   Date: Tue, 14 Feb 95 13:10:41 -0500 (EST)   From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~  Subject: Re: Re: superflip in quarter turn metric

Ok, let's try, try again. I was incorrect in my response to der Mouse,
and Martin Schoenert's correction is appreciated.

The original issue was as follows: suppose you have created a data base
for N levels of God's algorithm, beginning with Start at the root of
a tree. With quarter-turns as generators, there is 1 position at
level zero, 12 positions at level one, 114 positions at level two,
etc.

Now, suppose you want to create a data base for N levels of God's
algorithm, starting with X at the root of the tree. Can you simply
compose your first tree with X on an element by element basis in order
to obtain the X-rooted data base? (You do have to be careful about
pre-multiplying vs. post-multiplying as Martin indicated!)

I stated essentially that the superflip was fairly unique in that
you could compose the superflip with the Start-rooted data base
in order to obtain the superflip-rooted data base, but that for
X-rooted data bases in general you would have to perform a complete
search starting with X. der Mouse (correctly) noted that the
same procedure would work for any position X as for the superflip.
I (incorrectly) took exception with der Mouse, citing my fallacious
distance argument.

However, the way my data bases work, I still think that the
superflip is fairly unique in its ability to be composed with
a Start-rooted data base. der Mouse was on the right track in
his first post when he questioned whether the issue was the
fact that the data bases only store representative elements of
M-conjugacy classes. I responded that the storage of representative
elements was not the issue, but in fact it is.

For example, when storing only representative elements of M-conjugacy
classes, consider an F-rooted data base. Strictly speaking, we would
have to speak of a Repr{m'Fm}-rooted data base, because F might not be
the representative element of {m'Fm} -- it could be any of the twelve
elements of Q. When storing only representative elements for
a Start-rooted data base, there is 1 element at level zero, 1 element
at level one, and 5 elements at level two. For a Repr{m'Fm}-rooted
data base, there is 1 element at level zero, and six (!) elements
at level one -- namely those same elements that are at level zero and
level two of the Start-rooted data base.

Hence, we cannot take the Start-rooted data base and pre-multiply
each element by Repr{m'Fm} to obtain a Repr{m'Fm}-rooted data base.
And in general, we cannot take the Start-rooted data base and
pre-multiply each element by an arbitrary Repr{m'Xm} to obtain
a Repr{m'Xm}-rooted data base.

But for the superflip E, we *can*
take the Start-rooted data base and pre-multiply each element
by Repr{m'Em} to obtain a Repr{m'Em}-rooted data base. In fact,
note that |{m'Em}|=1, so we must have Repr{m'Em}=E. However, note
that for each Y in the Start-rooted data base, it is not sufficient
to calculate (EY). Rather, we must calculate Repr{m'(EY)m}. That is,
we know by definition that we have Y=Repr{m'Ym}, but we do not
necessarily have (EY)=Repr{m'(EY)m}.

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Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax     