Forum for Two-eye-formation

Superfluous clause? [#2179]

Unkx80: Superfluous clause? (2010-02-04 15:42) [#7159]

I repeat the definition of two-eye-definition:

A two-eye-formation is a set of one or several strings of the same player and exactly two empty intersections so that:
(a) each of the strings is adjacent to each of the two intersections,
(b) none of the strings is adjacent to another empty intersection, and
(c) each of the two intersections is adjacent only to the strings.

I can't help but feel that clause (b) is superfluous. I think I can loosen the definition thus:

A two-eye-formation is a set of one or several strings of the same player and exactly two empty intersections so that:
(a) each of the strings in the set is adjacent to both empty intersections, and
(b) each of the two empty intersections is adjacent only to the strings in the set.

I think this definition also captures the notion of uncaptureable groups with two one-space eyes, as long as that said player does not play in any of the two intersections. Did I miss anything?

HermanHiddema: Re: Superfluous clause? (2010-02-04 15:58) [#7160]

I think you can loosen the definition as far as Benson's Algorithm, while retaining the same basic benefits for a rules context. The main benefit of the two-eye-formation, IMO, is that it is simpler to check for (by humans, computers couldn't care less ^_^).

Unkx80: Re: Superfluous clause? (2010-02-05 02:49) [#7167]

Thanks for that pointer; I have forgotten about its existence. I have read parts of the original paper and I quite liked it, although I agree that it is not easy for a human to apply.

RobertJasiek: Re: Superfluous clause? (2010-02-04 17:43) [#7161]

The definition cannot be relaxed without changing its meaning. Its current meaning is intended and has been applied, e.g., in important propositions (the most remarkably [ext] Chris Dams' theorem) or other, higher level definitions.

It is possible to study different things for different purposes. If that is your intention, then define new terms like, e.g., n-eye-formation (n>1) or two-eye-formation-with-optionally-outer-extra-liberties (or whichever name you would like to use).

Unkx80: Re: Superfluous clause? (2010-02-05 02:43) [#7166]

Your point is taken. Thanks.

ThorAvaTahr: Re: Superfluous clause? (2010-02-04 17:45) [#7162]

In order to excluded formations with 3 or more eyes, you will need the clause:

the phrase set of one or several strings of the same player and exactly two empty intersections is not sufficient, since you can take the necessary ingredients from a formation with three eyes to compose such a set.

xela: Re: Superfluous clause? (2010-02-05 00:58) [#7165]

No, I think the definition is correct as it stands. Compare these two examples:

In example 1, the black string together with the two a intersections makes a two-eye formation. In example 2, the black string together with the two a intersections does not make a two-eye formation, because the black string is also adjacent to the empty intersection b.

HermanHiddema: Re: Superfluous clause? (2010-02-05 12:29) [#7169]

I think that was the point that ThorAvaTahr was trying to make (i.e. where he says "you will need the clause", he is referring to the clause (b) that unkx80 suggested was superfluous, and the next line is an explanation of why).

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