Smith wording improvement

[Mike Ossipoff]

Where I said "...some member of the Smith set...", I could instead
say "...any one or more members of the Smith set..."

Still, "some member of the Smith set" seems pretty unmistakable in its
meaning. If I'm a candidate, and there's a candidate known to be in the
Smith set, and I don't have more defeats than he/she does, then I can
say that I don't have more defeats than some member of the Smith set,
and I don't know how anyone could construe the wording so as to say
that isn't so.

***

Of course I never said that my suggested procedure was an algorithm
or a computer program. That's why I didn't say in what order to 
check the alternatives not known to be in the Smith set to find out
if they beat or tie a known member of the Smith set--It makes no difference
in what order it's done, and I was stating a way of determining the Smith
set members, not writing an algorithm. Plainly if one wrote a computer
program one would have to specify that those alternatives be checked in
some particular order.

Likewise, it doesn't matter if we test alternatives for not being
beaten by more alternatives than 1 or more known members of the Smith
set first, or whether we first check them for whether they beat or
tie a known Smith set member. It doesn't matter because, either way,
if an alternative passes that test it's established as a Smith set
member.

But if I were specifying the order in which to do those things, I'd
say:
1. Order the alternatives according to how many alternatives they're
beaten by.

2. The alternative beaten by fewest other alternatives is a member
of the Smith set (as are any alternatives tied for that distinction).

3. Determine by how many alternatives is beaten the known member of
the Smith set which is beaten by the most alternatives.

4. Any alternative not yet named as a member of the Smith set, which
isn't beaten by more alternatives than the number in paragraph 3
shall be named as a member of the Smtih set.

5. Check each alternative not yet named as a Smith set member
(starting with the one beaten with the most alternatives, and
then checking the one beaten by the next largest number of alternatives,
etc.) by comparing it one at a time to each named mamber of the Smith
set (starting with the named Smith set member beaten by the most other
alternatives, then comparing to the known Smith set member beaten
by the next largest number of alternatives, etc.) to determine if
that alternative not yet named as a Smith set member beats or ties
that known Smith set member.

6. If any new Smith set members were found in the most recent
carrying out of paragraphs 5 &/or 6, then go to paragraph 3. Else
stop.

***

It makes it much longer when one specifies the order in which to
do those comparisons. That's why I didn't, in my previous
letters, since, as I said, it doesn't matter about the order, except
for speeding up the process. Since, when it's time to determine if
any more alternatives beat or tie a known Smith set member, each 
has to be so checked, and since finding one beaten by lots of alternatives
would yield more new Smith set members in paragraph 4, I started with
the ones beaten by the most alternatives. The reason why I'd compare
an alternative not yet named as a Smith set member to the known Smith
set members starting with the known Smith set member beaten by the most
alternatives is that such an alternative has a better chance of being
beaten or tied, and it's only necessary to beat or tie one of them.

Anyway, it was for brevity & simplicity that I didn't specify the
order of the comparisons in previous letters.

***

One thing I left out of the detailed procedure in this letter is:
When I said to start with the alternative that is beaten by the
most alternatives, and then do the one beaten by the next highest
number of alternatives, etc., I didn't specify what to do when
2 are tied in that respect. So let me specify it now: The one with
the lowest "alternative number" goes first. (I'd, at the beginning
of the count, or on the blackboard in a committee vote, give each
alternative an "alternative number". 

There now, I've specified every detail of what to do in what order.

And please note that I've done it in English. I still have no idea
what it is that Bruce is saying to do, since it's all in language
intended to be decrypted only by other mathematicians.

***

But if you aren't into formulese, I've posted, first, a brief
description of a Smith set procedure, and, now, one which details
the order in which to do things.

***

Mike


 



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