[r-t] Extension (was Proving)

[Philip Saddleton]

Philip Earis said  on 12/04/2006 09:21:
> 
> 
> Tony Cox:
> "I think it's time to step back from the ever more complex (and still
> highly flawed) decision on extension and take a wider view... "
> 
> I agree, though I suspect my step back would be bigger than your step
> back. 
> 
> Philip Saddleton was telling me about some similar ideas of his
> yesterday after the ASCY meeting - perhaps he would like to comment?
> 
> 

The idea behind extension is that related methods should contain the 
same features - everything in an extension should have its counterpart 
in the parent and vice versa. The difficulty comes in putting this 
formally. The current Decision works reasonably well for Plain and 
Treble Dodging methods, but in other cases are too restrictive, or do 
not cover the situation at all.

I think that in order to cover the more general case, enumerating the 
possibilities in a similar way to the current Decision is a non-starter. 
Instead we need a general construction that can be applied to any block 
of changes. Any extension ought to fit within such a construction. 
Further restrictions may then be applied if required (e.g. some 
relationship between the number of cycles of working bells at different 
stages).

Here is a first attempt, which I think encompasses the current Decision, 
as well as most of the non-extensions that have been mentioned here:


Define a Stage Series S as an infinite arithmetic series of integers 
s0+k.n (n=0,1,...).

An Extension Space is defined, given S and a linear function L(s), as 
all ordered integer triples (s,c,p) where s is in S, 0<=c<L(s) and 1<=p<=s.

A subset R of S is defined as Ringable if for any fixed s and c the 
values of p where (s,c,p) is in R satisfies
either s is even and the subset is empty
or if the values in ascending order are p(1)...p(m), then p(1) is odd, 
p(i+1)-p(i) is odd, and s-p(m) is even.

The elements of a Ringable subset R are called Places. For fixed s and 
c, the subsets of R each define a valid place notation, and as c varies 
from 0 to L(s)-1 there is a natural mapping onto a block of changes of 
length L(s) on s  bells.

Now define a Line as an infinite set of places (s(i),c(i),p(i)), 
i=0,1,... where s(i)=s(0)+A.i, c(i)=c(0)+B.i mod L(s), p(i)=p(0)+C.i.

We say two places Correspond if there is a Line in R containing both places.

A Construction E is defined as a Ringable subset of an Extension Space 
such that for any Place in E, and any s in S there is a corresponding 
Place (s,c,p).




-- 
Regards
Philip
http://www.saddleton.freeuk.com/