Apart from assigning Left to Black and Right to White by convention, is there any difference between (1|2) and (2|1)? Is one of them, by convention, a number and the other not or is neither a number?

i don't get the question
but that fine i probably won t have the answer anyway

_________________
In theory, there is no difference between theory and practice. In practice, there is.

CGT stands for Combinatorial Game Theory

http://senseis.xmp.net/?path=CGTPath&pa ... ctionToCGT

Robert's question did not use the notation that seems most common on SL. If the curly-bracket notation on that page is what Robert intends, then I'm fairly certain there's a difference between (1|2) and (2|1), since {0|} = 1 and {|0} = -1. Specifically, {1|2} means Black can move to 1 while White can move to 2, while {2|1} means Black can move to 2 while White can move to 1.

If something different is meant, then I am not familiar with the notation he's using.

{1|2} is a number, it's 3/2. {2|1} is not a number, it's "hot", meaning both players want to play there. I've linked to this elsewhere, but there's a nice java program that will calculate CGT values for you, plot thermographs, and do some other cool stuff:

http://cgsuite.sourceforge.net/

{1|2} doesn't make a lot of sense as a go position, it describes a situation where neither player wants to play in a region, because the opponent will gain a point (maybe a kind of very small seki?). Many CGT games force players to make moves in unfavorable regions, so the theory has to handle this (the red-blue hackenbush game from the CGT books, for example). {2|1} is just a standard 1 pt gote.

In a somewhat confusing choice (I think), the "Chilling gets the last point" book uses a technique called "chilling" to turn hot plays into cold ones, I guess because the analysis is more in line with other cool CGT games. Basically you imagine that every move on the board costs you 1 point to make. So {2|1} actually becomes {2-1 | 1+1} = {1|2} ! As I said, I found this very confusing on my first read through that book

Hope that helps.

Consider this 6x12 board under area scoring.

Click Here To Show Diagram Code

[go]$$ Evaluate this board. (Area scoring)
$$ -------------------------
$$ | . . X O O . O O . O X . |
$$ | O X X O . O X O O O X X |
$$ | O O X O O X X X X X O . |
$$ | O . X O X X . X X O O O |
$$ | X X X O X . X . X X O . |
$$ | O O O O X X X X X O O O |
$$ -------------------------[/go]

What about its value under territory scoring? Not Japanese/Korean, but Lasker-Maas/Spight/Ikeda-something-or-other?

If you chill area scoring you get territory scoring.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

That has nothing to do with convention, the term "number" in the CGT sense is strictly defined.


A "game" is defined as some {L|R}, where L and R are sets of games.

Let X := {XL|XR} and Y := {YL|YR}. X <= Y if and only if:
There is no xL in XL such that Y <= xL, and
there is no yR in YR such that yR <= X.

A "numeric game" is defined as a game {L|R} where L and R contain only numeric games, and each l in L is strictly less than each r in R.

Two numeric games X and Y are "equivalent" iff X <= Y and Y <= X.

A "number" is defined as an equivalence class of numeric games.


Because 1 < 2, {1|2} (or more accurately it's equivalence class) is a number. And of course 2 < 1 is false, so {2|1} is not a number.
Note that 0, 1, 2 and so on are just abbreviations for certain numbers. 0 := {|}, 1 := {0|} and 2 := {1|}.
If you want you can check the axioms to see that 1 < 2 is really true .


See http://en.wikipedia.org/wiki/Surreal_number, this article is a lot better than the articles on senseis. In this article, "games" are called "forms", but it's really the same thing. Conway defined the numbers first, and applied them to game theory later.

Thank you and emeraldemon!

IMO, Conway's definition of numbers in On Numbers and Games, which I read a couple of years ago, is the most elegant. At university we still learned the traditional ways via, e.g., peano axioms though.

I guess I asked because I thought of {1|2} as being the complement of {2|1} but, of course, Right prefers negative numbers. Do I get it right that -{1|2} = {-2|-1}?

Peano axioms define the Natural numbers. On top of them, you can of course constructively define more types of "numbers", like rational numbers, real numbers, complex numbers and so on.

Conway's "numbers" are entirely different mathematical objects, even though they share the same name and some of their properties (essentially, forming an ordered field).



Yes.

Click Here To Show Diagram Code

[go]$$ Evaluate this board. (Area scoring)
$$ -------------------------
$$ | . . X O O . O O . O X . |
$$ | O X X O . O X O O O X X |
$$ | O O X O O X X X X X O . |
$$ | O . X O X X . X X O O O |
$$ | X X X O X . X . X X O . |
$$ | O O O O X X X X X O O O |
$$ -------------------------[/go]

No bites yet?

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

How very nice! Very well constructed!

Thanks, Robert.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

In the number farm what makes Conway's "numbers" less equal than the numbers derived from Peano? And why aren't Conway's "numbers" derivable from Peano? Maybe question1 = question2 !

Conway numbers are special kinds of games. Peano numbers are natural numbers, aka counting numbers. Counting is implicit in Conway numbers, but they are not counting numbers.

Peano numbers are usually augmented to include zero. We can write the Peano numbers this way.

{} = 0
{0} = 1
{1} = 2
***

Conway defines 0 as a game that the second player wins. A player wins if her opponent has the move but has no play. The two players are called Left and Right. The simplest game is one in which neither player has a move. It is 0. We can write it this way.

{|} = 0

The vertical bar separates options to which Left can move (to the left) from options to which Right can move (to the right). Neither player has an option, and hence, no move. Whoever has the move loses.

We also have

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

+1 is the game where Black to play can move to 0, while White has no play. -1 is the opposite.

Note that Conway numbers skip the natural numbers as go directly to the integers.

We also have

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

It is also true that {|+1} = 0 = {-1|} = {-1|+1}. All are second player wins.

For more see Winning Ways, by Berlekamp, Conway, and Guy or On Numbers and Games, by Conway.

----

We can use the Peano numbers to define the integers. One way to do that is like this:

(1,1) = 0
(1,0) = +1
(0,1) = -1

I.e., the integer is the difference between the first Peano number and the second.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Thx Bill. I know about Peano and about Conway. But your answer does not explain to me why say complex numbers are completely different mathematical objects from Conways `numbers`. As Flover claims. Both are numbers aren´t they? Wait! Could it be that there is no multiplication of Conway´s numbers?

You can multiply Conway numbers. (I would have to look up the definition, though. )



flOvermind does not mean that rational numbers, real numbers, complex numbers, etc., are different from Conway numbers. Conway numbers include the rationals and reals. To get complex numbers, you add i to Conway numbers, just as you add it to numbers based on natural numbers. (Or you can define a complex number as an ordered pair of real numbers.)

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Or there's my favorite way of taking the ring of real polynomials in one variable and quotienting by the maximal ideal generated by x^2+1 ... I'll show myself to the door.

Why?

By counting number I mean natural number, one that conceptually rests upon the sensory-motor operation of counting. Conway numbers start with the integers.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

It is an advantage that Conway does not restrict himself to a special case set of numbers:)