[r-t] Practical Extension

Robin Woolley robin at robinw.org.uk
Mon Jul 30 15:05:14 UTC 2018

```Hi All,

A brief history of indefinacy.

In the old days, indefinacy didn't matter so no problem there.

At the time of the first big revision of Extension rules circa mid-90's,
a comparative new idea that Extensions working at more stages would be
preferred to one working at fewer stages - within the space available,
of course, which has practical limit of c20.

Then, in the early 2000's, all valid extensions had to be indefinite.

At the time, I went on record against this on the grounds that (i)
extensions above a certain stage would never be rung and (ii) I couldn't
see how indefinite could be proved in general.

As regards (i), no peals above stage 22 still have never been rung.

As regards (ii), it is possible that a given method only has a valid
extension at, say, every 50 bells, i.e., n = 22, 72, 122,... and I
couldn't see anyway of demonstrating this theorectically - or every 100
bells, etc....

On this latter point, I was assured by certain list-members that it
would be theoretically possible to show indefinacy. However, these
persons have decided since that they were wrong in this case. (I have no
doubt it is in some cases, but not all.)

To cut a long story short, indefinacy has been redfined in the framework:

"There is no known approach for demonstrating mathematically that a
given extension construction will work indefinitely. Therefore the
following practical test is applied:
An extension construction is deemed to work indefinitely if:
(1) It has been tested up to stage 32;
(2) At least one valid extension is produced in the range of stage 25 to
stage 32.
E.g. an extension construction of a Royal method that produces valid
extensions at stages 18 and 26 is deemed to work indefinitely."

This seems an ideal compromise and is ideal for computer implementation,
although there are extensions which work in much greater increases.
There is one of Surfleet which increases by 24 bells each time.

Best wishes
Robin
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