# [r-t] RE: Various

Sun Nov 21 12:48:55 UTC 2004

Robin Woolley <robin at robinw.org.uk> wrote at 09:02:00 on Sun, 21 Nov
2004
>I think that the use of 'countable' means that there is a simple
>algorithm to count the members of a set.

A set is countable if there is a mapping from the integers onto the set
(i.e. without missing any). All finite sets are countable.

>Looking at extensions themselves. Classical extension had (2n-3) possible
>paths below (worst case scenario) and 2(n-2) possible paths above, giving a
>maximum set size of 2(n-2)(2n-3), which is of the order of n^2. Quantum
>extension (extension by modes) has (n-1)(n-2) both above and below which is
>of the order of n^4. It's interesting to see the increase in the limit of
>possible extension paths quantified. Classical extension of minor can be
>carried out by hand on the back of the RW address label.

You forget that we can now extend by more than two stages, giving
n(n-1)(n-2)/4 constructions each above and below (for even n). In most
practical cases the number of distinct constructions will be much less.

>
>As regards a program to do this, is it more efficient to generate the 400
>extensions of any minor parent, pick off the ones with (say) pb lead ends
>and then apply the restrictions, or apply the restrictions then generate the
>extensions?

Generate the distinct constructions, including any restrictions, then
check for pb lead ends (don't forget that you have to do this at an
infinite number of stages).

>
>Here's a case of 'uncountably finite' in that there's no algorithm to give
>the number of true extensions, but it is finite.

How about 'generate all extensions, then count them'.

>
>Back to the old topic:
>Part of the MC's justification for (G)B1 is "This new restriction gives us
>the confidence to introduce several new constructions without the risk that
>they will increase the number of unsatisfactory extensions".  If an
>extension is 'unsatisfactory', which I take to mean almost quite, but not
>totally, unlike the parent, would any ringers wish to disregard it as an
>extension anyway? (Or am I being naive on this?) Taking Roker as my usual
>example, the two extensions to royal aren't particularly similar to the
>parent.

London Major isn't particularly similar to the parent - nowhere in the
Minor is there an equivalent to turning round in 4ths below the treble.

>On families, Sir Martin Rees, the Astronomer Royal, reminds us that, whilst
>there is an infinite set of plane regular figures, three dimensions give us,
>as pointed out already, just five regular solids. In four dimensions there
>are six, but just three at all possible higher stages.

So these three are extensions - the others don't extend!

--
Regards
Philip