[r-t] Extension

[Philip Saddleton]

Graham John wrote:
> Don wrote:
>
>   
>> I'd also be interested to see it presented as a pair of clear,
>> unambiguous, directly implementable algorithms in pseudo code
>> or something like that:
>>     
>
> I hope that this is what PABS might have been referring to, when he said I'm
> working on it.
>
> Graham
Not quite.

There is no way in the current Decision of extending Principles, 
Differentials or Little Methods by increasing the length of the lead. 
Allowing this, along with unifying the treatment of different types of 
Method are the drivers behind what follows. This is not intended to be a 
formal treatment, but:

Take a method that's lead has been extended, preferably by several 
applications of the relevant formula, e.g. Cambridge Maximus. Plot the 
places in a lead in 2-d, with changes along one axis, and places along 
the other. There are several straight lines in this plot, with equal 
steps between the points on the lines. E.g. starting with the lead end 
places, and ignoring alternate changes (where no places are made), we have

|1  2
|   3T
|  14
| 125T
|  36
| 147T
|  58
| 169T
|  70
| 18
|  9  T
| 10
|  E  T
etc

(View in a fixed font and ignore the vertical bars, which I hope will 
keep the columns aligned). The "lines" are given in columns 2-5.

If we do the same at another stage of the same method, a similar pattern 
occurs, although the lines have different lengths. There are some places 
which don't occur on a varying-length line - there are the same number 
of these at each stage.

Now extend our plots to 3-d, where the third dimension is the stage. 
There are straight lines in this 3-d plot, joining

a) for places not on a line, corresponding places at each stage

b) each end of corresponding lines at each stage

e.g. writing as triplets (change, place, stage), with change 0 at the 
lead  end there is a line for Maximus

{(0,2,12), (2,3,12), (4,4,12), ... (16,10,12)}

The starts and ends of corresponding lines at stages 6 upwards are:

{(0,2,6), (0,2,8), (0,2,10), (0,2,12), ... }
{(4,4,6), (8,6,8), (12,8,10), (16,10,12), ... }

Turning this around, we have a set of criteria for extending a general 
set of places. If this results in a valid place notation at an 
indefinite number stages we say that it is a Candidate Place Notation 
Extension (CPNE).

I believe that all extensions covered by the current Decision (G) are 
CPNEs by this definition. At its simplest, we could say that a CPNE that 
produces a method of the same Class at an indefinite number of stages is 
a valid extension. However, this is probably too loose a requirement.

For one thing, odd and even places can occur on the same line. These 
have quite a different feel to them for a right-place method, as even 
places are made from the front, and odd ones from the back. We could 
require that the steps along the place axis must always be even (our 
extension of Cambridge is still valid, but there are more lines to be 
considered). However, this would also mean that extensions must always 
be by an even number of stages (we will come back to this later). I 
prefer a different approach: if we look at the blue line for the changes 
either side of a given place there are nine types of place, according to 
whether the line moves up, down or stays level before and after the 
place. Our additional requirement is that lines used to define a CPNE 
can only join places of the same type.

Other restrictions are suggested by parts of the existing Decision 
(assuming we want to keep them):

- external places remain external - reasonable enough? Without it
Cambridge  S Minor is extends to Lincolnshire
Ipswich S Minor extends to Ipswich S Major
London S Minor extends indefinitely

- adjacent places remain adjacent? I have a problem with this in the 
current Decision: first it only applies where both places are made by 
working bells - a penultimate place at the half-lead of a hunter does 
not necessarily have to remain in an extension; and second, it is the 
'characteristic' that is retained in an extension - where corresponding 
places are duplicated only one occurrence have to remain adjacent. 
Either this should apply to all repetitions of all adjacent places (the 
corresponding lines must be parallel) or it should be dropped.

- symmetry should be retained

So far we have been dealing with extension of a general block of place 
notation. When we come to methods we should require that an extension is 
of the same Class as its parent (or it could not in any case have the 
same title) - this is part of Decision (E)D. But particularly when we 
come to Differentials or Differential Hunters there ought to be some 
relationship between the cycles of working bells at different stages. 
(G)B.4 goes some way towards this, but I think that this should go further:

Since there has to be the same number of cycles at different stages, it 
should be possible to draw a correspondence between cycles. 
Corresponding cycles should have similar properties in terms of
- symmetry (or lack of), regardless of the symmetry of the method as a 
whole, and e.g. where the work of one cycle is the same as, or a 
reflection or rotation of another this characteristic should be retained
- transitivity (i.e. whether the line covers all positions in the row, 
and if each is visited the same nuber of times)
The same should apply to hunt bells (i.e. cycles of length 1)
Each place belongs to a particular cycle. An extension should only be 
permitted if the lines making up a CPNE connect places from 
corresponding cycles.

Note that in some circumstances extension by an odd number of stages is 
possible: extension of a Plain method by adding an extra hunt bell is a 
CPNE, as could be other hunt bell paths where all places made cause a 
change of direction and no other places are made beyond them. Certain 
principles also naturally extend by an odd number of stages, if we 
consider the change  number modulo the lead length (should this only be 
for fixed-length leads, as there is otherwise an ambiguity?):

Original
"Duffield" Caters:  7.3.9.3.7.1
"Saturn" Major:  36-36.18

That is the bare bones. Can anyone come up with bizarre extensions that 
would be permitted by this scheme that suggest more restrictions are 
required? Alternatively, are there examples that it should be relaxed to 
permit?

Philip