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Infinitesimals
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Here are some infinitesimals in chilled go.

[Diagram]
Diag.: Infinitesimals


Let's take the top one first.

If Black takes, the local score is +2 (which chills to +1). If White connects, the local score is 0 (which chills to +1). The chilled game looks like this:

                    { 1 | 1 }

If a game is not a number, we may add a number to it by adding that number to both its Left and Right followers, so

                    { 1 | 1 } = 1 + { 0 | 0 } = 1*

The top game chills to 1 plus * (STAR). What the number is does not affect correct play at temperature 1, where we are simply trying to get tedomari, so it is customary to ignore the number and just call this position *.

(I have introduced * and ^ (UP) in Chilling.)

Now for the middle one.

Black can move to a score of 3, which chills to 2. White can move to the first position, which chills to 2* (1* by itself, but White has made a move, which adds one to the chilled score). The chilled game looks like this:

                    { 2 | 2* } = 2 + { 0 | * } = 2^

So the middle game chills to 2 plus ^ (UP). Again, we ignore the number and just call this position ^.

Now for the bottom one.

Black can move to a score of 4, which chills to 3. White can move to the middle position, which chills to 3^. The chilled game looks like this:

                    { 3 | 3^ } = 3 + { 0 | ^ }

The infinitesimal, { 0 | ^ }, it is the canonical form of ^^* (Double UP STAR). So the bottom game chills to 3 plus ^^*, and we just call it ^^*.


Why { 0 | ^ } = ^^*

White (Right) can play to ^^ or to ^ + * + * = ^. Since ^ < ^^, ^ dominates.

Black (Left) can play to ^* or to ^^. Both reverse. From ^* White can play to 0. We know that ^^* > 0, so the play to ^* reverses. However, 0 has no Left option, so we eliminate that option. From ^^ White plays to ^*, ^^* > ^* because ^ > 0, so that play reverses. From ^* Black can play to ^ or *. Both reverse. From * White plays to 0, so we eliminate that option, as above. From ^ White plays to *. ^^* > *, so that play reverses. From * Black moves to 0. Whew!


The chilled value of the whole position is just the sum of its parts: 1* + 2^ + 3^^* = 6 + ^^^ + * + * = 6^^^. Ignoring the number, as usual, we just call it ^^^ (Triple UP).

Note that the infinitesimals of the corridors form an arithmetic series. (By contrast, the fractions of the corresponding empty corridors form a geometric series.)

     * + ^* = ^ + * + * = ^
     ^ + ^* = ^^*

Does the next corridor equal ^^^ ?

     ^^* + ^* = ^^^

Yes, it does. :-)

Etc., etc.

This means that when White plays in a corridor (except to connect to the stone) she gains ^* in the chilled game, which is about as good as you can do in the fight for tedomari. (Connecting gains *, which may be no gain at all!)

Each infinitesimal is roughly equivalent to some number of ^s, called its atomic weight. Atomic weight is similar to the liberty count in a semeai, but the contest is over tedomari. An advantage of at least 2 in atomic weight is enough to guarantee tedomari.

Tinies and Minies

What do these games chill to?

[Diagram]
Diag.: Tinies


The top one has the following game tree:

                    { 4 || 3 | * }

(An unchilled * is a dame.) It chills to

                    { 3 || 3 | 2 } = 3 + { 0 || 0 | -1}

{ 0 || 0 | -1} is called TINY-1. Its negative is MINY-1. A game with the canonical form { 0 || 0 | -G}, where G is a game greater than some positive number, is called TINY-G. You can also write it as a plus sign with a subscript, G, but SL does not have subscripts. You can write a miny with a minus sign plus subscript.

The bottom one has the game tree:

                    { 5 || 4 | 0 }

which chills to

                    { 4 || 4 | 2 } = 4 + TINY-2

As usual, we ignore the integer and just call these TINY-1 and TINY-2.

Tinies and minies are, of course, sente. Tinies are positive, as you may confirm. Tinies are called tinies because they are positive, but less than ^.

We can confirm this with the difference game, ^ - TINY-1 = ^ + MINY-1, which is greater than 0.

[Diagram]
Diag.: ^ - TINY-1


To make the value infinitesimal, let's make the mean value 0 by saying that Black has captured 1 White stone. We are mainly interested in the tedomari at temperature 1.

[Diagram]
Diag.: ^ - TINY-1 Black first

Black wins. The board score is the same, but Black has the captured stone. However, Black has made one more play than White, which costs one point in the chilled game, so the score is 0. Black wins because he gets tedomari.


[Diagram]
Diag.: ^ - TINY-1 White first

Black wins. The score is even, but Black got tedomari.

Since Black wins no matter who goes first, the difference is greater than 0, and ^ is larger.



You may verify that ^ > 2 x TINY-1. In fact, UP is greater than any number of tinies. Furthermore, TINY-1 is greater than any number of TINY-2's, etc.

Atomic Weight of Tinies and Minies

Tinies are less than ^, so their atomic weight is 0, as is the atomic weight of minies.

But what about { 0 | tiny-1 } ? Let's compare it with ^ via a difference game.

[Diagram]
Diag.: ^ - {0|tiny-1}

White has two captured Black stones.


[Diagram]
Diag.: ^ - {0|tiny-1} Black first

Black gets tedomari and wins the chilled game.


[Diagram]
Diag.: ^ - {0|tiny-1} White first

White gets tedomari and wins the chilled game.



Since the player with sente wins the difference game, { 0 | tiny-1 } is confused with ^, and has an atomic weight of 1.[1] (You may verify that it is greater than *.)

Similarly, { 0 || 0 | tiny-1 } has atomic weight, 2. Etc. We may write such games as 0(n)tiny-x, where n is the number of zeroes, and also the atomic weight. (We should really use superscripts, but that's not easy on SL.)

More in Playing Infinitesimals, Other Infinitesimals, and More Infinitesimals.

Further discussion in Infinitesimals Discussion.

--BillSpight


[1] This is not a proof. There are other games that are confused with ^ that do not have an atomic weight of 1. Atomic weights are not necessarily numbers.



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This is a copy of the living page "Infinitesimals" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.