[r-t] Lead-based methods [was: Poll on consecutive blows in the same position]

[Don Morrison]

On Mon, Dec 29, 2014 at 7:10 PM, Iain Anderson <iain at 13to8.co.uk> wrote:
> I can't help feeling that it would be useful to have a universal way of
> defining non-β methods

I have grave doubts that any universal system can be devised to
describe all non-β methods. However, it is clearly +possible+ to
describe any non-β method that is an α-method of finite length: simply
enumerate all the sequences of permutations, one sequence for each of
the N! possible starting rows. If you disallow jump-changes those
sequences of permutations can even be encoded as place notation, if
you like.

Of course, that's an absurdly voluminous encoding of an arbitrary
finite α-method, particularly at higher stages. There might be some
way to make it a bit more compact by partitioning the starting rows
into sets that all result in the same changes, but it's still completely
impractical. And, of course, there are potential α-methods for which
there is no such partitioning, each of the N! starting rows resulting
in a different course length.

But, while possible, I sincerely doubt a +practical+, compact notation
can be found that can describe an arbitrary, finite α-method. I
suppose some clever person could work out the minimum number of bits
that must be required for such a notation, as a function of stage and
maximum lead length to be considered (probably a sensible maximum lead
length for such a back of the envelope endeavor might be 5,040?). I'm
guessing the result is not going to be an encouraging lower bound.

On the other hand, I would not be surprised if a suitable notation
could be found for a broad, but restricted, subset of non-β, finite
α-methods, that might be a practical class (perhaps such a subset is a
suitable thing to apply the name "Dixonoid" to, rather than applying
it to all non-β methods). For example, perhaps methods that have an
underlying β-method (that's Plain Bob in Dixon's) with some changes
altered based solely on the positions of one or more bells in the row
to which the change is being applied. Dixon's would be a member of
such a class, as, I suspect, are all "rule-based methods" that have to
date been rung (deliberately).



-- 
Don Morrison <dfm at ringing.org>
"Enum is actually a generic class defined as Enum<T extends Enum<T>>.
This circular definition is probably the most confounding generic type
definition you are likely to encounter. We're assured by the type
theorists that this is quite valid and significant, and that we should
simply not think about it too much, for which we are grateful."
           -- Ken Arnold and David Holmes, _The Java Programming Language_