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# Equivalences

March 19, 2008

In all of the following, GLOP notation is used:

No mark between two spots at the same boundary; a dot (.) at the end of each boundary; a } at the end of each region; a ] at the end of each position. There is no use of ] in this file as only have partial positions are examined. A dot (.) is also used at the start of a partial position to emphasize that we mean a whole boundary and not a part of it.

As a result, .1. == .22. means that 1. can be replaced with 22. when they are boundaries in a position. For example, all these positions are equivalent:

0.0.0.1.1.1.12.1A.}1.A.}]
== 0.0.0.1.1.22.12.}1.A.}]
== 0.0.0.1.22.22.12.}1.A.}]
== 0.0.0.22.22.22.12.}1.A.}]
== 0.0.0.1.1.1.12.1A.}22.A.}]
== 0.0.0.1.1.22.12.}22.A.}]
== 0.0.0.1.22.22.12.}22.A.}]
== 0.0.0.22.22.22.12.}22.A.}]

But 0.0.1A.}ABC.}BC.}] cannot be thought equivalent
to 0.0.22A.}ABC.}BC.}]

Similarly a } is used at the start of a partial position to emphasize that we mean a whole region and not a part of it.

Type 1 - Branch Equivalencies
-----------------------------

aa == 2

In case it's not clear what I mean with the above, lets say we have the position: 0.0.1aAa.}1.A.}] After we connect the 1 to A (in the }1.A.} region), we have: 0.0.1aa.}2bb.}] But most of us (GLOP and AB for sure) would consider this as: 0.0.12.}22.}] in other words replace aa and bb with 2.

Comment: Everybody has been using this, just thought it would be nice if we put it in a separate category (type). It would be great if more equivalencies of this type were found but I'm not optimistic on this. But it could be really exciting if we found something like a22a == 222 (not true, just an example of what we may find).

Type 2 - Border Equivalencies
-----------------------------

.1. == .22.

I was really surprised to find this one, or better, that nobody had considered this before!

It would also be great if we found more of these. Josh says that this equivalence alone may reduce the size of his database by 4% !

Type 3 - Region Equivalencies
-----------------------------

Regions with 2 liberties can be combined in one border (or disconnected to 1 liberty borders) :
------------------------------------------------------

}2A.} == }2.A.}

}AB.} == }A.B.}

}22.} == }2.2.} == }1.} (these are not very useful)
== 1^0

Regions with 3 liberties can be combined in one border (or disconnected to 1 liberty borders) :
------------------------------------------------------

}2.2.A.} == }22.A.} == }2.2A.} == }22A.} == }1.A.} == }1A.}

}2.A.B.} == }2.AB.} == }2A.B.} == }2B.A.} == }2AB.}

}A.B.C.} == }AB.C.} == }A.BC.} == }AC.B.} == }ABC.}

}2.2.2.} == }22.2.} == }222.} == }1.2.} == }12.} == }0.} (no big deal here either)
== 0^1

(GLOP uses the above 2 and 3 liberties equivalences converting them to }ABC.} type, e.g. all spots in one boundary. I would prefer the }A.B.C.} type, e.g. as more boundaries as possible.)

Regions with 4 liberties can SOMETIMES be combined (or disconnected to 1 liberty borders) :
--------------------------------------------------

}2.2.2.A.} == }22.2A.} == }1.2A.}

}2.2.A.B.} == }22.AB.} == }2A.2B.} == }1.AB.}

}2.A.B.C.} == }2A.BC.} == }2B.AC.} == }2C.AB.} (notice something similar with the previous?)

}A.B.C.D.} == }AB.CD.} == }AC.BD.} == }AD.BC.} (and again...)

}2.2.2.2.} == }22.22.} == }1.1.} (is this getting boring?)
== 1^0

More Region Equivalencies:
--------------------------

}2.A.} == }2.2.2.A.} == }2.2.2.2.2.A.}} == ... == }2.2.2....2.A.} (odd number of 2. s)

== }0.A.} == }22.2A.} == }1.2A.} == }2A.BC.}2BC.} == }2AB.}22B.}

(the above is from Sprouts theory group discussions, just converted to GLOP notation)

}2.2.A.} == }2.2.2.2.A.} == }2.2.2.2.2.2.A.} == ... == }2.2....2.A.} (even number of 2. s)

== }1A.} == }2aAa.} == }2AB.}2B.} == }ABC.}BD.}CD.} == }0.2A.}

(also from Sprouts theory group discussions)

}2AB.} == }aAaB.} (a rare jewel)

}2.} == }2.2.2.} == }2.2.2.2.2.} == ... == }2.2....2.} (odd number of 2. s)
== 0^1

}2.2.} == }2.2.2.2.} == }2.2.2.2.2.2.} == ... == }2.2....2.} (even number of 2. s)
== 1^0

And now the good and recent stuff. delivered just today from yper's oven:
------------------------------------------

}2.A.B.} == }2.2.2.A.B.} == }2.2.2.2.2.A.B.} == ... == }2.2.2....2.A.B.} (odd number of 2. s)

}2.2.A.B.} == }2.2.2.2.A.B.} == }2.2.2.2.2.2.A.B.} == ... == }2.2....2.A.B.} (even number of 2. s)

and more:

}2.A.B.C.} == }2.2.2.A.B.C.} == }2.2.2.2.2.A.B.C.} == ... == }2.2.2....2.A.B.C.} (odd number of 2. s)

}2.2.A.B.C.} == }2.2.2.2.A.B.C.} == }2.2.2.2.2.2.A.B.C.} == ... == }2.2....2.A.B.C.} (even number of 2. s)

and even more:

..................................

}2.A.B.C....X.} == }2.2.2.A.B.C....X.} == ... == }2.2.2....2.A.B.C....X.} (odd number of 2. s)

}2.2.A.B.C....X.} == }2.2.2.2.A.B.C....X.} == ... == }2.2....2.A.B.C....X.} (even number of 2. s)

Yes, it works for any number of letters and any mumber of (more than one) 2. !!!

In general if you have a region where all the boundaries are 1-liberty boundaries, you can remove a pair of (2.) as many times as you like, as long as you leave at least one (2.) boundary.

Even more good and even more recent stuff delivered just today from yper's oven:
------------------------------------------

}2.2.2A.} == }2.2.2.2.2A.} == }2.2.2.2.2.2.2A.} == ... == }2.2....2.2A.} (even number of 2. s)

}2.2.2.2A.} == }2.2.2.2.2.2A.}} == ... == }2.2.2....2.2A.} (odd number of 2. s)

CAUTION: the }2.2A.} is different from }2.2.2.2A.} So in these cases, we have to leave at least two (2.) boundaries.

And a last one for today:

}2.2.2.AB.} == }2.2.2.2.2.AB.} == ... == }2.2.2....2.AB.} (odd, three or more, number of 2. s)

}2.2.2.2.AB.} == }2.2.2.2.2.2.AB.} == ... == }2.2....2.AB.} (even, four or more number of 2. s)

AGAIN CAUTION: the }2.AB.} is different from }2.2.2.AB.} and the }2.2.AB.} is different from }2.2.2.2.AB.}

So in these cases, we have to leave at least three (2.) boundaries.

I'm pretty convinced that more of these can be found. More in the future article on "Periodic Table of Sprouts Elements".

Please send me any errors or typos you may find. (thnx Jeff) I would also like to be informed if you find that any of these equivalencies is wrong, if for example you find that they produce two positions that have different Grundy values. All my calculations are by hand and there may be an error.