Re: [gnugo-devel] Half and false eye with ko.

[Daniel Bump]

> Yes, this case seems to be fine. What's worse is the difference
> between my case (b) and the modification (b') below
> 
>     .X..
> (b) ..OO
>     OO.O
>     O.OO
>     XOX.
>     .X..
> 
>      .X..
> (b') ..OO
>      OO.O
>      OXOO
>      X.X.
>      .X..
> 
> Notice that while (b') is a 2.5 topological eye, (b) is a real eye
> regardless of ko. I actually had intended to have (b') in my last
> message but mistakenly took the wrong diagram. I'm not sure how much
> this complicates things, but it certainly makes the analysis more
> messy.

This seems to be OK. 

I think there is an arithmetic with 1/4 eyes in Howard Landsman's
paper. The following table converts diagonal sums to
1/4-fractional eyes. The total number of eyes to live would be
2. For example, a group with 8 quarter eyes should be alive, the
example at the end shows. If at any point B takes a ko, W just takes
some other one.

diagonal     number of     meaning
sum          eyes
--------------------------------------------------
2.0        1.0           true eye 

2.5         .75          half eye if X wins ko, true eye otherwise

3.0         .5           half eye regardless

3.5         .25          half eye if O wins ko, false eye otherwise

4.0        0.0           false eye

(;GM[1]FF[4]
SZ[19]HA[0]KM[0.5]
;
AW[ec][fc][hc][ic][kc][lc][nc][oc][cd][dd][fd][gd][id][jd][ld][md][od][ce][ee][fe][he][ie][ke][le][ne][oe][of][og][oh][oi][fj][gj][ij][jj][lj][mj][oj][pj][dk][ek][gk][hk][jk][kk][mk][nk][pk][dl][fl][gl][il][jl][ll][ml][ol][pl]
AB[db][eb][fb][gb][hb][ib][jb][kb][lb][mb][nb][ob][pb][ac][bc][cc][dc][gc][jc][mc][pc][pd][de][ge][je][me][pe][af][bf][cf][ef][ff][hf][if][kf][lf][nf][pf][dg][eg][gg][jg][mg][ng][pg][eh][fh][gh][hh][ih][jh][kh][lh][mh][nh][ph][ci][di][ei][hi][ki][ni][pi][qi][bj][ej][hj][kj][nj][qj][bk][qk][bl][el][hl][kl][nl][ql][cm][dm][fm][gm][im][jm][lm][mm][om][pm][qm][dn][en][gn][hn][in][kn][ln][nn][on]
;B[df];W[gf];B[jf];W[mf];B[nm];W[km];B[hm];W[em];B[ge];W[me];B[kl];
W[el]
)

Dan