[r-t] Candidate definition #10

Tue Aug 12 16:01:20 UTC 2008

What if B is one row long?

MJC

-----Original Message-----
From: Don Morrison <dfm at ringing.org>
Sent: 12 August 2008 16:03
To: Ringing Theory List <ringing-theory at bellringers.net>
Subject: [r-t] Candidate definition #10

Based on all the past candidates and discussion, I propose
consideration of the following:

---------------------------------------------------

A block, B, is a sequence of rows. Every bell strikes exactly once in
each row. During the performance of B no bell strikes except as part
of exactly one of the rows of B. The number of rows contained within B
is its length. One or more bells may ring in the same position in each
and every row of B, and such bells are non-changing bells. The stage,
S, of B, is the number bells ringing that are not non-changing bells.

Consider a block B of stage S. There are S factorial possible
different rows, which are called the extent at stage S. B is true if
every possible distinct row of the extent at stage S occurs at least N
times in B and no more than N+1 times, for some N>=0. B is complete if
every row of the extent at stage S occurs exactly N times, for some
N>=1. B is incomplete if it is not complete.

A touch is a sequence of one or more blocks, rung in order, all on all
of the same bells, with no bell striking that is not a part of exactly
one block. The length of the touch is the sum of the lengths of its
constituent blocks.

A touch is a peal if it has length >=5000, and all its constituent
blocks are true, and no more than one of its constituent blocks is
incomplete.

---------------------------------------------------

Unless I've made a mistake (which as you all know is not unlikely;
please point it out) I think I am pretty happy with the above. If it
all works and hangs together the way I think it does, I believe all of
the following are true:

Everything acceptable as a peal under the current Decisions
remains a peal within this definition.

This definition captures a stringent definition of truth, that even
Mark will be happy with. We won't know for sure until he returns, of
course. It is, I think, similar to what Martin proposed yesterday,
just tightened up to be unambiguous in those cases Martin left
unspecified, and without the special case regarding length. Some of us
would be happy with a more relaxed notion of truth, but I personally
am content with this as a suitable compromise. I hope others happy
with a more relaxed notion of truth are likely to feel similarly to me
in finding this an acceptable compromise. Please let us know if you
don't.

This definition applies at all stages and is free of ad hoc special
cases. This means it avoids the anomalies and inconsistencies that
afflict the current Decisions and some of the proposed alternatives
when you deal with long lengths of middle sized stages. It deals
gracefully with the mixture of stages minor and below, stirred
together with stages triples and above, which have been problematic in
several other alternatives.

This definition requires no recourse to methods or other ways of
generating rows, and so is flexible enough to handle Dixonoids or
anything else we care to throw at it. It also handles jump changes.

This definition is more liberal about lengths than some people are
comfortable with. If the consensus within the ringing community is
that we would like to exclude certain length ranges for certain
stages, for historical continuity, these are better added as explicit
special cases outside the definitions. Something like "The Council does
not recognize peals of triples of length shorter than 5040" or
whatever. That's more honest and clearer, and doesn't muddy the
definitions.

This definition contains fewer than 250 words. I believe it replaces a
little over 1500 words in the current Decisions. Though if further
special case restrictions are added that 250 will obviously increase
commensurately.

In this definition we do have to give up those invariants that my
intuition cries out for. But perhaps they are not achievable in a
definition that others will find acceptable. While those invariants do
not hold for a peal in this definition, they do, I believe, hold for
each of the constituent blocks that make up the peal. This may be some
consolation.

This definition does not cover cylindricals. While I and at least a few
others would prefer to see such things included, I think excluding
them is reasonable. They are highly contentious. To the best of my
knowledge no one has actually successfully rung any to even quarter
peal length, despite several bands, at least one very advanced,
putting some effort into trying--they are very difficult in practice.
And they are a mine field of definitional problems: for example, I
think you can quickly get into horrible confusion about exactly where
a row begins and ends. And as Richard has noted there are strange
interactions with jump changes, which latter people really do ring
today, even in some peals the Council has declined to accept (OEDJ*).
I think leaving cylindricals out is a sensible compromise with the
practicalities of what we can do, and how hard it would be to convince
others.

I worry that as I've phrased this definition it may sound a little
intimidating to some. I hope (though I may be disappointed in this)
that it's actually pretty clear and straightforward, if read slowly
and carefully. I welcome suggestions for improving it for the benefit
of those who find such language intimidating.

Not terribly relevant, but I believe that if you substitute 1250 for
5000, it is exactly the definition I current apply implicitly for
quarters that I choose to call, though until we'd had the discussion
of the past week or two there's no way I could have phrased it so
clearly.

What do others think?

* OEDJ = Or the Euphemism Du Jour

-- 
Don Morrison <dfm at ringing.org>
"Prospering as a forager is a more difficult problem than doing
calculus or playing chess."   -- Stephen Pinker, _How the Mind Works_

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