about "unconditional life" rather than "life", then it seems to me that Benson's theorem may be the best available purely textual definition along the lines you're looking for.
In the above, I should have used the term pass-alive rather than unconditional life because that would have been clearer.
Anyway deMangler, I saw your SL page and I wish you all the best in your journey! Don't worry at all about losing games in the beginning. You may have heard of proverbs like the best way to start at Go is to lose 100 games as quickly as possible.
You wrote in your SL page that you lose many games through not recognising false eyes. That's a common thing for beginners. There are some common simple situations that one can become familiar with quickly. I recommend having a look at Capture Three To Make An Eye - and I was thinking of a similar, related page Capture Two To Make A Half Eye? which doesn't exist at the moment - it may or may not exist at some point in the future.
I saw Xela's comments which are very sensible. I feel that for most beginners, trying to get precise / rigorous / formal definitions for some things is not be the best approach. Indeed, the New Zealand rules are short and precise - and they don't mention concepts such as life and death; they don't need to.
I'm sorry to tell you that what you're looking for simply doesn't exist! I looked for it too when I started learning go :-)
First you have the theory. Yes, it's true that if all stones are "connected" to two liberties, roughly speaking, then the group is alive. (Unkx80 and Malcolm have linked to more precise versions of this statement.) But this is not the only way to live! Exceptions in real games are rare, but if you want your definitions to be logically correct then you have to deal with things like moonshine life and hanezeki.
Then there's the practical side. It's common to end the game with some groups that surround large empty spaces but don't have two distinct eyes. Usually, two experienced players will agree that these groups are alive beyond all reasonable doubt, but won't require a mathematical or logical proof that they are 100% alive. (Unless the surrounded space is so huge that an invasion is possible, in which case one player won't pass but will have a go at invading.)
If you're playing with Chinese rules (or some other form of area scoring), you could of course fill in most of the space until you've created a pass-alive group, and there you have your mathematical proof. But this is time-consuming and annoying. If you make a habit of doing this past the beginner stage, people won't want to play with you. And under Japanese rules or territory scoring, you will also lose a lot of points by doing this. So at some point you have to get used to living with a tiny amount of ambiguity.
If you like simple logical, textual definitions, then you'll probably enjoy the New Zealand rules! They're not quite what you're looking for, but they are in a sense very beautiful.
My use of the phrase "directly or indirectly" is redundant now that I think of it. Connected is connected. Thanks for the feedback that really helps, the effort in trying to create a definition has helped and learning more why this is not straightforward is helping more. :) Also it is great to be reminded that go is always rewarding of more attention and analysis, and there is so much to discover from other peoples work on it. Not many games like that!
Yes, we've got something like a 2,000 year history behind us, plus just a few years since AI turned everything upside down. No fear of running out of interesting things to learn :-) Keep enjoying yourself, and keep asking questions!
No worries. :) If there is a definition of alive or pass-alive that uses construction rather than the inability to capture, then the community would have come out with one by now. Therefore, I suspect a definition by construction doesn't really exist.
However, I do admit that operationally, the resist capture definition is difficult to apply. Therefore people came out with various heuristics, of which many are of the form of one-sided implications (if certain properties are satisfied, then the group is alive). Examples include:
- If a chain has two real eyes, then it is pass-alive. (See recognizing an eye for a heuristic for determining whether an eye is real.)
- If a group can obtain two real eyes with perfect play by the defender, then the group is alive. (The part on perfect play is typically omitted and implied in discussions.)
- If a chain is part of a seki, then it is alive but not pass-alive.
You can start by learning and applying such constructive heuristics first, then expanding your knowledge by figuring out how to make or destroy eyes (this is a gigantic topic), and learning exceptions such as false eye life. I would say, the ability to make two eyes or to make seki covers the vast majority of the cases, so gun for those first.
All the best! :)
Oh, another thing. In this diagram, all black stones are connected to two liberties, but the group is not alive.
![[Diagram]](diagrams/24/2206e45e2fd062fe2b39ec436ad332c9.png)
![[Diagram]](diagrams/46/36646226d98039c1beeca7681289f5c4.png)
Not alive by your definition? It's not alive in the strict sense of being pass-alive, but in an actual game you would stop adding stones to it at this point and treat it as alive anyway.
![[Welcome to Sensei's Library!]](images/stone-hello.png "Sensei's Library")