[r-t] Method extension

[Philip Saddleton]

On 22/04/2015 13:05, Tim Barnes wrote:
> On Jun 12, 2014 6:20 PM, "Philip Saddleton" <pabs at cantab.net 
> <mailto:pabs at cantab.net>> wrote:
>
> > The existing Decision (G) is inadequate, but I don't think it should 
> be scrapped without putting something in its place. I have ideas on 
> this that are too complex to put here, but in principle:
> > - an extension construction can be defined for any block of changes
> > - work at different stages ought to be clearly related, with nothing 
> occurring in the extension that does not occur in the parent (the 
> minimum definition of 'work' being a place and the blows either side 
> of it)
> > - the construction should lead to a valid block of changes at an 
> infinite series of stages
>
> Philip - we were reminded of this post of yours from last summer.  Are 
> your more detailed ideas in a form that you could post here?
>
> Tim
>
>
In semi-formal terms:

A block of changes can be uniquely defined by its stage, length and a 
set of places, where a place can be described by an ordered pair (c,p), 
where c is the number of the change relative to the start of the block, 
and p the number of the place made. If the block is a lead of a method 
then c is taken to be modulo the length of the lead. There are 
restrictions on the subsets with the same c since the combination of 
places must represent a change.

Certain places may form a line, i.e. a set of places (c+i.dc, p+i.dp) 
for i=0..n-1, including the trivial case where n=1.

An extension construction is defined at a set of stages that is an 
infinite arithmetic progression (the parent is the lowest stage in this 
set):
a) The block length is a linear function of the stage (which may be 
constant);
b) The places in the parent are grouped into lines;
c) Lines extend by making c, p and n linear functions of the stage (dc 
and dp being fixed).
d) The construction is valid at each stage in the set (i.e. the set of 
places generate changes)

That is basically it, and I think that this covers all extensions 
permitted by (G), except where a place at the lead head or lead end 
expands into the grid (one place maps to three or more that are not in a 
line - this could be catered for by permitting lines to overlap, but I 
don't think that omitting this possibility would be a great loss). It 
also permits a lot more.

The requirement for the set of stages to be infinite (or at least 
greater than two) is necessary, or practically anything could be an 
extension.

I would also add two more requirements:
- symmetry (or lack of it) should be the same at all stages.
- the effect of a place on the blue line depends on the places made in 
the changes either side. There are nine types of place depending whether 
the bell moves up, down or makes a place either side of the place in 
question. All places in a line should have the same type at all stages.

Further restrictions could be added along the lines of (G)B but I think 
that the fewer the better. The existing Decision only really works for 
treble dodging methods with a single hunt bell.

PABS
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